text:
\begin{document}

\title{Virasoro algebra and wreath product convolution}

\author{Igor B.~Frenkel and Weiqiang Wang
}
\date{}

\maketitle

\begin{abstract}
We present a group theoretic construction of the Virasoro algebra
in the framework of wreath products. This can be regarded as a
counterpart of a geometric construction of Lehn in the theory of
Hilbert schemes of points on a surface.
\end{abstract}

\section*{Introduction}
It is by now well known that a direct sum $\bigoplus_{n \geq 0}
R(S_n)$ of the Grothendieck rings of symmetric groups $S_n$ can be
identified with the Fock space of the Heisenberg algebra of rank
one. One can construct vertex operators whose components generate
an infinite-dimensional Clifford algebra, the relation known as
boson-fermion correspondence \cite{F} (also see \cite{J}). A
natural open problem which arises here is to understand the group
theoretic meaning of more general vertex operators in a vertex
algebra \cite{B, FLM}.

A connection between a direct sum $\RG = \bigoplus_{n \geq 0}
R(\G_n)$ of the Grothendieck rings of wreath products $\Gn = \G
\sim S_n$ associated to a finite group $\G$ and vertex operators
has been realized recently in \cite{W} and \cite{FJW} (also see
\cite{Z, M} for closely related algebraic structures on $\RG$).
When $\G$ is trivial one recovers the above symmetric group
picture. On the other hand, this wreath product approach turns out
to be very much parallel to the development in the theory of
Hilbert schemes of points on a surface (cf. \cite{W, N2} and
references therein). As one expects that new insight in one theory
will shed new light on the other, this refreshes our hope of
understanding the group theoretic meaning of general vertex
operators.

The goal of this paper is to take the next step in this direction
to produce the Virasoro algebra within the framework of wreath
products. Denote by $\G_*$ the set of conjugacy classes of $\G$,
and by $c^0$ the identity conjugacy class. Recall \cite{M, Z} that
the conjugacy classes of the wreath product $\Gn$ are
parameterized by the partition-valued functions on $\G_*$ of
length $n$ (also see Section~\ref{sect_wreath}). Given $c \in
\G_*$, we denote by $\laz $ the function which maps $c$ to the
one-part partition $(2)$, $c^0$ to the partition $( 1^{n-2})$ and
other conjugacy classes to $0$. We will define an operator $\Dec$
in terms of the convolution with the characteristic class function
on $\Gn$ (for all $n$) associated to the conjugacy class
parameterized by $\laz$. We show that this operator can be
identified with a differential operator which is the zero mode of
a certain vertex operator when we identify $\RG$ as in \cite{M}
with a symmetric algebra. A group theoretic construction of
Heisenberg algebra has been given in \cite{W} (also see
\cite{FJW}) which acts on $\RG$ irreducibly. The commutator
between $\Dec$ and the Heisenberg algebra generators on $\RG$
provides us the Virasoro algebra generators.

Our construction is motivated in part by the work of Lehn \cite{L}
in the theory of Hilbert schemes. Among other results, he showed
that an operator defined in terms of intersection with the
boundary of Hilbert schemes may be used to produce the Virasoro
algebra when combined with earlier construction of Heisenberg
algebra due to Nakajima and independently Grojnowski \cite{N1,
Gr}. It remains an important open problem to establish a precise
relationship between Lehn's construction and ours.

We remark that the convolution operator in the symmetric group
case (i.e. $\G$ trivial), when interpreted as an operator on the
space of symmetric functions, is intimately related to the
Hamiltonian in Calegero-Sutherland integrable system  and to the
Macdonald operator which is used to define Macdonald polynomials
\cite{AMOS, M}.

After we discovered our group theoretic approach toward the
Virasoro algebra, we notice that our convolution operator in the
symmetric group case has been considered by Goulden \cite{Go} when
studying the number of ways of writing permutations in a given
conjugacy class as products of transpositions. We regard this as a
confirmation of our belief that the connections between $\RG$ and
(general) vertex operators are profound. It is likely that often
time when we understand something deeper in this direction, we may
realize that it is already hidden in the vast literature of
combinatorics, particularly on symmetric groups and symmetric
functions for totally different needs. Then the virtue of our
point of view will be to serve as a unifying principle which
patches together many pieces of mathematics which were not
suspected to be related at all.

The plan of this paper is as follows. In Sect.~\ref{sect_vertex},
we recall some basics of Heisenberg and Virasoro algebras from the
viewpoint of vertex algebras. In Sect.~\ref{sect_wreath} we set up
the background in the theory of wreath products which our main
constructions are based on. In Sect.~\ref{sect_conv} we present
our main results. Some materials in the paper are fairly standard
to experts, but we have decided to include them in hope that it
might be helpful to the reader with different backgrounds.

\section{Basics of Heisenberg and Virasoro algebras} \label{sect_vertex}
In the following we will present some basic constructions in
vertex algebras which give us the Virasoro algebra from a
Heisenberg algebra.

Let $L$ be a rank $N$ lattice endowed with an integral
non-degenerate symmetric bilinear form. Indeed we will only need
the case $L = \Z^N$ with the standard bilinear form.

Denote by $\fh = \C \bigotimes_{\Z} L$ the vector
space generated by $L$ with the bilinear from $\langle-,-\rangle$
induced from $L$. We define the Heisenberg algebra
$$
  \hh = \C [t, t^{-1}] \bigotimes \fh \bigoplus \C C
$$
with the following commutation relations:

\begin{eqnarray*}
   [ C, a_n ]     & =& 0  , \\
   {[ a_n, b_m ]} & =& n \delta_{n, -m} \langle a,b \rangle C,
\end{eqnarray*}
where $a_n $ denotes $t^n \otimes a, a \in \fh$.

We denote by $ \SL$ the symmetric algebra generated by
$\hh^{-} = t^{-1} \C [t^{-1}] \bigotimes \fh$.
It is well known that $\SL$ can be given the structure
of an irreducible module over Heisenberg algebra $\hh$
by letting $a_{-n}, n>0$ acts as multiplication
and letting

\begin{eqnarray*}
  a_n . a_{-n_1}^1 a_{-n_2}^2 \ldots a_{-n_k}^k
  = \sum_{i =1}^k \delta_{n,n_i} \langle a, a^i \rangle
      a_{-n_1}^1 a_{-n_2}^2 \ldots
      \check{a}_{-n_i}^i \ldots a_{-n_k}^k ,
\end{eqnarray*}

where $n \geq 0, n_i > 0, a, a^i \in \fh$ for $i =1, \ldots , k$,
and $\check{a}_{-n_i}^i $ means the very term is deleted.
A natural gradation on $\SL$ is defined by letting
$$
 \deg (a_{-n_1}^1 a_{-n_2}^2 \ldots a_{-n_k}^k)
  = n_1+  \ldots  +n_k.
$$ We say an operator on $\SL$ is of degree $p$ if it maps any
$n$-th graded subspace of $\SL$ to $(n +p)$-th graded subspace.

The space $\SL$ carries a natural structure of a vertex algebra
\cite{B, FLM}. It is convenient to use the generating function in
a variable $z$: $$
 \az = \sum_{n \in \Z} a_n z^{ -n -1}, \quad a \in \fh.
$$
In the language of vertex algebras, this is the vertex operator
associated to the vector $a_{-1} \in \SL$.
The normally ordered product between two vertex operators
$a(z)$ and $b(z)$ is defined as
\[
  : a(z) b(z) : =
  \sum_{n <0} a_n z^{ -n -1} b(z)
  + b(z) \sum_{n \geq 0} a_n z^{ -n -1}.
\]
We remark that the commutation relations between $a_n, b_m, n, m
\in \Z$ are encoded in the following operator product expansion
(OPE) (cf. \cite{FLM}):
\[
 a(z) b(w) \sim \frac{ \langle a, b \rangle }{ (z -w)^2}.
\]

We recall that the Virasoro algebra is spanned by
$L_n, n \in \Z$ and a central element $C$, with
the following commutation relations:
\[
  [ L_n, L_m] = (n -m) L_{n +m} + \frac{n^3 -n}{12} \delta_{n,m} C.
\]
It is convenient to denote
\[
 \Lz = \sum_{n \in \Z} L_n z^{ -n -2}.
\]
Given an orthonormal basis $ a^i, i =1, \ldots, n$ of $\fh$, we
define a series of operators $\widetilde{L}^i_n$ $ (i =1, \ldots
N, n \in \Z) $ acting on $\SL$ by

\begin{eqnarray*}
 \widetilde{L}^i (z)
     \equiv \sum_{n \in \Z} \widetilde{L}^i_n z^{ -n -2}
                   =  \hf : a^i (z) a^i (z)  : .
\end{eqnarray*}

The following proposition is well known (cf. e.g. \cite{FLM}).

\begin{proposition}
The operators $\widetilde{L}^i_n$ $(i =1, \ldots N, n \in \Z)$
generate $N$ commutative copies of Virasoro algebra of central
charge $1$. The operators $\widetilde{L}_n, n \in \Z $ generate
the Virasoro algebra with central charge $N$. Namely we have
  \begin{eqnarray*}
   {[\widetilde{L}^i_n, \widetilde{L}^j_m  ]}
     = \delta_{ij} (n -m) \widetilde{L}^i_{n +m}
     + \frac{n^3 -n}{12} \delta_{ij} \delta_{n,m} .
 \end{eqnarray*}
\end{proposition}
From now on we will simply write $\widetilde{L}^i_n$ as $L^i_n$.

We introduce operators
$$
 \De^i = \frac16 \int :a^i (z)^3: z^2 dz.
$$
These are well-defined
operators of degree $0$ acting on $\SL$. One can
easily check that
$$
 \De^i =\frac12 \sum_{n, m > 0}
  ( a^i_{ -n -m} a^i_n a^i_m + a^i_{-n}a^i_{-m}a^i_{n+m} ).
$$ Here we omit on the right hand side the terms involving $a_0$
since $a_0$ acts as $0$ on the Fock space $\SL$.

\begin{lemma}  \label{lem_ham}
   We have
 \[
  [\De^i, a_n ] = -n \langle a^i, a \rangle  L^i_n.
 \]
\end{lemma}

\begin{demo}{proof}
 It is clear that $[\De^i, a^j_n ] = 0$ for $i \neq j$.
 So it suffices to prove that
 \[
  [\De^i, a^i_n ] = -n L^i_n.
 \]

We calculate that
 \begin{eqnarray*}
   {[\De^i, a^i (z) ]} & =&
    {\rm Res}_{w=0} \frac16 [ :a^i ( w)^3:, a^i (z) ] w^2  \\
   & =& - \hf {\rm Res}_{w=0}:a^i ( w)^2: w^2 \sum_{m \in \Z}  m w^{m -1} z^{-m -1} \\
   & =& -{\rm Res}_{w=0} \sum_{n \in \Z} L^i _n w^{ -n -2} w^2
        \sum_{m \in \Z}  m w^{m -1} z^{-m -1} \\
   & =& -\sum_{n \in \Z} n L^i _n z^{-n -1},
 \end{eqnarray*}

 Therefore
 $$
  [\De^i, a^i_n ] = {\rm Res}_{z=0} [\De^i, a^i (z)] z^n  = -n L^i _n.
 $$
\end{demo}

\section{Representation rings of wreath products}  \label{sect_wreath}

  Given a finite group $\G$, we denote by $\G^*$ the set of complex
irreducible characters and by $\G_*$ the set of conjugacy classes.
We denote by $\Rz$ the $\Z$-span of irreducible characters of $
\G$. Denote by $c^0$ the identity conjugacy class. We identify
$R(\G) = \C \otimes_{\Z } R_{\Z}(\G)$ with the space of class
functions on $\G$.

For $c \in \G_*$ let $\zeta_c$ be the order of the centralizer of
an element in the class $c$. Denote by $|\G |$ the order of $\G$.

The usual bilinear form $\langle -, - \rangle_{\G}$ on $R(\G )$ is
defined as follows (often we will omit the subscript $\G$):

\begin{eqnarray}  \label{eq_bilin}
 \langle f, g \rangle = \langle f, g \rangle_{\G}
    = \frac1{ | \G |}\sum_{x \in \Gamma}
          f(x) g(x^{ -1})
 = \sum_{c \in \Gamma_*} \zeta_c^{ -1} f(c) g(c^{ -1}),
\end{eqnarray}
where $c^{ -1}$ denotes the conjugacy class
$\{ x^{ -1}, x \in c \}$. Clearly $\zeta_c = \zeta_{c^{-1}}$.
It is well known that
\begin{eqnarray}
  \langle \g, \g' \rangle
   &= & \delta_{\g, \g'}, \quad \g, \g' \in \G^*\nonumber \\
  \sum_{ \g \in \G^*} \g (c ')  \g ( c^{ -1})
    &= & \delta_{c, c '} \zeta_c, \quad c, c ' \in \G_*. \label{eq_stand}
\end{eqnarray}

One may regard $\CG$ as the space of functions on $\G$, and thus
$R(\G)$ as a subspace of $\CG$. Given $f, g \in \CG$, the  {\em
convolution} $f * g \in\CG$ is defined by
\[
  f * g (x) =

    \sum_{y \in \G} f(x y^{-1}) g(y),
  \quad f, g \in \CG, x \in \G.
\]
In particular if $f, g \in R(\G)$, then so is $f *g$. It is well
known that
\begin{eqnarray}  \label{eq_idem}
   \g ' * \g = \frac{|\G|}{d_{\g}} \delta_{\g ,\g '} \g,
    \quad \g', \g \in \G^*,
\end{eqnarray}
where $d_{\g}$ is the {\em degree} of the irreducible character
$\g$.

Denote by $K_c$ the sum of all elements in the conjugacy class
$c$. By abuse of notation, we also regard $K_c$ the class function
on $\G$ which takes value $1$ on elements in the conjugacy class
$c$ and $0$ elsewhere. It is clear that $K_c, c \in \G_*,$ form a
basis of $R(\G)$. The elements $K_c, c \in \G_*$ actually form a
linear basis of the center in the group algebra $\CG$. But we will
not need this fact.

For wreath products we basically follow the excellent presentation
of Macdonald \cite{M}, Appendix B, Chapter 1, with the exception
of Theorem \ref{th_heis} which is quoted from \cite{W} (also see
\cite{FJW}). Given a positive integer $n$, let $\Gamma^n = \Gamma
\times \cdots \times \Gamma$ be the $n$-th direct product of
$\Gamma$. The symmetric group $S_n$ acts on $\Gamma^n$ by
permutations: $\sigma (g_1, \cdots, g_n)
  = (g_{\sigma^{ -1} (1)}, \cdots, g_{\sigma^{ -1} (n)}).
$
The wreath product of $\Gamma$ with $S_n$ is defined to be
the semi-direct product
$$
 \Gamma_n = \{(g, \sigma) | g=(g_1, \cdots, g_n)\in {\Gamma}^n,
\sigma\in S_n \}
$$
 with the multiplication
$$
(g, \sigma)\cdot (h, \tau)=(g \, {\sigma} (h), \sigma \tau ) .
$$

Let $\la=(\la_1, \la_2, \cdots, \la_l)$ be a partition of integer
$|\la|=\la_1+\cdots+\la_l$, where $\la_1 \geq \dots \geq \la_l
\geq 1$. The integer $l$ is called the {\em length} of the
partition $\la $ and is denoted by $l (\la )$. We will identify
the partition $(\la_1, \la_2, \cdots, \la_l)$ with $(\la_1, \la_2,
\cdots, \la_l, 0, \cdots, 0)$. We will also make use of another
notation for partitions: $$ \la=(1^{m_1}2^{m_2}\cdots) , $$ where
$m_i$ is the number of parts in $\la$ equal to $i$.

For a finite set $X$ and $\rho=(\rho(x))_{x \in X}$ a family of
partitions indexed by $X$, we write $$\|\rho\|=\sum_{x  \in
X}|\rho(x)|.$$ Sometimes it is convenient to regard
$\rho=(\rho(x))_{x \in X}$ as a partition-valued function on $X$.
We denote by ${\cal P}(X)$ the set of all partitions indexed by
$X$ and by ${\cal P}_n (X)$ the set of all partitions in ${\cal P}
(X)$ such that $\|\rho\| =n$.

The conjugacy classes of ${\Gamma}_n$ can be described in the
following way. Let $x=(g, \sigma )\in {\Gamma}_n$, where $g=(g_1,
\ldots , g_n) \in {\Gamma}^n,$ $ \sigma \in S_n$. The permutation
$\sigma $ is written as a product of disjoint cycles. For each
such cycle $y=(i_1 i_2 \cdots i_k)$ the element $g_{i_k} g_{i_{k
-1}} \cdots g_{i_1} \in \Gamma$ is determined up to conjugacy in
$\Gamma$ by $g$ and $y$, and will be called the {\em
cycle-product} of $x$ corresponding to the cycle $y$. For any
conjugacy class $c$ and each integer $i \geq 1$, the number of
$i$-cycles in $\sigma$ whose cycle-product lies in $c$ will be
denoted by $m_i(c)$. Denote by $\rho (c)$ the partition $(1^{m_1
(c)} 2^{m_2 (c)} \ldots )$, $c \in \G_*$. Then each element $x=(g,
\sigma)\in {\Gamma}_n$ gives rise to a partition-valued function
$( \rho (c))_{c \in \G_*} \in {\cal P} ( \G_*)$ such that
$\sum_{i, c} i m_i(c) =n$. The partition-valued function $\rho =(
\rho(c))_{ c \in G_*} $ is called the {\em type} of $x$. It is
known that any two elements of ${\Gamma}_n$ are conjugate in
${\Gamma}_n$ if and only if they have the same type.

Given a partition $\lambda = (1^{m_1} 2^{m_2} \ldots )$,
we define
$
  z_{\la } = \prod_{i \geq 1}i^{m_i}m_i!.
$
We note that $z_{\la }$ is the order of the centralizer of an
element of cycle-type $\la $ in $S_{|\la |}$. The order of the
centralizer of an element $x = (g, \sigma) \in {\Gamma}_n$ of the
type $\rho=( \rho(c))_{ c \in \G_*}$ is

\begin{eqnarray}   \label{eq_centr}
Z_{\rho}=\prod_{c\in \G_*}z_{\rho(c)}\zeta_c^{l(\rho(c))}.
\end{eqnarray}

Recall that $R(\Gn) = R_{\Z} (\Gn) \otimes_{\Bbb Z} \Bbb C$. We
set

\begin{eqnarray*}
  \RG = \bigoplus_{n \geq 0} R (\Gn).
\end{eqnarray*}

A symmetric bilinear form on $\RG$ is given by
\begin{eqnarray}   \label{eq_series}
\langle u, v \rangle
 = \sum_{ n \geq 0} \langle u_n, v_n \rangle_{\Gn } ,
\end{eqnarray}

where
$u = \sum_n u_n$ and $v = \sum_n v_n$ with $u_n, v_n\in \Gn$.

Since $\Rz$ may be regarded as an integral lattice with
non-degenerate symmetric bilinear form given by (\ref{eq_bilin})
and with an orthonormal basis $\G^*$, we can apply the
constructions in Sect.~\ref{sect_vertex} to the lattice $\Rz$. We
denote the corresponding Heisenberg algebra by $\hg$ with
generators $a_n (\g ), n \in \Z, \g \in \G^*$, and its irreducible
representation by $\SG$.

By identifying $a_{-n} (\g)$ $(n >0, \g \in \G^*)$ with the $n$-th
power sum in a sequence of variables parameterized by $\g \in
\G^*$, we may regard $\SG$ as the algebra of symmetric functions
parameterized by $\g \in \G^*$. In particular the operator $a_n
(\g)$ $(n >0, \g \in \G^*)$ acts as the differential operator $n
\frac{\partial}{\partial a_{-n}(\g)}$.

For $m \in \Bbb Z, c \in \G_*$ we define
\begin{eqnarray}  \label{eq_change}
 a_{ m}( c) = \sum_{ \g\in \G^*} \gamma(c^{-1}) a_m( \g ).
\end{eqnarray}

From the orthogonality of the irreducible characters
(\ref{eq_stand}) it follows that
\begin{eqnarray*}
  a_m( \g )
   = \sum_{c \in \G_*} \zeta_c^{ -1}
   \g  (c) a_m(c).
\end{eqnarray*}

Thus $ a_{n}(c)$ ($ n\in \Bbb Z, c \in \G_*$) and $C$ form a new
basis for the Heisenberg algebra $\hg$.

Given a partition $ \la = ( \la_1, \la_2, \ldots )$
and $c \in \G_*$, we define
\begin{eqnarray*}
  p_{ \la } (c )  = p_{ \la_1}(c) p_{ \la_2 } (c) \cdots.
\end{eqnarray*}
For any $\rho = ( \rho (c) )_{ c \in \G_* } \in
 {\cal P } ( \G_* )$, we further define

\begin{eqnarray*}
 P_{ \rho} = \prod_{ c \in \G_*} p_{ \rho (c)} (c).
\end{eqnarray*}
The elements $P_{ \rho} ,
\rho \in {\cal P}(\G_* )$ form a $ \C$-basis for $\SG $.

We define a bilinear form
$\langle \ ,  \ \rangle $ on the space $ \SG $ by letting
\begin{eqnarray}   \label{eq_sg}
  \langle P_{\rho}, P_{\sigma} \rangle
  = \delta_{\rho, \sigma} Z_{\rho}.
\end{eqnarray}

Let $\Psi : \Gn \rightarrow \SG$ be the map defined by $\Psi (x) =
P_{ \rho}$ if $x \in \Gn$ is of type $\rho$. We define a $\Bbb
C$-linear map $\ch: \bigoplus_{ n \geq 0} \C [\Gn] \longrightarrow
\SG$ by letting
\begin{eqnarray}
 \ch (f )
 & =& \frac 1{| \Gn|} \sum_{x \in \Gn} f(x) \Psi (x)
                              \label{eq_charmap}   \\
 & =& \sum_{\rho \in {\cal P} (\G_*)}
  Z_{\rho}^{-1} f_{\rho} P_{  \rho } \quad \mbox{if } f \in \RG,
    \nonumber
\end{eqnarray}
where $f_{\rho}$ is the value of $f$ at elements of type $\rho$.
Most often we will think of $\ch$ as a map from $\RG$ to $\SG$ as
in \cite{M}, usually referred to as the {\em characteristic map}.
It is well known that $\ch : \RG \rightarrow \SG$ is an isometry
for the bilinear forms on $\RG$ and $\SG$ defined in
(\ref{eq_series}) and (\ref{eq_sg}).

As we identify $a_{-n}(\g)$ $(n >0, \g \in \G^*)$ with the $n$-th
power sum $p_n (\g)$ and the space $\SG$ with the space of
symmetric functions indexed by $ \G^*$, we may regard the Schur
function $s_{\lambda} (\g)$ associated to $\g$ and a partition
$\lambda$ a corresponding element in $\SG$. For $\lambda \in {\cal
P} (\G^*)$, we denote
\begin{eqnarray*}
  s_{\lambda} = \prod_{\g \in \G^*} s_{\lambda(\g)} (\g) \in \SG.
\end{eqnarray*}

Then $s_{\lambda}$ is the image under the characteristic map
$\ch$ of the character of an irreducible
representation $\chi^{\lambda}$ of $\Gn$ (cf. \cite{M}).

Denote by $c_n (c \in \G_*)$ the conjugacy class in $\Gn$ of
elements $(x, s) \in \Gn$ such that $s$ is an $n$-cycle and the
cycle product of $x$ lies in the conjugacy class $ c$. Denote by
$\sigma_n (c )$ the class function on $\Gn$ which takes value $n
\zeta_c$ (i.e. the order of the centralizer of an element in the
class $c_n$) on elements in the class $c_n$ and $0$ elsewhere. For
$\rho = \{ m_r (c) \}_{r \geq 1, c \in \G_*} \in {\cal P}_n
(\G_*)$, $\sigma_{\rho} = \prod_{r \geq 1, c \in \G_*} \sigma_r
(c)^{m_r (c)}$ is the class function on $\Gn$ which takes value
$Z_{\rho}$ on the conjugacy class of type $\rho$ and $0$
elsewhere. Given $\g \in R(\G)$, we denote by $\sigma_n (\g )$ the
class function on $\Gn $ which takes value $n \g (c) $ on elements
in the class $c_n, c \in \G_*$, and $0$ elsewhere.

We define an operator $ \widetilde{a}_{ -n} (\gamma), n >0$ to be
a map from $\RG$ to itself by the following composition
\[  R (\Gm) \stackrel{ \sigma_n ( \g ) \otimes }{\longrightarrow}
  R(\Gn) \bigotimes R (\Gm)  \stackrel{{Ind} }{\longrightarrow}
  R ( {\Gamma}_{n +m}).
\]
We also define another operator
$ \widetilde{a}_{ n} (\gamma), n >0$ to be a map from $\RG$
to itself (which is the adjoint of $\widetilde{a}_{ -n} (\gamma)$
with respect to the bilinear form (\ref{eq_series}) )
as the composition
\[
  R (\Gm)  \stackrel{ Res }{\longrightarrow}
   R(\Gn)\bigotimes R ( {\G }_{m -n})
 \stackrel{ \langle \sigma_n ( \g), \cdot \rangle }{\longrightarrow}
 R ( {\G }_{m -n}).
\]

The following theorem was established in \cite{W} (also see
\cite{FJW}).

\begin{theorem} \label{th_heis}
  The space $\RG$ affords a representation of the Heisenberg algebra
 $\hg $  by letting $ a_n (\g )$
 $( n \in \Bbb Z \backslash 0)$ act as $ \widetilde{a}_{ n} (\g )$,
 and $C$ as $1$. The characteristic map
 $\ch$ is an isomorphism of $\RG$
 and $\SG$ as representations over the Heisenberg algebra.
\end{theorem}

\section{Virasoro algebra and group convolution}  \label{sect_conv}

We first look at the case when $\G$ is trivial and so $\Gn$
becomes the symmetric group $S_n$. We simply write the $i$-th
power sum $p_i (\g)$ as $p_i$.

We consider the convolution product on the space of class
functions on $S_n$ with the class function $K_{(1^{n -2} 2)}$,
which takes value $1$ at elements of cycle type $ (1^{n -2} 2)$
and $0$ otherwise. It follows from (\ref{eq_charmap}) that $$ \ch
(K_{(1^{n -2} 2)} ) = \frac1{2 (n -2)!} p_1^{n -2} p_2 . $$ We
denote
\begin{eqnarray}   \label{eq_delta}
  \De = \hf \sum_{i, j \geq 1}
      \left( i j p_{i +j} \diffi \diffj
         + ( i +j) p_i p_j \diffij
      \right) .
\end{eqnarray}

Given $f, g \in R(S_n)$, it is natural to define the
convolution of two symmetric functions
$\ch (f)$ and $\ch (g)$ as
$$
  \ch (f) * \ch (g)  := \ch (f * g).
$$

The next theorem describes the effect of the convolution with
$K_{(1^{n -2} 2)}$ on the space of symmetric polynomials by means
of the characteristic map.

\begin{theorem} \label{th_symm}
 For any $f \in \C [S_n ]$, we have
 $$
   \ch (K_{(1^{n -2} 2)} * f) = \De \ch (f).
 $$
 Equivalently we have
 $$
   \frac 1{2 (n -2)!} p_1^{n -2} p_2 * \ch (f) = \De \ch (f).
 $$
\end{theorem}

\begin{demo}{proof}
 Take a transposition $(a,b)$ and a permutation $\tau$.

If $a$ and $b$ lie in different cycles of $\tau$, the product
$(a,b) \tau$ will have the effect of combining the two cycles, say
of cycle length $i$ and $j$ respectively, in $\tau$ containing
respectively $a$ and $b$ into a single cycle. For example, let
$n=7, a=3, b=5$ and $\tau = (1, 3) (2, 6, 5)(4, 7)$, then $(3, 5)
\tau = (1,5,2,6,3) (4, 7)$. Thus one $p_i p_j$ in $\ch (\tau )$ is
replaced by one $p_{i +j}$ in $\ch ( (a, b)\tau )$. Among all
transpositions  $(a, b) \in S_n$, there are exactly $i j$ of which
have the effect of replacing one $p_i p_j$ by one $p_{i +j}$.

On the other hand, if $a$ and $b$ lie in a same cycle, say of
cycle length $k$, of $\tau$, then the product $(a, b) \tau$ will
have the effect of splitting this cycle in $\tau$ containing $a$
and $b$ into two disjoint cycles. For example, let $n = 8,
a=3,b=5$ and $\tau = (8, 7, 3,1, 4, 5) (2, 6)$, then $(3, 5) \tau
= (8, 7, 5) (1,4, 3) (2, 6)$. More precisely the cycle of $\tau$
splits into two disjoint cycles of length $i$ and $k -i$ if $a$
and $ b$ in the cycle of $\tau$ are separated by $i-1$ and $k-i-1$
elements. Thus one $p_{k}$ in $\ch (\tau )$ is replaced by one
$p_i p_{k -i}$ in $\ch ( (a, b)\tau )$. We can see easily among
all possible transpositions there are $k$ (resp. $k/2$) of them
which have the effect of replacing one $p_{k}$ by one $p_i p_{k
-i}$ when $k/2 \neq i$ (resp. $k/2 = i$).

Combining the above considerations together, we have proved the
theorem.
\end{demo}

\begin{remark}  \rm
Our purpose of studying this convolution is to find a group
theoretic construction of the Virasoro algebra. It turns out that
such a convolution appeared earlier in a paper of Goulden
(\cite{Go}, Proposition~3.1) in his study of the number of ways of
writing permutations in a given conjugacy class as products of
transpositions.
\end{remark}

Recall that $s_{\la}$ denotes the Schur functions associated to
the partition $\la$. We denote by $f^{\la}$ the degree of
$s_{\la}$, i.e. the dimension of the irreducible representation of
$S_n$ corresponding to $s_{\lambda}$.

\begin{proposition}  \label{prop_reform}
  We have
\begin{eqnarray*}
\De (\schu ) = \frac {n (n -1)}{2 \f } \langle p_1^{n -2} p_2,
\schu \rangle \schu.
\end{eqnarray*}
\end{proposition}

\begin{demo}{Proof}
By the orthogonality relation of characters (\ref{eq_stand}) we
have $\langle s_\mu , s_\la \rangle = \delta_{\mu, \la}$, and thus
$p_1^{n -2} p_2 =\sum_{\mu} \langle p_1^{n -2} p_2, s_{\mu}
\rangle s_{\mu}.$ In addition Eq.~(\ref{eq_idem}) implies that
$s_{\mu} * s_{\la} = \frac{n!}{f^{\la}}\delta_{\mu,\la} s_{\la}.$
Thus by Theorem~\ref{th_symm}, we have
\begin{eqnarray*}
\De (\schu ) &=& \frac1{2 (n-2)!} p_1^{n -2} p_2 * \schu \\
 &=& \frac1{2 (n-2)!} \sum_{\mu} \langle p_1^{n -2} p_2, s_{\mu}
\rangle s_{\mu} * \schu \\
 &=&\frac {n (n -1)}{2 \f } \langle p_1^{n -2} p_2,
\schu \rangle \schu.
\end{eqnarray*}
\end{demo}

Now we return to the general case of wreath product $\Gn$. Given
$c \in \G_*$, we denote by $\laz \in {\cal P}_n (\G_*)$ the
function which maps $c$ to the one-part partition $(2)$, $c^0$ to
the partition $( 1^{n-2})$ and other conjugacy classes to $0$. We
denote by $K_{\laz}$ the sum of all the elements in the conjugacy
class corresponding to $\laz$.

We introduce the operator acting on $\RG$
$$
\De^{\g } =  \frac16 \int :\widetilde{a}^{\g} (z)^3 : z^2  dz
$$
where $ \widetilde{a}^{\g} (z)
 = \sum_{n \in \Z} \widetilde{a}_n (\g) z^{ -n -1}.$

We define
\begin{eqnarray} \label{eq_convop}
 \Dec = \sum_{\beta \in \G^*}
  \frac{|\G|^2 \beta (c^{-1})}{d_{\beta}^2 \zeta_c}
  \Delta^{\beta}.
\end{eqnarray}
The operator $\Dec$ reduces to (\ref{eq_delta}) for $\G$
trivial. By the characteristic map, the operator $\De^{\g}$ can be
identified with the differential operator $$  \hf \sum_{i, j \geq
1}
      \left( i j p_{i +j}(\g) \diffig \diffjg
         + ( i +j) p_i(\g) p_j(\g) \diffijg
      \right).
$$
Note that
$$
 \ch (K_{\laz}) =
  \frac{1}{Z_{\laz }} p_1 (c^0)^{n -2} p_2 (c),
$$
where
\begin{eqnarray}  \label{eq_order}
 Z_{\laz} = 2 (n -2)! | \G |^{n -2} \zeta_c
\end{eqnarray}
is the order of the centralizer of an element in the conjugacy
class associated to ${\laz}$, cf. (\ref{eq_centr}). The following
theorem generalizes Theorem~\ref{th_symm}.

\begin{theorem} \label{th_main}
 Given $f \in  \RG$, we have
 $$
  \ch (K_{\laz} * f) = \Dec \ch (f) .
 $$
 Equivalently we have
 $$
  \frac{1}{Z_{\laz }} p_2 (c) p_1(c^0)^{n -2} * \ch (f)
  = \Dec \ch (f).
 $$
\end{theorem}

We need some preparation for the proof of the theorem. Recall that
the image of the irreducible character associated to $\la \in
{\cal P}_n (\G^*)$ under the characteristic map is $\schu =
\prod_{\g} \sge$. We set $n_{\g} = |\la (\g)|$. The first lemma
below is straightforward.

\begin{lemma}  \label{lem_zero}
 Given $\be \in \G^*$ and a sequence of non-negative
 integers $m_{\g}, \g \in \G^*$ such that
 $m_{\g} \neq n_{\g}$ for at least one $\g $,
 we have

 \begin{eqnarray*}
   \left\langle p_1 (\be)^{m_{\be} -2} p_2 (\be)
        \prod_{\g \neq \be} p_1 (\g)^{m_{\g}} ,
        \schu
   \right\rangle  = 0 .
 \end{eqnarray*}
\end{lemma}

\begin{lemma}  \label{lem_comp}
Given $\be \in \G^*$, $\la \in {\cal P}_n (\G^*)$, and setting
$n_{\g} = |\la (\g)|$ (for all $\g \in \G^*$), we have
 \begin{eqnarray*}
  & & \frac {n_{\be} (n_{\be} -1)}{2}
  \left\langle p_1 (\be)^{n_{\be} -2} p_2 (\be)
        \prod_{\g \neq \be} p_1 (\g)^{n_{\g}} ,
        \schu
  \right\rangle \schu      \\
  & =&\left( \prod_{\g} \fge
       \right)
       \De^{\be} \sbe \cdot
       \prod_{\g \neq \be} \sge.
 \end{eqnarray*}
\end{lemma}

\begin{demo}{Proof}
 It follows from
 Proposition~\ref{prop_reform} that
 \begin{eqnarray}  \label{eq_factor}
  \De^{\be} (\sbe ) =
  \frac {n_{\be} (n_{\be} -1)}{2 \fbe}
  \left\langle p_1 (\be)^{n_{\be} -2} p_2 (\be), \sbe
  \right\rangle \sbe.
 \end{eqnarray}

    Noting also that
 \begin{eqnarray}   \label{eq_deg}
   \left\langle p_1 (\g)^{n_{\g}}, \sge  \right\rangle = \fge,
 \end{eqnarray}

 we calculate that

 \begin{eqnarray*}
  & & \frac {n_{\be} (n_{\be} -1)}{2}
  \left\langle p_1 (\be)^{n_{\be} -2} p_2 (\be)
        \prod_{\g \neq \be} p_1 (\g)^{n_{\g}} ,
        \schu
  \right\rangle \schu                       \nonumber   \\
  & =& \frac {n_{\be} (n_{\be} -1)}{2}
   \left\langle p_1 (\be)^{n_{\be} -2} p_2 (\be), \sbe
   \right\rangle \sbe \cdot                        \nonumber  \\
   & & \cdot \prod_{\g \neq \be}
       \left\langle p_1 (\g)^{n_{\g}}, \sge
       \right\rangle \sge                       \nonumber   \\
  & =& \fbe  \De^{\be} \sbe \cdot
       \prod_{\g \neq \be} \fge \sge
        \quad \mbox{ by Eqs. } (\ref{eq_factor})
             \mbox{ and }(\ref{eq_deg}),
                                 \nonumber   \\
  & =& \left( \prod_{\g \in \G^*} \fge
       \right)
       \De^{\be} \sbe \cdot
       \prod_{\g \neq \be} \sge.
 \end{eqnarray*}
 This proves the lemma.
\end{demo}

\begin{demo}{Proof of Theorem~\ref{th_main} }
Consider the irreducible character associated to $\la \in {\cal
P}_n (\G^*)$ which maps by the characteristic map $\ch$ to $\schu
= \prod_{\g} \sge$. We set $n_{\g} = |\la (\g)|$.

 We calculate that

 \begin{eqnarray}   \label{eq_expan}
  & & \frac 1{2 (n -2)!} p_1(c^0)^{n -2} p_2 (c)   \nonumber \\
  & =& \frac 1{2 (n -2)!}
    \left( \sum_{\g \in \G^*} d_{\g} p_1 (\g)
    \right)^{n -2}
    \left(\sum_{\be \in \G^*} {\be}(c^{-1}) p_2 (\be)
    \right)  \quad \mbox{ by Eq.}~(\ref{eq_change}), \nonumber  \\
  & =& \hf
   \left(
       \sum_{ \{ m_{\g}\} }
       \prod_{\g  \in \G^*}
       \frac{ (d_{\g} p_1 (\g))^{m_{\g}} }{m_{\g}!}
   \right)
    \left(\sum_{\be  \in \G^*} {\be}(c^{-1}) p_2 (\be)
    \right)    \nonumber \\
  & =& \hf \sum_{ \{ m_{\g}\} } \sum_{\be \in \G^*}
    {\be}(c^{-1}) p_2 (\be)
    \frac{(d_{\be} p_1 (\be))^{m_{\be} }}{(m_{\be})!}
    \prod_{\g \neq \be}
       \frac{ (d_{\g} p_1 (\g))^{m_{\g}} }{m_{\g}!} ,
 \end{eqnarray}
 where the sum $\sum_{ \{ m_{\g}\} }$ is taken over all
 sequences of nonnegative integers $m_{\g}, \g \in \G^*$
 such that $\sum_{\g} m_{\g} =n -2$.

 Recall \cite{M, Z} that the degree of the irreducible character
 associated to $\lambda \in {\cal P}_n (\G^*)$ is
 \begin{eqnarray}   \label{eq_total}
  \f = n ! \prod_{\g} d_{\g}^{n_{\g}} \fge / n_{\g} !.
 \end{eqnarray}
 We also obtain by using (\ref{eq_idem})
 and $|\Gn| = n! |\G|^n$ that

 \begin{eqnarray}  \label{eq_apply}
  && \frac{\f}{n! | \G |^n}
   \left(
       p_2(\be) p_1(\be)^{n_{\be} -2}
       \prod_{\g \neq \be} p_1(\g)^{n_{\g}}
         \right)  * \schu       \nonumber   \\
   & =&
   \left \langle
       p_2(\be) p_1(\be)^{n_{\be} -2}
       \prod_{\g \neq \be} p_1(\g)^{n_{\g}} ,  \schu
   \right \rangle \schu.
 \end{eqnarray}

 We then have

 \begin{eqnarray*}
  &  & \frac 1{Z_{\laz}}
        p_1(c^0)^{n -2} p_2 (c) * \schu  \\
  & =& \frac{1}{2 |\G|^{n -2} \zeta_c}\sum_{\beta \in \G^*}
    {\be}(c^{-1}) p_2 (\be)
    \frac{(d_{\be} p_1 (\be))^{n_{\be} -2}}{(n_{\be} -2)!}
    \prod_{\g \neq \be}
       \frac{ (d_{\g} p_1 (\g))^{n_{\g}} }{n_{\g}!}  * \schu  \\
  & &   \quad \quad \quad \quad
       \mbox{ by Eqs.}~(\ref{eq_order}), (\ref{eq_expan}),
       \mbox{ and Lemma}~\ref{lem_zero}, \\
  & =& \frac{1}{ |\G|^{n -2}\zeta_c }
      \prod_{\g} \frac{d_{\g}^{n_{\g}}}{n_{\g}!}
      \sum_{\be \in \G^*}
      \frac{n_{\be} (n_{\be} -1) \be(c^{-1})}{2 d_{\be}^2}
     \left(
       p_2(\be) p_1(\be)^{n_{\be} -2}
       \prod_{\g \neq \be} p_1(\g)^{n_{\g}}
         \right)  * \schu  \\
  & =& \frac{\f}{n!|\G|^{n -2}\zeta_c}  \sum_{\be \in \G^*}

       \frac{n_{\be} (n_{\be} -1) \be(c^{-1})}{2 d_{\be}^2
        (\prod_{\g} \fge)}
     \left(
       p_2(\be) p_1(\be)^{n_{\be} -2}
       \prod_{\g \neq \be} p_1(\g)^{n_{\g}}
         \right)  * \schu \\
         &&\qquad \qquad \qquad \qquad \mbox{ by Eq.}~(\ref{eq_total}), \\
  & =& \frac{|\G|^2}{\zeta_c}  \sum_{\be \in \G^*}

       \frac{n_{\be} (n_{\be} -1) \be (c^{-1})}{2 d_{\be}^2(\prod_{\g} \fge)}
     \left\langle
       p_2(\be) p_1(\be)^{n_{\be} -2}
       \prod_{\g \neq \be} p_1(\g)^{n_{\g}} ,
         \schu \right\rangle s_{\la} \\
         &&\qquad \qquad \qquad \qquad  \mbox{ by Eq.}~(\ref{eq_apply}), \\
  & =&  \sum_{\be \in \G^*}
       \frac{|\G|^2\be (c^{-1})}{d_{\be}^2 \zeta_c}
     \left(
             \De^{\be} \sbe \cdot
       \prod_{\g \neq \be} \sge
         \right)  \\
         &&\qquad \qquad \qquad \qquad  \mbox{ by Lemma}~\ref{lem_comp}, \\
  & =& \Dec \schu.
\end{eqnarray*}
The last equation holds since $\De^{\be} \sge = 0$ for $\g \neq
\be$. This finishes the proof, since $\schu, \lambda \in {\cal P}
(\G^*)$ form a basis of the space $\RG$.
\end{demo}

\begin{corollary}
For the identity conjugacy class $c^0 \in \G_*$, we have
$\De_{c^0} = \sum_{\be \in \G^*} \frac{|\G|}{d_{\be}} \De^{\be}.$
\end{corollary}

Combining (\ref{eq_convop}), Lemma~\ref{lem_ham},
Theorem~\ref{th_heis} and Theorem~\ref{th_main}, we obtain the
following.

\begin{theorem}
The operator $\Dec$ acting on $\RG$ is realized by the convolution
with $K_{\laz}$. The commutation relation between $\Dec$ and the
Heisenberg algebra generator $\widetilde{a}_{n} (\g)$ (constructed
in a group theoretic manner) is given by
 \[
 [ \Dec, \widetilde{a}_{n} (\g)]
  = -\frac{n|\G |^2 \g (c^{-1})}{d_{\g}^2 \zeta_c} L_{n}^{\g},
\]
where the operators $L_n^{\g}$ acting on the space $\RG$ satisfy
the Virasoro commutation relations:
 \[
    [ L_n^{\be}, L_m^{\g} ]
    = \delta_{\be  \g} L_{n +m}^{\g} +
     \frac{n^3 -n}{12} \delta_{\be  \g} \delta_{n, -m}.
 \]
\end{theorem}

\frenchspacing


TITLE: Integrability - conditions of lax pairs
QUESTION [1 upvotes]: I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation:
$$
\partial_t U - \partial_x V + [U,V]=0
$$
where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ are matrix-valued functions and $\lambda$ is a parameter.
The motivation behind this question is that the Lax pairs for the KdV equation:
$$
u_t + 6uu_x - u_{xxx} = 0
$$
is given by:
$$
U = \begin{pmatrix} 0 & 1 \\ \lambda + u & 0 \end{pmatrix} \text{ and } V = \begin{pmatrix} u_x & 4 \lambda - 2u \\ 4 \lambda^2 + 2 \lambda u + u_{xx} - 2u^2 & - u_x \end{pmatrix}
$$
Now, it is not too difficult to verify that this indeed satisfies the zero-curvature representation, but I'm trying to figure out why we cannot use the Lax pairs:
$$
U = \begin{pmatrix} 0 & 0 \\ \lambda + u & 0 \end{pmatrix} \text{ and } V = \begin{pmatrix} 0 & 0 \\ \lambda + 3 u^2 - u_{xx} & 0 \end{pmatrix}
$$
These matrices clearly satisfy the zero-curvature representation, but for some reason none of the notes I've been reading use them. What is the reason that they are not a valid Lax pair for the KdV equation?
I've also asked this question here (I hope that's ok): http://www.physicsoverflow.org/26475/integrability-conditions-of-lax-pairs

REPLY [2 votes]: In addition to the answer by @issoloroap, the article Prolongation structures of nonlinear evolution equations by Allan Fordy (here is the Mathscinet link and here is its first page on Google books) explains why "good" zero-curvature representations should live in (semi)simple Lie algebras.
Another important point is that the spectral parameter should be essential, i.e., it should not be removable by gauge transformations, cf. e.g. this paper and references therein, and for the discussion of related issues in the case of dispersionless sysems in more than two independent variables see e.g. this article, and this one, and references therein.


TITLE: What is the reason for so much big prof of Cayley-Hamilton theorem in Linear Algebra?
QUESTION [0 upvotes]: As I was studying linear algebra from various books I came to know about Cayley -Hamilton theorem. It states that:
An $n×n$ matrix satisfies it's own characteristic equation.
I see in various books that they have given a big proof for this. But this seems quite obvious to me. Because characteristic equation of an $n×n$ matrix A is given by:
$det(xI_n-A)=0$, where $I_n$ is the $n×n$ identity matrix. If we put $x=A$ in the lhs of the characteristic equation we get:
$det(A.I_n-A)=det(A-A)=det(O_n)=0$, where $O_n$ is the zero matrix of order n.
Is there any mistake in my idea? If not then why are they giving so much bigger proofs.

REPLY [1 votes]: I just discovered the answer from a similar post in stack:
You can't just replace $x$ by $A$ since the operation between $x$ and $I_n$ in $det(xI_n-A)$ is the scalar multiplication. But when you put $x=A$ the operation becomes matrix multiplication.


TITLE: Show that $ImT^t=(kerT)°$
QUESTION [2 upvotes]: Let $T:V→W$ be linear transformation and V have a finite dimension. Show that $ImT^t=(kerT)°$
I have to prove it by mutual inclusion. I have proven the first inclusion but I don't know how to prove that $(kerT)°$ is contained in $ImT^t$.
And I don't know the theory for orthogonal complements yet, so the problem has to be solved using the inclusions I mentioned or dimensions.
It's the problem number 11.38 in the Schaum's Outline for Linear Algebra by S. Lipschutz, and the theory I have to use has to be what's given until the chapter 11 of that book.

REPLY [1 votes]: Suppose $g\in \ker(T)^{\circ}$, then
$$
\forall x\in V, Tx = 0 \Rightarrow g(x) = 0
$$
Hence, we get a well-defined map
$$
f : \text{Im}(T) \to \mathbb{C} \text{ given by } Tx \mapsto g(x)
$$
Extend $f$ trivially to the rest of $W$ to get $f \in W^{\ast}$ which satisfies $T^t(f) = g$. Hence,
$$
\ker(T)^{\circ} \subset \text{Im}(T^t)
$$


TITLE: Finding electric field of a line charge
QUESTION [0 upvotes]: I have a bit of an issue setting up the integral for finding an electric field of a particular scenario. The scenario is that a line charge of length R is a distance R from the left side of the origin. Furthermore, the line charge starts below the x-axis so that only the top end touches the x-axis.
So I know that $dq=\lambda dysin\theta$.
Then $dE=k\frac{dq}{\sqrt{r^2}}=k\lambda \frac{ dy sin\theta}{\sqrt{y^2+R^2}}$.
Now since $sin\theta=\frac{R}{\sqrt{y^2+R^2}}$, $dE=k\lambda \frac{Rdy}{\sqrt{(y^2+R^2)^3}}$.
Then I would integrate from $-R$ to $0$.
Is that the correct set up? Any help is appreciated.

REPLY [1 votes]: No, there are several mistakes in your derivation, although you miraculously end up with the right expression.
The term $\sin \theta$ comes from taking the horizontal (X) component of the electric field - not from the expression you used for $dq$. The diagram I envisage for your problem is this:

So I would say that
$$dq = \lambda dy$$
and then
$$dE = k\frac{dq}{r^2}$$
(no square root)
Now for the horizontal component, we need to take account of the factor $\frac{x}{r}$, making the final expression
$$dE = k\frac{\lambda\ dy}{r^2}\frac{x}{r}\\
= k\frac{\lambda\cdot x\cdot dy}{(x^2 + y^2)^{3/2}}$$
Now we can put $x=R$ and integrate from $-R$ to $0$:
$$E = \int_{-R}^{0}k\frac{\lambda R dy}{(R^2+y^2)^{3/2}}$$


TITLE: Euler's Formula Question.
QUESTION [0 upvotes]: Find
$\cos\theta + \cos3\theta + ... + \cos((2n+1)\theta$,
and
$\sin\theta + \sin3\theta + ... + \sin((2n+1)\theta$.
Where $\theta \in$ Reals.

REPLY [1 votes]: At first,
$$e^{i\theta}+e^{3i\theta}+\cdots +e^{(2n+1)i\theta}=e^{i\theta}\frac{1-e^{2(n+1)i\theta}}{1-e^{2i\theta}}=\frac{2 (e^{2(n+1)i\theta}-1)}{\sin\theta}$$
for $\sin\theta \neq 0$. And
$$\operatorname{Re }(e^{i\theta}+e^{3i\theta}+\cdots +e^{(2n+1)i\theta}) = \cos\theta+\cos3\theta +\cdots + \cos(2n+1)\theta$$
$$\operatorname{Im }(e^{i\theta}+e^{3i\theta}+\cdots +e^{(2n+1)i\theta}) = \sin\theta+\sin3\theta +\cdots + \sin(2n+1)\theta$$
So
$$\cos\theta+\cos3\theta +\cdots + \cos(2n+1)\theta = \frac{2(\cos 2(n+1)\theta -1)}{\sin\theta}$$
$$\sin\theta+\sin3\theta +\cdots + \sin(2n+1)\theta = \frac{2\sin2(n+1)\theta}{\sin\theta}$$
for $\sin\theta\neq 0$.


text:
\begin{document}

\maketitle

\begin{abstract}
  In \citet{Haller-Beron-13} we developed a variational principle
  for the detection of coherent Lagrangian vortex boundaries. The
  solutions of this variational principle turn out to be closed
  null-geodesics of the Lorentzian metric induced by a generalized
  Green--Lagrange strain tensor family. This metric interpretation
  implies a mathematical analogy between coherent Lagrangian vortex
  boundaries and photon spheres in general relativity. Here we give
  an improved discussion on this analogy.
\end{abstract}

\section{The main results of \citet{Haller-Beron-13}}

We consider a two-dimensional velocity field $v(x,t)$, with $x$
labeling the location within a two-dimensional region $U$ of interest
and with $t$ referring to time. Fluid trajectories generated by
$v(x,t)$ are denoted $x(t;t_{0},x_{0}$), with $x_{0}$ referring to
the initial position of the trajectory at time $t_{0}$. These
trajectories solve the differential equation
\begin{equation}
  \dot{x} = v(x,t),
  \label{eq:uxt}
\end{equation}
and generate the flow map
\begin{equation}
  F_{t_{0}}^{t}(x_{0}) := x(t;t_{0},x_{0}),
\end{equation}
which takes an initial position $x_{0}$ at time $t_{0}$ to its
current position at time $t$.

The right Cauchy--Green strain tensor field associated with the
flow map is defined as $C_{t_{0}}^{t}(x_{0}) = \nabla
F_{t_{0}}^{t}(x_{0})^\top \nabla F_{t_{0}}^{t}(x_{0})$, with
eigenvalues $\lambda_{i}(x_{0})$ and eigenvectors $\xi_{i}(x_{0})$
satisfying
\begin{equation}
C_{t_{0}}^{t}\xi_{i} = \lambda_{i}\xi_{i},\quad
\left|\xi_{i}\right| = 1,\quad i=1,2;\quad
0<\lambda_{1}\leq\lambda_{2},\quad
\xi_{1}\perp\xi_{2}.
\end{equation}
In \citet{Haller-Beron-13} we sought the time $t_{0}$ positions
of Lagrangian vortex boundaries as closed stationary curves of the
averaged Lagrangian strain. Such curves turned out to coincide with
the zero-energy solutions of a one-parameter family of variational
problems defined as
\begin{equation}
  \delta\mathcal{E}_{\lambda}(\gamma)=0,\quad
  \mathcal{E}_{\lambda}(\gamma)
  = \oint_{\gamma}\langle r^{\prime}(s),
  E_{\lambda}(r(s))r^{\prime}(s)\rangle\,\mathrm{d}s,\quad
  \lambda\in\mathbb{R}^{+}.
  \label{eq:qdef-1}
\end{equation}
Here the strain energy functional $\mathcal{E}_{\lambda}(\gamma)$
is defined through the generalized Green--Lagrange strain tensor
family
\begin{equation}
  E_{\lambda}(x_{0}) = \frac{1}{2}[C_{t_{0}}^{t}(x_{0})-\lambda^{2}I].
  \label{eq:Elambda}
\end{equation}

Consider the flow domain
\begin{equation}
  U_{\lambda} = \{x_0 \in U \mid \lambda_1(x_0) < \lambda^2
  < \lambda_2(x_0)\},
  \label{eq:Ulda}
\end{equation}
where the tensor field $E_{\lambda}$ has two nonzero eigenvalues
of opposite sign.  Then the quadratic function
\begin{equation}
  g_{\lambda}(u,u) = \left\langle u,E_{\lambda}u\right\rangle
  \label{eq:g}
\end{equation}
defines a Lorentzian metric \citep{Beem-etal-96} on $U_{\lambda}$,
with signature $(-,+)$ inherited from the eigenvalue configuration
of $E_{\lambda}$. The zero-energy solutions of (\ref{eq:qdef-1})
are therefore precisely the closed null-geodesics of the Lorentzian
metric $g_{\lambda}$, which satisfy one of the two differential
equations
\begin{equation}
   r^{\prime}(s) = \eta_{\lambda}^{\pm}(r(s)),\quad
	\eta_{\lambda}^{\pm}(r) =
	\sqrt{\frac{\lambda_{2}(r)-\lambda^{2}}{\lambda_{2}(r)-\lambda_{1}(r)}}\,\xi_{1}(r)\pm
	\sqrt{\frac{\lambda^{2}-\lambda_{1}(r)}{\lambda_{2}(r)-\lambda_{1}(r)}}\,\xi_{2}(r).
	\label{eq:ode-1}
\end{equation}
In \citet{Haller-Beron-13} we concluded that closed orbits of
(\ref{eq:ode-1}) (termed \emph{closed $\lambda$-lines}) must
necessarily encircle metric singularities of $g_{\lambda}$. Such
singularities occur at points $x_{0}$ where $\lambda_{1}(r)=\lambda_{2}(r)$
holds for the eigenvalues of the Cauchy--Green strain tensor.

The Lorentzian metric interpretation discussed above implies a
geometric analogy between coherent Lagrangian vortex boundaries and
photon spheres in cosmology. Below we give more detail on this
analogy, followed by an improved version of its summary with a more
relevant reference.

\section{More on the analogy with photon spheres}

In the vicinity of any $\lambda$-line in $U_{\lambda}$, the vector
fields $\xi_{i}$ define a curvilinear coordinate system with pointwise
orthogonal coordinate lines. Direct substitution of $\xi_{i}$ into
the metric (\ref{eq:g}) gives
\begin{equation}
  g_{\lambda}(\xi_{i},\xi_{i}) = \frac{1}{2}(\lambda_{i} -
  \lambda^{2}),
\label{eq:xi_sign}
\end{equation}
showing that $g_{\lambda}(\xi_{1},\xi_{1})<0$ and
$g_{\lambda}(\xi_{2},\xi_{2})>0$ everywhere in $U_\lambda$.  This
shows that $\xi_{1}$-trajectories form the time-like coordinates
and the $\xi_{2}$-trajectories form the space-like coordinates of
the metric $g_{\lambda}$ in $U_{\lambda}$ \citep{Beem-etal-96}. We
further note that
$g_{\lambda}(\eta_{\lambda}^{\pm},\eta_{\lambda}^{\pm})=0$, and
hence a $\lambda$-line is \emph{nowhere space-like} in the language
of Lorentzian geometry. Given that our closed null-geodesics are
nowhere space-like hypersurfaces built out of null-geodesics, they
are \emph{photon surfaces} by the general definition of
\citet{Claudel-etal-01}.

Next we note that
\begin{equation}
  \left\langle \eta_{\lambda}^{\pm}(r),\xi_{1}(r)\right\rangle =
  \sqrt{\frac{\lambda_{2}(r)-\lambda^{2}}{\lambda_{2}(r)-\lambda_{1}(r)}}\in(0,1),\quad
  r\in U_{\lambda}.
\end{equation}
Consequently, trajectories of the $\xi_{1}(r)$ line field ($\xi_{1}$
coordinate lines) intersect any closed $\lambda$-line $\gamma$
transversally, with an angle of intersection that is always less
than $\frac{1}{2}\pi$, as sketched qualitatively in Figure \ref{fig:proj}.
In the same figure, we also show a representative trajectory of the
$\xi_{2}(r)$ line field (a $\xi_{2}$ coordinate line) which is
pointwise orthogonal to the $\xi_{1}$ coordinate lines by construction.
We conclude that the projection of $\gamma$ onto a space-like
submanifold along the time-like coordinates results in a periodic
(albeit discontinuous) space-like orbit (\cf Figure
\ref{fig:proj}).

\begin{figure}
  \centering
  \includegraphics[width=.7\textwidth]{blkholesadd-fig01}
  \caption{A closed null-geodesic $\gamma$ of the Lorentizan metric
  $g_{\gamma}$ has a periodic space-like projection along the
  time-like coordinate lines. This space-like projection has a
  jump-discontinuity due to the low dimensionality of the space-time,
  but evolves periodically along with the underlying closed
  null-geodesic. This is in contrast to general $\lambda$-lines
  that have aperiodic space-like projections.\label{fig:proj}}
\end{figure}

In summary, a closed orbit $\gamma$ of the $\eta_{\lambda}^{\pm}(r)$
vector field is a photon surface of the $(U_{\lambda},g_{\lambda})$
space-time. Geodesics forming this photon surface have
periodically moving projections on the space-like coordinates, with
the projection taken along the time-like coordinate lines.

The orbit $\gamma$, therefore, satisfies a plausible extension of
the Claudel--Virbhadra--Ellis definition of a \emph{photon sphere}
\citep{Claudel-etal-01} from symmetric higher-dimensional space-times
to non-symmetric two-dimensional space-times. This extension relaxes
the requirement for a rotational symmetry and smoothness of the
projected spatially periodic orbits. Both of these features are
unattainable in curved, two-dimensional space-times, and hence can
be reasonably waived.

\section{Revised wording of the cosmological analogy}

In view of the above discussion, the wording of the mathematical
analogy between coherent Lagrangian eddy boundaries and photon
spheres in cosmology \citep[][p.\ 731, para.\ 2]{Haller-Beron-13}
should be revised as follows:
\begin{quotation}
  The closed $\lambda$-lines we have been seeking are therefore
  closed null-geodesics of $g_{\lambda}$. Trajectories spanning
  these geodesics project along the local time-like coordinates
  (trajectories of the $\xi_{1}$ vector field) onto periodic
  trajectories on the local space-like coordinates (trajectories
  of the $\xi_{2}$ vector field). In cosmology, the projection of
  some families of null-geodesics from space-time onto the space-like
  variables also produces closed trajectories. Such families of
  null-geodesics are often referred to as \emph{photon spheres},
  as their spatial projections trap photons orbiting around black
  holes \citep{Claudel-etal-01}. In the cosmological context,
  null-geodesics are tangent to \emph{light cones}, which in our
  case are formed by the two vectors $\eta_{\lambda}^{\pm}(x_{0})$
  at $x_{0}$ (Figure 2 of Haller \& Beron-Vera 2013).
\end{quotation}

We acknowledge comments by Angus Prain and Valerio Faraoni that
made us realize the need for this clarifying note.  We are grateful
to Mohammad Farazmand for his numerical work in connection with
Figure 1. FJBV acknowledges support by NSF grant CMG0825547 and by
a grant from the BP/Gulf of Mexico Research Initiative.

\bibliographystyle{jfm}

config: arxiv


TITLE: Einstein field equations and SEM tensor + Alcubierre
QUESTION [1 upvotes]: I wonder how I can find, using the Einstein field equations, the SEM tensor in a region of space with a function $k(x,y,x)$ that describes the curvature of space in that region at a moment (so it is not a function of time). Is there a way to do it?
An example would be with the Alcubierre metric, we can fing $T_{00}$ by defining the metric and working out the Ricci tensor. Is defining the metric a little bit like defining my function $k$?

REPLY [1 votes]: If your goal is to consider the Alcubierre metric then you'll notice the energy density is sometimes negative. The classical vacuum has an energy density that is zero.
So this tells you that you either need quantum theory (which is wishful thinking since if you have quantum gravity you need a different theory than GR and so there might not be Alcubierre metrics in that theory) or else you need to have matter.
Exotic matter. Matter with a negative energy density.


TITLE: Change of basis, where the new basis is unknown
QUESTION [0 upvotes]: I'm being asked a basis $B$ such that
$|f|_B =
\begin{bmatrix}
0&0\\
0&1\\
\end{bmatrix}
$
$
f: R^2 \rightarrow R^2
$
is linear transformation that is a projection such that
$
f([x_1 x_2]^T) = [2x_1+x_2 \ \ \ -2x_1-x_2]^T
$
Then, I define as the basis I will work in...
$
N = \{(1,0)(0,1)\}
$
Then, the linear transformation does...
$
f([1 0]^T) =  [2 \ \ -2]_B
$
$
f([0 1]^T) =  [1 \ \ -1]_B
$
The matrix $A$ associated to the linear transformation Ax=b is then:
$A=
\begin{bmatrix}
2        &  1\\
-2         & -1\\
\end{bmatrix}
$
If I'm understanding right, I'm being asked a pair of vectors $v_1 , v_2 \in B$ such that
$
Av_1 = [0 0]^T
\\
Av_2 = [0 1]^T
$
But when trying to solve it this way, I end up with a couple of equations that I'm not understading well, also, I'm not using the additional information that this linear transformation is a projection.
Edit because of hint
So I can choose whatever solution I want to...
Let $v_1 = (a,b)$ and $v_2 = (c,d)$
$
\begin{bmatrix}
2&1\\
-2&-1\\
\end{bmatrix}
$
$
\begin{bmatrix}
a\\
b\\
\end{bmatrix}
$=
$
\begin{bmatrix}
0\\
0\\
\end{bmatrix}
$
I can say $v_1 = (0,0)$
$
\begin{bmatrix}
2&1\\
-2&-1\\
\end{bmatrix}
$
$
\begin{bmatrix}
c\\
d\\
\end{bmatrix}
$=
$
\begin{bmatrix}
0\\
1\\
\end{bmatrix}
$
I can say $v_2 = (-1/4,-1/2)$
Therefore
$B=\{(0,0)(-1/4,-1/2)\} = \{(-1/4,-1/2\}$

REPLY [1 votes]: Hint  You are being asked to find two vectors $v_1$ and $v_2$ such that $f(v_1)=0$ and $f(v_2)=v_2$. Let $v_1=(x_1,x_2)$ and solve $f(x_1,x_2)=(0,0)$ and choose one solution (there are infinitely many solutions). The same for $v_2$.


TITLE: How do traveling waves pass through a standing wave node, if the node doesn't move?
QUESTION [7 upvotes]: I'm having trouble with the explanation that a standing wave in a string is the superposition of traveling waves.

(source: physicsclassroom.com)
The nodes in the diagram above are points where the particles of the string's medium undergo zero displacement, i.e. they do not move at all. But if they do not move, how is the disturbance of (any internal traveling wave) propagated past the node?
The usual explanation for how a wave is propagated is that when one particle is disturbed (say, moved up), it exerts a pull on another, which in turn exerts a pull on the next one, and so on. In other words, to exert a pull or push on the next particle there must be some movement/disturbance of the previous one. But the particle(s) at the node point do not move at all, so how does the disturbance of a traveling wave propagation pass through them? (I'm trying to understand the picture in terms of the mechanical forces between particles).

REPLY [0 votes]: This is a great paradox! I've wondered this myself as a trumpet player, and it took me a while to figure out where the confusion comes from.
I think people hear that standing waves are the sum of a traveling wave and its reflection, and get irritated because +1 + -1 is still zero. It just seems like a math trick.
But there's a big difference between an object with no force on it, and an object with equal and opposite forces on it, even though they both have zero acceleration. The difference is that the object with equal and opposite forces is probably (assuming it's elastic, blah blah blah) transferring those forces across itself.
Force detour
Consider an aluminum block (negligible mass) sitting on a concrete floor. It has no(-ish) force on it. Now stand on the block. The position of the block hasn't changed, but your weight is now transferred through the block to the floor, and the floor's 3rd Law response is transferred through the block to you. You can feel this, in your feet (sitting certainly feels better!). It's not just a math trick, it's real.
Applied to waves
Likewise, standing waves show up when a traveling wave interferes with its own reflection. Meaning nodes aren't points of zero force, they're points where the (non-zero) forces for each wave sum to zero acceleration. But they still transfer the force of each wave to neighboring particles. And for those neighboring particles, the accelerations don't cancel out (at least not most of the time).
Transferring force without deformation
user45848 pointed out that the aluminum block mentioned ought to deform a bit, and that was bugging me too. But I don't think making the block incompressible changes anything: the top layer should be able to transfer force to lower layers even if it doesn't move at all.
That still sounds a bit nonsensical, but it helped when I imagined this scenario:
Imagine three adjacent particles in a line with no force on them, then replace the outer particles with oppositely charged particles. None of the particles have moved, but now force is being transferred through the center particle.


text:
\begin{document}

\allowdisplaybreaks

\newcommand{\arXivNumber}{1806.05375}

\renewcommand{\PaperNumber}{016}

\FirstPageHeading

\ShortArticleName{The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_r\varphi_s(\bm{a};\bm{b};q,x)$}

\ArticleName{The $\boldsymbol{q}$-Borel Sum of Divergent Basic Hypergeometric\\ Series $\boldsymbol{{}_r\varphi_s(\bm{a};\bm{b};q,x)}$}

\Author{Shunya ADACHI}

\AuthorNameForHeading{S.~Adachi}

\Address{Graduate School of Education, Aichi University of Education, Kariya 448-8542, Japan}
\Email{\href{mailto:s217m064@auecc.aichi-edu.ac.jp}{s217m064@auecc.aichi-edu.ac.jp}, \href{mailto:s.adachi0324@icloud.com}{s.adachi0324@icloud.com}}

\ArticleDates{Received June 15, 2018, in final form February 24, 2019; Published online March 05, 2019}

\Abstract{We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic expansion. Such an actual solution is obtained by using $q$-Borel summability, which is a $q$-analog of Borel summability. Our result shows a $q$-analog of the Stokes phenomenon. Additionally, we show that letting $q\to1$ in our result gives the Borel sum of classical hypergeometric series.
The same problem was already considered by Dreyfus, but we note that our result is remarkably different from his one.}

\Keywords{basic hypergeometric series; $q$-difference equation; divergent power series solution; $q$-Borel summability; $q$-Stokes phenomenon}

\Classification{33D15; 39A13; 34M30}

\section{Introduction}
We set $q\in\mathbb{C}^*$ and $0<|q|<1$ throughout this paper. Basic hypergeometric series with the base $q$ is defined by
\begin{gather}\label{qHGS}
{}_r\varphi_s\left(\bm{a};\bm{b};q,x\right)=
{}_r\varphi_s\left(\begin{matrix}\bm{a}\\ \bm{b}\end{matrix};q,x\right)
:=\sum_{n\ge0} \frac{(a_1,a_2,\ldots,a_r;q)_n}{(b_1,\ldots,b_s;q)_n(q;q)_n}\big\{(-1)^nq^{\frac{n(n-1)}{2}}\big\}^{1+s-r}x^n,
\end{gather}
where $x\in\mathbb{C}$, $\bm{a}=(a_1,a_2,\ldots,a_r)\in\mathbb{C}^r$, $\bm{b}=(b_1,\ldots,b_s)\in\mathbb{C}^s$ and $b_1,\ldots,b_s\notin q^{-\mathbb{N}}$.

Basic hypergeometric series \eqref{qHGS} is a $q$-analog of the classical hypergeometric series
\begin{gather}\label{HGS}
{}_rF_s\left(\bm{\alpha};\bm{\beta};x\right)=
{}_rF_s\left(\begin{matrix}
\bm{\alpha}\\
\bm{\beta}
\end{matrix};x\right)
:=\sum_{n\ge0}\frac{(\alpha_1)_n(\alpha_2)_n\cdots(\alpha_r)_n}{(\beta_1)_n\cdots(\beta_s)_nn!}x^n,
\end{gather}
where $\bm{\alpha}=(\alpha_1,\alpha_2,\ldots,\alpha_r)\in\mathbb{C}^r$, $\bm{\beta}=(\beta_1,\ldots,\beta_s)\in\mathbb{C}^s$ and $\beta_1,\ldots,\beta_s\notin \mathbb{Z}_{\le0}$.

The radius of convergence of the series \eqref{qHGS} and \eqref{HGS} are both $\infty$, $1$ or $0$ according to whether $r<s+1$, $r=s+1$ or $r>s+1$. In this paper, we assume $r>s+1$ and $a_1a_2\cdots a_rb_1b_2\cdots b_s\neq0$. Therefore the series \eqref{qHGS} and \eqref{HGS} are divergent in this paper.

Basic hypergeometric series \eqref{qHGS} formally satisfies the following linear $q$-difference equation:
\begin{gather}\label{qHGEq}
\left[x(-\sigma_q)^{1+s-r}\prod_{j=1}^r(1-a_j\sigma_q)-(1-\sigma_q)\prod_{k=1}^s\left(1-\frac{b_k}{q}\sigma_q\right)\right]y(x)=0,
\end{gather}
where $\sigma_q$ is the $q$-shift operator defined by $\sigma_qy(x)=y(qx)$. As applying the local theory of linear $q$-difference equations (cf.\ Adams~\cite{Adams1929}), we see that the equation~\eqref{qHGEq} has a fundamental system of solutions around infinity\footnote{In his paper, Adams used the function $q^{-\frac{1}{2}(\log_q x)(\log_q x-1)}$ instead of $\theta_q(x)$. We remark that two functions $q^{-\frac{1}{2}(\log_q x)(\log_q x-1)}$ and $\theta_q(x)$ play the same role in constructing formal solutions since they satisfy the same $q$-difference equation
\begin{gather*}
\sigma_q^n y=x^{-n}q^{-\frac{n(n-1)}{2}}y.
\end{gather*}}:
\begin{gather}\label{qHGEq_Sol}
y_i^{(\infty)}(x)=\frac{\theta_q(-a_ix)}{\theta_q(-x)}{}_{r}\varphi_{r-1}
\left(\begin{matrix}
a_i,\frac{a_iq}{b_1},\frac{a_iq}{b_2},\ldots,\frac{a_iq}{b_s},0,\ldots,0\\
\frac{a_iq}{a_1},\ldots,\frac{a_iq}{a_{i-1}},\frac{a_iq}{a_{i+1}},\ldots,\frac{a_iq}{a_r}
\end{matrix};q,\frac{qb_1\cdots b_s}{a_1\cdots a_rx}\right),\qquad\!\! 1\le i\le r,\!\!\!
\end{gather}
which are convergent in $\vert x\vert>\vert qb_1\cdots b_s/a_1\cdots a_r\vert$.

In this paper, we construct an actual solution of \eqref{qHGEq} which admits \eqref{qHGS} as an asymptotic expansion and give its analytic continuation, by using $q$-Borel--Laplace transform (see Theorem~\ref{Theorem:Main1}). This result shows a $q$-analog of Stokes phenomenon. After that, we consider to take the limit $q\to1$ in Theorem~\ref{Theorem:Main1} (see Theorem~\ref{Theorem:Main2}).

The motivation of this study comes from as follows: Ichinobe \cite{Ichinobe2001} obtained the Borel sum of divergent classical hypergeometric series.
\begin{Theorem}[Ichinobe {\cite[Theorem 2.1]{Ichinobe2001}}]\label{Theorem:Ichinobe}
Let $\bm{\alpha}=(\alpha_1,\alpha_2,\ldots,\alpha_r)\in\mathbb{C}^r$ and $\bm{\beta}=(\beta_1,\beta_2,\ldots,$ $\beta_s)\in\mathbb{C}^s$. Assume $\alpha_i-\alpha_j\notin\mathbb{Z}$, $i\neq j$. Then ${}_rF_s(\bm{\alpha};\bm{\beta};x)$ is $1/(r-s-1)$-summable in any direction $d$ such that $d\neq0$ $({\rm mod}~2\pi)$ and its Borel sum $f(x)$ is given by
\begin{gather*}\label{Ichinobe_result1}
f(x)=C_{\bm{\alpha\beta}}\sum_{j=1}^rC_{\bm{\alpha\beta}}(j)(-x)^{-\alpha_j}{}_{s+1}F_{r-1}\left( \begin{matrix}
\alpha_j,1+\alpha_j-\bm{\beta}\\
1+\alpha_j-\widehat{\bm{\alpha}_j}
\end{matrix}
;\frac{(-1)^{1+s-r}}{x}\right),
\end{gather*}
where $x\in S(\pi,(r-s+1)\pi,\infty):=\{x\in\mathbb{C}^*;\left|\pi-\arg x\right|<(r-s+1)\pi/2)\}$ and $\widehat{\bm{\alpha}_j}\in\mathbb{C}^{r-1}$ is the vector which is obtained by omitting the $j$-th component from $\bm{\alpha}$ and
\begin{gather}\label{Ichinobe_result2}
C_{\bm{\alpha\beta}}=\frac{\Gamma(\bm{\beta})}{\Gamma(\bm{\alpha})},\qquad C_{\bm{\alpha\beta}}(j)=\frac{\Gamma(\alpha_j)\Gamma(\widehat{\bm{\alpha}_j}-\alpha_j)}{\Gamma(\bm{\beta}-\alpha_j)}.
\end{gather}
Here we use the following abbreviations
\begin{gather*}
\Gamma(\bm{\alpha})=\prod_{l=1}^r\Gamma(\alpha_l),\qquad \Gamma(\widehat{\bm{\alpha}_j}-\alpha_j)=\prod_{l=1,\,l\neq j}^r\Gamma(\alpha_l-\alpha_j).
\end{gather*}
\end{Theorem}
Since basic hypergeometric series \eqref{qHGS} is a $q$-analog of hypergeometric series \eqref{HGS}, whether there exists a $q$-analog of Theorem \ref{Theorem:Ichinobe} is a natural question. To answer this question, we use the theory of $q$-Borel summability. It is a $q$-analog of the theory of Borel summability and studied by many authors (cf.~Ramis \cite{Ramis1992}, Zhang \cite{Zhang2002,Zhang2005}, Di Vizio--Zhang \cite{Vizio-Zhang} and Dreyfus--Eloy~\cite{DreyfusEloy2016}). Dreyfus~\cite{Dreyfus2015_1} proved that for every formal power series solution of a linear $q$-difference equation with rational coefficients, we may construct a meromorphic solution of the same equation applying several $q$-Borel and $q$-Laplace transformations of appropriate orders and appropriate direction.

Our results are generalizations of Zhang \cite{Zhang2002} and Morita \cite{Morita2014}. Zhang studied the $q$-Borel summability of divergent basic hypergeometric series ${}_2\varphi_0$ which is the case of $r=2$ and $s=0$ in our result. Later, Morita obtained similar results with Zhang for ${}_3\varphi_1$, which is the case of $r=3$ and $s=1$ in our result. The common point of their assumptions is that $r$ and $s$ satisfy $r-s=2$. Our result gives the resummation of ${}_r\varphi_s$ in the case of $r-s\ge2$.

This paper is organized as follows. We fix our notions and review the theory of $q$-Borel summability in Section \ref{Section:Preliminary}. Main results of this paper are given in Section \ref{Section:Main Results}. In Sections~\ref{Section:Proof_Main1} and~\ref{Section:Proof_Lemma:Main1}, we give proofs of Theorem~\ref{Theorem:Main1} and Lemma~\ref{Lemma:Main1} respectively. A~proof of Theorem~\ref{Theorem:Main2} is given in Section~\ref{Section:Proof_Main2}.

\subsection*{On an article of Dreyfus}
We note that Dreyfus \cite{Dreyfus2015_2} treated the resummation of basic hypergeometric series \eqref{qHGS} via $q$-Borel summability. But his result is remarkably different from our result: In Dreyfus' paper, the order of $q$-Laplace transform and $q$-Borel transform are different. Therefore the function he obtained doesn't satisfy the same equation with \eqref{qHGS}. To explain this more detail, we use the following notations which are used in Dreyfus' paper \cite{Dreyfus2015_2} only in this subsection.
\begin{itemize}\itemsep=0pt
\item $q>1$ and $p=1/p$,
\item $\underline{q}:=q^{1/(r-s-1)}$ and $\underline{p}:=p^{1/(r-s-1)}$,
\item the definitions of $\hat{B}_q$ and $L_{q}^{[d]}$ are essentially same with this paper (see Section~\ref{Review}).
\end{itemize}
Under these notations, Dreyfus considered the resummation of ${}_r\varphi_s(\bm{a};\bm{b};\underline{p},z)$. He said that~\cite[pp.~478--479]{Dreyfus2015_2}
\begin{gather*}
\text{\dots, we can compute a solution of the same linear $\displaystyle\sigma_{\underline{q}}$-equation than ${}_r\varphi_s\!\left(
\begin{matrix}
a_1,\ldots,a_r\\
b_1,\ldots,b_s
\end{matrix};\underline{p},z
\right)$}\\
\text{$=\sum\limits_{n\in\mathbb{N}}\!\frac{(a_1,{\ldots},a_r;\underline{p})_n}{(b_1,{\ldots},b_s;\underline{p})_n(\underline{p};\underline{p})_n}\big\{(-1)^n\underline{p}^{\frac{n(n-1)}{2}}\big\}^{1+s-r}z^n$ applying successively to it $\hat{B}_{\underline{q}}$~and~$L_{\underline{q}}^{[d]}$.}
\end{gather*}
However, since
\begin{gather*}
\left(\underline{p}^{\frac{n(n-1)}{2}}\right)^{1+s-r}=p^{-\frac{n(n-1)}{2}}=q^{\frac{n(n-1)}{2}},
\end{gather*}
we have to apply $q$-Borel transform $\hat{B}_q$ to ${}_r\varphi_s(\bm{a};\bm{b};\underline{p},z)$, not $\underline{q}$-Borel transform $\hat{B}_{\underline{q}}$. In addition, the computation actually written in \cite{Dreyfus2015_2} seems to be wrong. He said that \cite[p.~479]{Dreyfus2015_2}
\begin{gather*}
\text{Applying $\hat{B}_{\underline{q}}$ to ${}_r\varphi_s\left(
\begin{matrix}
a_1,\ldots,a_r\\
b_1,\ldots,b_s
\end{matrix};\underline{p},z
\right)$, we obtain for all $d\not\equiv(r-s-1)\pi[2\pi]$}\\
h(\zeta):={}_r\varphi_{r-1}\left(\begin{matrix}
a_1,\ldots,a_r\\
b_1,\ldots,b_s,0,\ldots,0
\end{matrix};\underline{p},(-1)^{1+s-r}\zeta
\right)\in\mathbb{H}_{\underline{q},1}^d.
\end{gather*}
But actually the function $h(\zeta)$ is equal to $\hat{B}_q ({}_r\varphi_s)$, not $\hat{B}_{\underline{q}} ({}_r\varphi_s)$ (see Section \ref{Section:Proof_Main1}). Here we remark that the series $\hat{B}_{\underline{q}} ({}_r\varphi_s)$ must be still divergent because of $q>\underline{q}$. After that, the $\underline{q}$-Laplace transform of $h(\zeta)$ was calculated in Lemma~7.2. As a result, the resummation denoted by~$\mathbb{S}_{\underline{q}}^{[d]}({}_r\varphi_s)$ in Theorem~7.3 is actually equal to
\begin{gather}\label{Dreyfus_wrong}
\big(L_{\underline{q}}^{[d]}\circ\hat{B}_{q}\big)({}_r\varphi_s).
\end{gather}
Since the orders of $q$-Borel and $q$-Laplace transformations are different, the function~\eqref{Dreyfus_wrong} doesn't satisfy the same equation with ${}_r\varphi_s$.

However, we remark that the limit $q\to1$ of $\mathbb{S}_{\underline{q}}^{[d]}({}_r\varphi_s)$ in Theorem~7.3 is accidentally same with the Borel sum of (classical) hypergeometric series~\eqref{HGS}. In other words, the statement of Theorem~7.4 itself is correct, but the proof is incorrect.

In this paper we give the correct result about the resummation of basic hypergeometric series~\eqref{qHGS} in Theorem~\ref{Theorem:Main1}. In addition, we consider the relation between the resummation and local solution around infinity~\eqref{qHGEq_Sol}. We remark that this topic wasn't treated by Dreyfus. In Theorem~\ref{Theorem:Main2}, we give the limit $q\to1$ of the resummation of \eqref{qHGS} in Theorem~\ref{Theorem:Main1}, with a correct proof.

\section{Preliminary}\label{Section:Preliminary}
\subsection{Basic notions}
Let us fix our notations. We denote by $\bm{0}_n$ the $n$-vector $(0,0,\ldots,0)$.

For $n=0,1,2,\ldots,$ we set the Pochhammer symbol as
\begin{gather*}
(a)_n:=
\begin{cases}
1,& n=0,\\
a(a+1)\cdots(a+n-1), &n\geq1,
\end{cases}
\end{gather*}
and the $q$-shift factorial is defined by
\begin{gather*}
(a;q)_n:=\prod_{j=0}^{n-1}\big(1-aq^j\big),\qquad (a;q)_\infty:=\prod_{j=0}^\infty \big(1-aq^j\big).
\end{gather*}
For any $a\in\mathbb{C}$, the infinite product $(a;q)_\infty$ is convergent. We set
\begin{gather*} (a_1,a_2,\ldots,a_r;q)_{n}:=\prod\limits_{j=1}^r (a_j;q)_n \qquad \text{for $n=0,1,2,\ldots$ or $n=\infty$}.
\end{gather*}

The theta function with the base $q$ is defined by
\begin{gather*}
\theta_q(x):=\sum_{n\in\mathbb{Z}}q^{\frac{n(n-1)}{2}}x^n,
\end{gather*}
which is holomorphic on $\mathbb{C}^*=\mathbb{C}\setminus\{0\}$. The following properties hold:
\begin{gather}
\theta_q(x) =(q,-x,-q/x;q)_\infty,\label{theta_tpf}\\
\theta_q(q^nx) =x^{-n}q^{-\frac{n(n-1)}{2}}\theta_q(x),\qquad n\in\mathbb{Z},\label{theta_q-diff}\\
\theta_q(qx) =\frac{\theta_q(x)}{x}=\theta_q\left(\frac{1}{x}\right)\label{theta_inverse}.
\end{gather}
For $\lambda\in\mathbb{C}^*$, we set the $q$-spiral $[\lambda;q]:=\lambda q^\mathbb{Z}=\{\lambda q^n ;\,n\in\mathbb{Z}\}$. From the equality~\eqref{theta_tpf}, we see
\begin{gather*}
\theta_q\left(-\frac{\lambda}{x}\right)=0\quad \Leftrightarrow \quad x\in[\lambda;q].
\end{gather*}
We remark that the $q$-spiral $[\lambda;q]$ is identified with an element of $\mathbb{C}^*/q^\mathbb{Z}$.

The $q$-gamma function $\Gamma_q(x)$ is defined by
\begin{gather*}
\Gamma_q(x):=\frac{(q;q)_\infty}{(q^x;q)_\infty}(1-q)^{1-x}.
\end{gather*}
When $q$ is a real number and satisfies $0<q<1$, the limit of $\Gamma_q(x)$ as $q\to1$ gives the gamma function (cf.\ Gasper--Rahman {\cite[Section~1.10]{Gasper&Rahman2004}})
\begin{gather}\label{qgamma_reduction}
\lim_{q\to1}\Gamma_q(x)=\Gamma(x).
\end{gather}

\subsection[Review of $q$-Borel summability]{Review of $\boldsymbol{q}$-Borel summability}\label{Review}
We review the theory of $q$-Borel summability (cf.~Zhang \cite{Zhang2002, Zhang2005} and Dreyfus--Eloy~\cite{DreyfusEloy2016}).

Let $\mathbb{C}[[x]]$ be the ring of formal power series of $x$ and $\hat{f}(x)=\sum\limits_{n\ge0}a_nx^n\in\mathbb{C}[[x]]$. For $\hat{f}(x)$, we define its $q$-Borel transform $\hat{\mathcal{B}}_{q;1}\colon \mathbb{C}[[x]]\to\mathbb{C}[[\xi]]$ by
\begin{gather*}
(\hat{\mathcal{B}}_{q;1}\hat{f})(\xi):=\sum_{n\ge0}a_nq^{\frac{n(n-1)}{2}}\xi^n.
\end{gather*}

We denote $\mathcal{M}(\mathbb{C}^*,0)$ the field of functions that are meromorphic on some punctured neigh\-borhood of~0 in $\mathbb{C}^*$. More explicitly $f\in\mathcal{M}(\mathbb{C}^*,0)$ if and only if there exists $V$, an open neighborhood of 0, such that $f$ is analytic on $V\setminus\{0\}$.

Let $[\lambda;q]\in\mathbb{C}^*/q^\mathbb{Z}$ and $f\in\mathcal{M}(\mathbb{C}^*,0)$. It is said that $f$ belongs to $\mathbb{H}_{q;1}^{[\lambda;q]}$ if there exist a positive constant $\varepsilon$ and a domain $\Omega\subset\mathbb{C}$ such that:
\begin{itemize}\itemsep=0pt
\item $\displaystyle\bigcup_{m\in\mathbb{Z}}\left\{x\in\mathbb{C}^* ;\,\left|x-\lambda q^m\right|<\varepsilon\left|q^m\lambda\right|\right\}\subset\Omega$.
\item The function $f$ can be continued to an analytic function on $\Omega$ with $q$-exponential growth at infinity, which means that there exist positive constants $L$ and $M$ such that for any $x\in\Omega\setminus\{0\}$, the following holds:
\begin{gather*}
|f(x)|<L\theta_{|q|}(M|x|).
\end{gather*}
We note that this estimate does not depend on the choice of $\lambda$.
\end{itemize}

Let $[\lambda;q]\in\mathbb{C}^*/q^{\mathbb{Z}}$. For $f\in\mathbb{H}_{q;1}^{[\lambda;q]}$, we define its $q$-Laplace transform $\mathcal{L}_{q;1}^{[\lambda;q]}\colon \mathbb{H}_{q;1}^{[\lambda;q]}\to\mathcal{M}(\mathbb{C}^*,0)$ by
\begin{gather*}
\big(\mathcal{L}_{q;1}^{[\lambda;q]}f\big)(x):=\frac{1}{1-q}\int_{0}^{\lambda\infty}\frac{f(\xi)}{\theta_q\big(\frac{\xi}{x}\big)}\frac{d_q\xi}{\xi}=\sum_{n\in\mathbb{Z}}\frac{f(\lambda q^n)}{\theta_q\big(\frac{\lambda q^n}{x}\big)}.
\end{gather*}
Here, this transformation is given in Jackson integral (cf.\ Gaspar--Rahman \cite[Section~1.10]{Gasper&Rahman2004}). For sufficient small $|x|$, the function $\big(\mathcal{L}_{q;1}^{[\lambda;q]}f\big)(x)$ has poles of order at most one that are contained in the $q$-spiral $[-\lambda;q]$. We remark that the well-definedness of $q$-Laplace transform can be shown in the same way as Th\'eor\`eme~1.3.2 of Zhang~\cite{Zhang2002}. If the function $\hat{f}(x)\in\mathbb{C}[[x]]$ satisfies $\big(\hat{\mathcal{B}}_{q;1}\hat{f}\big)\in\mathbb{H}_{q;1}^{[\lambda;q]}$, we call that $\hat{f}(x)$ is $[\lambda;q]$-summable and $\big(\mathcal{L}_{q;1}^{[\lambda;q]}\circ\hat{\mathcal{B}}_{q;1}\hat{f}\big)(x)$ is the $[\lambda;q]$-sum of~$\hat{f}(x)$.

We give some fundamental properties of $q$-Borel--Laplace transform without proofs.
\begin{Proposition}[Dreyfus--Eloy {\cite[Section~1]{DreyfusEloy2016}}] Let $[\lambda;q]\in\mathbb{C}^*/q^\mathbb{Z}$ and $f$ be an analytic function. Then, the followings hold:
\begin{itemize}\itemsep=0pt
\item $\big(\hat{\mathcal{B}}_{q;1}f\big)\in\mathbb{H}_{q;1}^{[\lambda;q]}$,
\item $\big(\mathcal{L}_{q;1}^{[\lambda;q]}\circ\hat{\mathcal{B}}_{q;1}f\big)=f$.
\end{itemize}
\end{Proposition}
\begin{Proposition}\label{prop_borel2} Let $[\lambda;q]\in\mathbb{C}/q^\mathbb{Z}$. For $\hat{f}(x)\in\mathbb{C}[[x]]$ and $g\in\mathbb{H}_{q;1}^{[\lambda;q]}$, the followings hold.
\begin{itemize}\itemsep=0pt
\item $\big(\hat{\mathcal{B}}_{q;1}\big(x^m\sigma_q^n\hat{f}\big)\big)=q^{\frac{m(m-1)}{2}}\xi^m\sigma_q^{n+m}\big(\hat{\mathcal{B}}_{q;1}\hat{f}\big)$,
\item $\big(\mathcal{L}_{q;1}^{[\lambda;q]}(\xi^m\sigma_q^ng)\big)=q^{-\frac{m(m-1)}{2}}x^m\sigma_q^{n-m}\big(\mathcal{L}_{q;1}^{[\lambda;q]}g\big)$.
\end{itemize}
\end{Proposition}
The first equality is shown by direct calculation. The second equality can be proved by using the property (2.2) of theta function. From this proposition, we have the following important corollary.
\begin{Corollary}\label{cor_q-diff}
Let $\hat{f}(x)\in\mathbb{C}[[x]]$ be a formal solution of a~linear $q$-difference equation. If $\hat{f}(x)$ is $[\lambda;q]$-summable, its $[\lambda;q]$-sum satisfies the same $q$-difference equation with $\hat{f}(x)$.
\end{Corollary}

At the end of this section, we give asymptotic properties of the $q$-Borel--Laplace transform.

Let $[\lambda;q]\in\mathbb{C}^*/q^\mathbb{Z}$, $\hat{f}(x)=\sum\limits_{n\ge0}a_nx^n\in\mathbb{C}[[x]]$ and $f\in\mathcal{M}(\mathbb{C}^*,0)$. Then we define
\begin{gather*}
f\sim_1^{[\lambda;q]}\hat{f},
\end{gather*}
if for any positive numbers $\varepsilon$ and $R$, there exist positive constants $C$ and $K$ such that for any $N\ge1$ and
\begin{gather*}
x\in\left\{x\in\mathbb{C}^* ;\,|x|<R\right\}\setminus\bigcup_{m\in\mathbb{Z}}\left\{x\in\mathbb{C}^* ;\,|x+\lambda q^m|<\varepsilon |q^m\lambda |\right\},
\end{gather*}
we have
\begin{gather*}
\left|f(x)-\sum_{n=0}^{N-1}a_nx^n\right|\le LM^N|q|^{-\frac{N(N-1)}{2}}|x|^N.
\end{gather*}
\begin{Proposition}[Dreyfus--Eloy {\cite[Section~1]{DreyfusEloy2016}}]\label{qBorel_AE}
Let $[\lambda;q]\in\mathbb{C}^*/q^\mathbb{Z}$ and $\hat{f}\in\mathbb{C}[[x]]$ with $\big(\hat{\mathcal{B}}_{q;1}\hat{f}\big)\in\mathbb{H}_{q;1}^{[\lambda;q]}$. Then $\big(\mathcal{L}_{q;1}^{[\lambda;q]}\circ\hat{\mathcal{B}}_{q;1}\hat{f}\big)\sim_{1}^{[\lambda;q]}\hat{f}$.
\end{Proposition}

\section{Main results}\label{Section:Main Results}
In this section, we give our main results which are about the resummation of the basic hypergeometric series
\begin{gather}\label{qHGS2}
\hat{f}(x)={}_r\varphi_s(\bm{a};\bm{b};q,x)={}_r\varphi_s\left( \begin{matrix}\bm{a}\\ \bm{b} \end{matrix};q,x\right).
\end{gather}
We remark that $\hat{f}(x)$ is the formal power series solution of the linear $q$-difference equation \eqref{qHGEq}. Our results are stated as follows.
\begin{Theorem}\label{Theorem:Main1}
We put $k=r-s-1$ and $p=q^k$. Assuming $a_i/a_j\notin q^{\mathbb{Z}}$, $i\neq j$, basic hypergeometric series \eqref{qHGS2} is $[\lambda;p]$-summable for any $\lambda\in\mathbb{C}^*\setminus\big[(-1)^{k};q\big]$ and its $[\lambda;p]$-sum ${}_rf_s(\bm{a};\bm{b};\lambda;q,x)=\big(\mathcal{L}_{p;1}^{[\lambda:p]}\circ\mathcal{\hat{B}}_{p;1}\hat{f}\big)(x)$ is given by
\begin{gather}
{}_rf_s(\bm{a};\bm{b};\lambda;q,x) =\frac{(a_2,\ldots,a_r,b_1/a_1,\ldots,b_{s}/a_1;q)_\infty}{(b_1,\ldots,b_{s},a_2/a_1,\ldots,a_r/a_1;q)_\infty}\frac{\theta_{p}\big(pa_1^kx/\lambda\big)}{\theta_{p}(px/\lambda)}
\frac{\theta_q\big((-1)^{1-k}a_1\lambda\big)}{\theta_q\big((-1)^{1-k}\lambda\big)}\nonumber\\
\hphantom{{}_rf_s(\bm{a};\bm{b};\lambda;q,x) =}{}
\times{}_r\varphi_{r-1}\left(\begin{matrix}a_1,a_1q/b_1,\ldots,a_1q/b_s,\bm{0}_k\\a_1q/a_2,\ldots,a_1q/a_r
\end{matrix} ;q,\frac{qb_1\cdots b_s}{a_1\cdots a_rx}\right)\nonumber\\
\hphantom{{}_rf_s(\bm{a};\bm{b};\lambda;q,x) =}{}
+\idem(a_1;a_2,\ldots,a_r),\label{Main1}
\end{gather}
where $x\in\mathbb{C}^*\setminus[-\lambda;p]$ and $|qb_1b_2\cdots b_s/a_1\cdots a_rx|<1$. Here the symbol ``$\,\idem(a_1;a_2,\ldots,a_r)$'' means the sum of $r-1$ terms which are obtained by interchanging $a_1$ with each $a_2,\ldots,a_r$ in the preceding expression.
The function ${}_rf_s(\bm{a};\bm{b};\lambda;q,x)$ has simple poles on the $p$-spiral $[-\lambda;p]$.
\end{Theorem}
From Corollary~\ref{cor_q-diff} and the well-definedness of $q$-Laplace transform, we see that the $[\lambda;p]$-sum ${}_rf_s(\bm{a};\bm{b};\lambda;q,x)=\big(\mathcal{L}_{p;1}^{[\lambda:p]}\circ\mathcal{\hat{B}}_{p;1}\hat{f}\big)(x)$ itself becomes the meromorphic solution of~\eqref{qHGEq} on some punctured neighborhood of the origin in $\mathbb{C}^*$. The expression~\eqref{Main1} gives the analytic continuation of ${}_rf_s(\bm{a};\bm{b};\lambda;q,x)$ to the neighborhood of infinity.
 In addition, this shows a $q$-analog of the Stokes phenomenon. Indeed, an actual solution~\eqref{Main1} has~\eqref{qHGS2} as its asymptotic expansion (see Proposition~\ref{qBorel_AE}) and its value changes depending on the choice of $[\lambda;p]\in\mathbb{C}^*/p^{\mathbb{Z}}$.
Using the fundamental system of solutions around infinity \eqref{qHGEq_Sol}, the $[\lambda;p]$-sum \eqref{Main1} can be expressed as
\begin{gather*}
{}_rf_s(\bm{a};\bm{b};\lambda;q,x)=\sum_{j=1}^r M_j(x,\lambda)y_j^{(\infty)}(x),
\end{gather*}
where $M_j(x,\lambda)=C_jT_j(x,\lambda)$ and
\begin{gather*}
C_j=\frac{(b_1/a_j,b_2/a_j,\ldots,b_s/a_j;q)_\infty}{(b_1,b_2,\ldots,b_s;q)_\infty}\prod_{1\le i\le r,\, i\neq j}\frac{(a_i;q)_\infty}{(a_i/a_j;q)_\infty},\\
T_j(x,\lambda)=\frac{\theta_{p}\big(pa_j^kx/\lambda\big)}{\theta_{p}(px/\lambda)}\frac{\theta_q\big((-1)^{1-k}a_j\lambda\big)}{\theta_q\big((-1)^{1-k}\lambda\big)}\frac{\theta_q(-x)}{\theta_q(-a_jx)}\label{T_j}.
\end{gather*}
The coefficients $M_j$ are called $q$-Stokes coefficients. Unlike the case of differential equation, the values of coefficients $M_j$ change continuously dependent on $\lambda\in\mathbb{C}^*\setminus\big[(-1)^k;q\big]$.

Next, we consider to take the limit $q\to1$ in above result. To consider taking the limit of~$q$, we restrict~$q$ to within a real number, and $0<q<1$. When~$q$ is a complex number, the situation is more complicated. For detail, see Sauloy \cite{Sauloy2000}.

\begin{Theorem}\label{Theorem:Main2}We assume that $q\in\mathbb{R}$ and $0<q<1$. Let $\bm{\alpha}=(\alpha_1,\alpha_2,\ldots,\alpha_r)\in\mathbb{C}^r$, $\bm{\beta}=(\beta_1,\beta_2,\ldots,\beta_s)\in\mathbb{C}^s$ and $\alpha_i-\alpha_j\notin\mathbb{Z}$, $i\neq j$. Take a parameter $\lambda\in\mathbb{C}^*\setminus (-1)^k\mathbb{R}_+$. Then for any $x\in\mathbb{C}^*\setminus(-1)^{k+1}\lambda\mathbb{R}_+$, the following equality holds:
\begin{gather}
\lim_{q\to1}{}_rf_s \left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^kx}{(1-q)^k}\right)\nonumber\\
\qquad {} =C_{\bm{\alpha\beta}}\sum_{j=1}^rC_{\bm{\alpha\beta}}(j)(-x)^{-\alpha_j}{}_{s+1}F_{r-1}\left(\!
\begin{matrix}
\alpha_j,1+\alpha_j-\bm{\beta}\\
1+\alpha_j-\widehat{\bm{\alpha}_j}
\end{matrix}
;\frac{(-1)^k}{x}
\right),\label{Main2}
\end{gather}
where $q^{\bm{\alpha}}=\big(q^{\alpha_1},q^{\alpha_2},\ldots,q^{\alpha_r}\big)$, $q^{\bm{\beta}}=\big(q^{\beta_1},q^{\beta_2},\ldots,q^{\beta_s}\big)$ and $\big|\arg{(-x)}-\arg{(-1)^{k-1}}\lambda\big|<\pi$. $C_{\bm{\alpha\beta}}$~and~$C_{\bm{\alpha\beta}}(j)$ are given by \eqref{Ichinobe_result2}.
\end{Theorem}
We remark that the left hand side of \eqref{Main2} formally converges to ${}_{r}F_{s}(\bm{\alpha};\bm{\beta};x)$ as $q\to1$. Therefore, Theorem~\ref{Theorem:Main2} can be seen as a $q$-analog of Theorem~\ref{Theorem:Ichinobe}.

\section{Proof of Theorem \ref{Theorem:Main1}}\label{Section:Proof_Main1}
We show that $\hat{f}(x)={}_r\varphi_s(\bm{a};\bm{b};q,x)$ is $[\lambda;p]$-summable for any $\lambda\in\mathbb{C}\setminus\big[(-1)^k;q\big]$. Let $g(\xi)$ be the $p$-Borel transform of $\hat{f}(x)$
\begin{gather*}
g(\xi)=(\hat{\mathcal{B}}_{p;1}\hat{f})(\xi)=\sum_{n\ge0}\frac{(a_1,a_2,\ldots,a_r;q)_n}{(b_1,\ldots,b_s;q)_n(q;q)_n}\big\{(-1)^nq^{\frac{n(n-1)}{2}}\big\}^{1+s-r}p^{\frac{n(n-1)}{2}}\xi^n.
\end{gather*}
Then, $g(\xi)$ is convergent in $|\xi|<1$ and again can be expressed by basic hypergeometric series:
\begin{gather*}
g(\xi)=\sum_{n\ge0}\frac{(a_1,a_2,\ldots,a_r;q)_n}{(b_1,\ldots,b_s;q)_n(q;q)_n}\big((-1)^{-k}\xi\big)^n={}_r\varphi_{r-1}\left(
\begin{matrix}
a_1,a_2,\ldots,a_r\\
b_1,b_2,\ldots,b_s,0,\ldots,0
\end{matrix}
;q;(-1)^{-k}\xi
\right).
\end{gather*}
To have the analytic continuation of $g(\xi)$, we use the following proposition.
\begin{Proposition}[Slater \cite{Slater1952} or Gaspar--Rahman {\cite[Section~4.5]{Gasper&Rahman2004}}] The analytic continuation of the basic hypergeometric series ${}_r\varphi_{r-1}(a_1,a_2,\ldots,a_r;b_1,\ldots,b_{r-1};q,\xi)$ is given by
\begin{gather}
{}_r\varphi_{r-1}\left(
\begin{matrix}a_1,\ldots,a_r\\
b_1,\ldots,b_{r-1}
\end{matrix};q,\xi\right)=\frac{(a_2,\ldots,a_r,b_1/a_1,\ldots,b_{r-1}/a_1;q)_\infty}{(b_1,\ldots,b_{r-1},a_2/a_1,\ldots,a_r/a_1;q)_\infty}\frac{\theta_q(-a_1\xi)}{\theta_q(-\xi)}\nonumber\\
\hphantom{{}_r\varphi_{r-1}\left(
\begin{matrix}a_1,\ldots,a_r\\
b_1,\ldots,b_{r-1}
\end{matrix};q,\xi\right)=}{} \times{}_r\varphi_{r-1}\left(
\begin{matrix}a_1,a_1q/b_1,\ldots,a_1q/b_{r-1}\\
a_1q/a_2,\ldots,a_1q/a_r
\end{matrix};q,\frac{qb_1\cdots b_{s}}{a_1\cdots a_r\xi }\right)\nonumber\\
\hphantom{{}_r\varphi_{r-1}\left(
\begin{matrix}a_1,\ldots,a_r\\
b_1,\ldots,b_{r-1}
\end{matrix};q,\xi\right)=}{} +\idem(a_1;a_2,\ldots,a_r).\label{r,r-1AC}
\end{gather}
\end{Proposition}
Letting $b_{s+1},\ldots,b_{r-1}\to 0$ in \eqref{r,r-1AC}, it is seen that $g(\xi)$ is continued analytically to $\xi\in\mathbb{C}^*\setminus\big[(-1)^{k};q\big]$ as follows:
\begin{gather}
g(\xi)=\frac{(a_2,\ldots,a_r,b_1/a_1,\ldots,b_{s}/a_1;q)_\infty}{(b_1\ldots,b_s,a_2/a_1,\ldots a_r/a_1;q)_\infty}\frac{\theta_q((-1)^{k-1}a_1\xi)}{\theta_q((-1)^{k-1}\xi)}\nonumber\\
\hphantom{g(\xi)=}{} \times{}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q,(-1)^k\frac{q^{k+1}b_1\cdots b_s}{a_1^{1-k}a_2\cdots a_r\xi}\right)\nonumber\\
\hphantom{g(\xi)=}{} +\idem(a_1;a_2,\ldots,a_r),\label{ac_borelim}
\end{gather}
where $\tilde{\bm{a}}=(a_1,a_1q/b_1,\ldots,a_1q/b_s)\in\mathbb{C}^{s+1}$ and $\tilde{\bm{b}}=(a_1q/a_2,\ldots,a_1q/a_r)\in\mathbb{C}^{r-1}$. We note that the basic hypergeometric series ${}_{s+1}\varphi_{r-1}$ in~\eqref{ac_borelim} is holomorphic on $\mathbb{C}^*$. In order to obtain \eqref{ac_borelim}, we use
\begin{gather*}
\lim_{b\to0}(aq/b;q)_nb^n=\lim_{b\to0}(b-aq)\big(b-aq^2\big)\cdots\big(b-aq^n\big)=(-a)^nq^{\frac{n(n+1)}{2}}.
\end{gather*}

Now, we shall show that $\big(\hat{\mathcal{B}}_{p;1}\hat{f}\big)(\xi)=g(\xi)\in\mathbb{H}_{p;1}^{[\lambda;p]}$ for any $\lambda\in\mathbb{C}^*\setminus\big[(-1)^k;q\big]$. The following lemma holds.
\begin{Lemma}\label{Lemma:Main1}
The function $g(\xi)$ has a $p$-exponential growth at infinity in
\begin{gather*}
\Omega_\delta:=\mathbb{C}^*\setminus\bigcup_{m\in\mathbb{Z}}\big\{\xi\in\mathbb{C}^* ;\big\vert\xi-(-1)^{k}q^m\vert<\delta\big\vert q^m\vert\big\}
\end{gather*}
for any $\delta>0$.
\end{Lemma}
A proof of this lemma will be given in Section~\ref{Section:Proof_Lemma:Main1}. For any $\lambda\in\mathbb{C}^*\setminus\big[(-1)^k;q\big]$, there exists positive constants $\delta$ and $\varepsilon$ such that
\begin{gather*}
\bigcup_{m\in\mathbb{Z}}\big\{\xi\in\mathbb{C}^* ;\,\vert\xi-\lambda p^m\vert<\varepsilon\vert p^m\lambda\vert\big\}\subset\Omega_\delta.
\end{gather*}
Therefore we obtain $g(\xi)\in\mathbb{H}_{p;1}^{[\lambda;p]}$ for any $\lambda\in\mathbb{C}^*\setminus\big[(-1)^k;q\big]$. In the other words, $\hat{f}(x)={}_r\varphi_s(\bm{a};\bm{b};q,x)$ is $[\lambda;p]$-summable for any $\lambda\in\mathbb{C}^*\setminus\big[(-1)^k;q\big]$.

We consider the $p$-Laplace transform of $g(\xi)$. For any $\lambda\in\mathbb{C}^*\setminus\big[(-1)^k;q\big]$, let ${}_rf_s(\bm{a};\bm{b};\lambda;q,x)$ be the $p$-Laplace transform of $g(\xi)$
\begin{gather*}
{}_rf_s(\bm{a};\bm{b};\lambda;q,x)=\big(\mathcal{L}_{p;1}^{[\lambda;p]}g\big)(x)=\sum_{m\in\mathbb{Z}}\frac{g\big(\lambda p^m\big)}{\theta_p\big(\frac{\lambda p^m}{x}\big)}.
\end{gather*}
Since
\begin{gather*}
\frac{\theta_q\big((-1)^{k-1}a_1\lambda p^m\big)}{\theta_q\big((-1)^{k-1}\lambda p^m\big)}=\frac{\theta_q\big((-1)^{k-1}a_1\lambda q^{km}\big)}{\theta_q\big((-1)^{k-1}\lambda q^{km}\big)}=\frac{1}{a_1^{km}}\frac{\theta_q\big((-1)^{k-1}a_1\lambda\big)}{\theta_q\big((-1)^{k-1}\lambda\big)},
\\
{}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q,(-1)^k\frac{q^{k+1}b_1\cdots b_s}{a_1^{1-k}a_2\cdots a_r\lambda p^m}\right)=\sum_{n\ge0}\frac{(\tilde{\bm{a}};q)_n}{(\tilde{\bm{b}},q;q)_n}p^{\frac{n(n-1)}{2}}\left(\frac{pqb_1\cdots b_s}{a_1^{1-k}a_2\cdots a_r\lambda p^m}\right)^n,
\end{gather*}
and
\begin{gather*}
\theta_p\left(\frac{\lambda p^m}{x}\right)=\left(\frac{\lambda}{x}\right)^{-m}p^{-\frac{m(m-1)}{2}}\theta_p\left(\frac{\lambda}{x}\right),
\end{gather*}
we have
\begin{gather*}
{}_rf_s(\bm{a};\bm{b};\lambda;q,x)= \frac{(a_2,\ldots,a_r,b_1/a_1,\ldots,b_{s}/a_1;q)_\infty}{(b_1\ldots,b_s,a_2/a_1,\ldots, a_r/a_1;q)_\infty}\frac{\theta_q\big((-1)^{k-1}a_1\lambda\big)}{\theta_q\big((-1)^{k-1}\lambda\big)}\frac{1}{\theta_p(\lambda/x)}\\
\hphantom{{}_rf_s(\bm{a};\bm{b};\lambda;q,x)=}{} \times\sum_{n\ge0}\frac{(\tilde{\bm{a}};q)_n}{(\tilde{\bm{b}},q;q)_n}\left(\frac{qb_1\cdots b_s}{a_1^{1-k}a_2\cdots a_r\lambda}\right)^n \sum_{m\in\mathbb{Z}}\left(\frac{\lambda}{a_1^kx}\right)^mp^{\frac{n(n-1)}{2}+n-mn+\frac{m(m-1)}{2}}\\
\hphantom{{}_rf_s(\bm{a};\bm{b};\lambda;q,x)=}{} +\idem(a_1;a_2,\ldots,a_r).
\end{gather*}
Here we remark that
\begin{gather*}
\sum_{m\in\mathbb{Z}}\left(\frac{\lambda}{a_1^kx}\right)^mp^{\frac{n(n-1)}{2}+n-mn+\frac{m(m-1)}{2}} =\left(\frac{\lambda}{a_1^kx}\right)^n\sum_{m\in\mathbb{Z}}p^{\frac{(m-n)(m-n-1)}{2}}\left(\frac{\lambda}{a_1^kx}\right)^{m-n}\\
\hphantom{\sum_{m\in\mathbb{Z}}\left(\frac{\lambda}{a_1^kx}\right)^mp^{\frac{n(n-1)}{2}+n-mn+\frac{m(m-1)}{2}} }{}
=\left(\frac{\lambda}{a_1^kx}\right)^n\theta_p\left(\frac{\lambda}{a_1^kx}\right).
\end{gather*}
Therefore we obtain
\begin{gather*}
{}_rf_s(\bm{a};\bm{b};\lambda;q,x)= \frac{(a_2,\ldots,a_r,b_1/a_1,\ldots,b_{s}/a_1;q)_\infty}{(b_1\ldots,b_s,a_2/a_1,\ldots a_r/a_1;q)_\infty}\frac{\theta_q\big((-1)^{k-1}a_1\lambda\big)}{\theta_q\big((-1)^{k-1}\lambda\big)}\frac{\theta_p\big(\lambda/a_1^kx\big)}{\theta_p(\lambda/x)}\\
\hphantom{{}_rf_s(\bm{a};\bm{b};\lambda;q,x)=}{} \times\sum_{n\ge0}\frac{(\tilde{\bm{a}};q)_n}{(\tilde{\bm{b}},q;q)_n}\left(\frac{qb_1\cdots b_s}{a_1a_2\cdots a_rx}\right)^n\\
\hphantom{{}_rf_s(\bm{a};\bm{b};\lambda;q,x)=}{} +\idem(a_1;a_2,\ldots,a_r).
\end{gather*}
From
\begin{gather*}
\frac{\theta_p\big(\lambda/a_1^kx\big)}{\theta_p(\lambda/x)}=\frac{\theta_p\big(pa_1^kx/\lambda\big)}{\theta_p(px/\lambda)}
\end{gather*}
(is seen from equality \eqref{theta_inverse}) and
\begin{gather*}
\sum_{n\ge0}\frac{(\tilde{\bm{a}};q)_n}{(\tilde{\bm{b}},q;q)_n}\left(\frac{qb_1\cdots b_s}{a_1a_2\cdots a_rx}\right)^n={}_r\varphi_{r-1}\left(
\begin{matrix}\tilde{\bm{a}},\bm{0}_k\\
\tilde{\bm{b}}
\end{matrix};q,\frac{qb_1\cdots b_s}{a_1a_2\cdots a_rx}\right),
\end{gather*}
we finish the proof.
\section{Proof of Lemma \ref{Lemma:Main1}}\label{Section:Proof_Lemma:Main1}
In this section, we give a proof of Lemma \ref{Lemma:Main1}, that is, we show
\begin{gather*}
\vert g(\xi)\vert\le L \theta_{\vert p\vert}(M\vert \xi\vert),\qquad \xi\in\Omega_\delta,
\end{gather*}
where $L$ and $M$ are positive constants.

As easily seen, the following inequality holds.
\begin{gather*}
\theta_{|p|}(A|\xi|)+\theta_{|p|}(B|\xi|)\le \theta_{|p|}(C|\xi|),
\end{gather*}
where $A, B>0$ and $C=A+B$. Hence, showing that the first term of \eqref{ac_borelim} has $p$-exponential growth at infinity in $\Omega_\delta$ is sufficient to prove Lemma \ref{Lemma:Main1}. We write the first term of \eqref{ac_borelim} as~$g_1(\xi)$, i.e.,
\begin{gather*}
g_1(\xi):=\frac{(a_2,\ldots,a_r,b_1/a_1,\ldots,b_{s}/a_1;q)_\infty}{(b_1\ldots,b_s,a_2/a_1,\ldots a_r/a_1;q)_\infty}\frac{\theta_q\big((-1)^{k-1}a_1\xi\big)}{\theta_q\big((-1)^{k-1}\xi\big)}\\
\hphantom{g_1(\xi):=}{}\times {}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q;(-1)^k\frac{q^{k+1}b_1\cdots b_s}{a_1^{1-k}a_2\cdots a_r\xi}\right),
\end{gather*}
where $\tilde{\bm{a}}=(a_1,a_1q/b_1,\ldots,a_1q/b_s)\in\mathbb{C}^{s+1}$ and $\tilde{\bm{b}}=(a_1q/a_2,\ldots,a_1q/a_r)\in\mathbb{C}^{r-1}$.

We show that basic hypergeometric series ${}_{s+1}\varphi_{r-1}$ has $p$-exponential growth. It holds that
\begin{gather*}
|(\alpha;q)_n| =|1-\alpha||1-\alpha q|\cdots\big|1-\alpha q^{n-1}\big|\le(1+|\alpha|)^n,\\
|(a_iq/a_j;q)_n| =|1-a_iq/a_j|\big|1-a_iq^2/a_j\big|\cdots\big|1-a_iq^n/a_j\big|\ge \left(\inf_{k\in\mathbb{N}^*}\big\{\big|1-a_iq^k/a_j\big|\big\}\right)^n>0
\end{gather*}
for any $\alpha\in\mathbb{C}$ and $1\le i,j\le r$. Therefore we obtain
\begin{align*}
\left|{}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q;(-1)^k\frac{q^{k+1}b_1\cdots b_s}{a_1^{1-k}a_2\cdots a_r\xi}\right)\right|&\le\sum_{n\ge0}\frac{|(\tilde{\bm{a}};q)_n|}{|(\tilde{\bm{b}},q;q)_n|}|p|^{\frac{n(n+1)}{2}}\left|\frac{qb_1\cdots b_s}{a_1^{1-k}a_2\cdots a_r\xi}\right|^n\\
&\le\sum_{n\ge0}|p|^{\frac{n(n+1)}{2}}\left(\frac{1}{M|\xi|}\right)^n\\
&\le\sum_{n\in\mathbb{Z}}|p|^{\frac{n(n+1)}{2}}\left(\frac{1}{M|\xi|}\right)^n=\theta_{|p|}(M|\xi|),
\end{align*}
where $M$ is a positive constant. Hence we have
\begin{gather}\label{g1_estimate_1}
|g_1(\xi)|\le C\left|\frac{\theta_q\big((-1)^{k-1}a_1\xi\big)}{\theta_q\big((-1)^{k-1}\xi\big)}\right|\theta_{|p|}(M|\xi|).
\end{gather}
Here $C$ is a positive constant.

Now, for any $\xi \in\Omega_\delta$, there exist an integer $n$ and
\begin{gather*}
\xi_0\in\Omega_\delta \cap\{\xi\in\mathbb{C}^* ;\,|p|\le|\xi|\le1\}
\end{gather*}
such that $\xi=p^n\xi_0$.
Substituting $\xi=p^n\xi_0$ into \eqref{g1_estimate_1}, we have
\begin{gather*}
|g_1(\xi)|\le C\left|\frac{\theta_q((-1)^{k-1}a_1\xi_0)}{\theta_q((-1)^{k-1}\xi_0)}\right|\big(M|a_1|^k\big)^{-n}|p|^{-\frac{n(n-1)}{2}}\theta_{|p|}(M|\xi_0|).
\end{gather*}
Since
\begin{gather*}
\big(M|a_1|^k\big)^{-n}|p|^{-\frac{n(n-1)}{2}}=\frac{\theta_{|p|}\big(M|a_1|^k|p|^n|\xi_0|\big)}{\theta_{|p|}\big(M|a_1|^k|\xi_0|\big)}=\frac{\theta_{|p|}\big(M|a_1|^k|\xi|\big)}{\theta_{|p|}\big(M|a_1|^k|\xi_0|\big)}
\end{gather*}
holds, we obtain
\begin{gather*}
|g_1(\xi)|\le C\left|\frac{\theta_q\big((-1)^{k-1}a_1\xi_0\big)}{\theta_q\big((-1)^{k-1}\xi_0\big)}\right|\frac{\theta_{|p|}(M|\xi_0|)}{\theta_{|p|}\big(M|a_1|^k|\xi_0|\big)}\theta_{|p|}\big(M|a_1|^k|\xi|\big).
\end{gather*}
Now, the function of $\xi_0$
\begin{gather*}
\left|\frac{\theta_q\big((-1)^{k-1}a_1\xi_0\big)}{\theta_q\big((-1)^{k-1}\xi_0\big)}\right|\frac{\theta_{|p|}(M|\xi_0|)}{\theta_{|p|}\big(M|a_1|^k|\xi_0|\big)}
\end{gather*}
is holomorphic on the compact set $\Omega_\delta \cap\{\xi\in\mathbb{C}^* ;\,|p|\le|\xi|\le1\}$. Hence there exists a positive constant $L$ such that
\begin{gather*}
\max_{\Omega_\delta \cap\{\xi\in\mathbb{C}^* ;\,|p|\le|\xi|\le1\}}\left|\frac{\theta_q\big((-1)^{k-1}a_1\xi_0\big)}{\theta_q\big((-1)^{k-1}\xi_0\big)}\right|\frac{\theta_{|p|}(M|\xi_0|)}{\theta_{|p|}\big(M|a_1|^k|\xi_0|\big)}\le L.
\end{gather*}

\section{Proof of Theorem \ref{Theorem:Main2}}\label{Section:Proof_Main2}
A proof of Theorem \ref{Theorem:Main2} is obtained from the following proposition.
\begin{Proposition}[Askey {\cite[Section~5]{Askey1978}}]
We assume that $q\in\mathbb{R}$ and $0<q<1$. For any $x\in\mathbb{C}^*\setminus\mathbb{R}_-$, we have
\begin{gather}\label{theta_reduction1}
\lim_{q\to1}\frac{\theta_q\big(q^\beta x\big)}{\theta_q(q^\alpha x)}=x^{\alpha-\beta}
\end{gather}
and
\begin{gather}\label{theta_reduction2}
\lim_{q\to1}\frac{\theta_p\big(\frac{p^\alpha x}{(1-q)^k}\big)}{\theta_p\big(\frac{p^\beta x}{(1-q)^k}\big)}(1-q)^{k(\beta-\alpha)}=x^{\beta-\alpha}.
\end{gather}
Here we assume $|\arg{x}|<\pi$.
\end{Proposition}
Let us give a proof of Theorem~\ref{Theorem:Main2}. Substituting $\bm{a}=q^{\bm{\alpha}}$, $\bm{b}=q^{\bm{\beta}}$ and $x=(-1)^kx/(1-q)^k$ into~\eqref{Main1}, we have
\begin{gather*}
{}_rf_s\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^kx}{(1-q)^k}\right) =\frac{\Gamma_q(\beta_1)\cdots\Gamma_q(\beta_s)\Gamma_q(\alpha_2-\alpha_1)\cdots\Gamma_q(\alpha_r-\alpha_1)}{\Gamma_q(\alpha_2)\cdots\Gamma_q(\alpha_r)\Gamma_q(\beta_1-\alpha_1)\cdots\Gamma_q(\beta_s-\alpha_1)}(1-q)^{-k\alpha_1}\\
\hphantom{{}_rf_s\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^kx}{(1-q)^k}\right) =}{} \times\frac{\theta_{p}\left(\frac{p^{\alpha_1+1}(-1)^kx}{\lambda(1-q)^k}\right)}{\theta_{p}\left(\frac{p(-1)^kx}{\lambda(1-q)^k}\right)}\frac{\theta_q((-1)^{1-k}q^{\alpha_1}\lambda)}{\theta_q((-1)^{1-k}\lambda)}\\
\hphantom{{}_rf_s\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^kx}{(1-q)^k}\right) =}{} \times{}_r\varphi_{r-1}\left(\begin{matrix}q^{\alpha_1},q^{1+\alpha_1-\beta_1},\ldots,q^{1+\alpha_1-\beta_s},\bm{0}_k\\q^{1+\alpha_1-\alpha_2},\ldots,q^{1+\alpha_1-\alpha_r}
\end{matrix} ;q;q^{\gamma}\frac{(1-q)^k}{(-1)^kx}\right)\\
\hphantom{{}_rf_s\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^kx}{(1-q)^k}\right) =}{} +\idem\big(q^{\alpha_1};q^{\alpha_2},\ldots,q^{\alpha_r}\big),
\end{gather*}
where $\gamma=1+\beta_1+\cdots+\beta_s-(\alpha_1+\cdots+\alpha_r)$. Since we have
\begin{gather*}
\lim_{q\to1}\frac{\Gamma_q(\beta_1)\cdots\Gamma_q(\beta_s)\Gamma_q(\alpha_2-\alpha_1)\cdots\Gamma_q(\alpha_r-\alpha_1)}{\Gamma_q(\alpha_2)\cdots\Gamma_q(\alpha_r)\Gamma_q(\beta_1-\alpha_1)\cdots\Gamma_q(\beta_s-\alpha_1)}\\
\qquad{} =\frac{\Gamma(\beta_1)\cdots\Gamma(\beta_s)\Gamma(\alpha_2-\alpha_1)\cdots\Gamma(\alpha_r-\alpha_1)}{\Gamma(\alpha_2)\cdots\Gamma(\alpha_r)\Gamma(\beta_1-\alpha_1)\cdots\Gamma(\beta_s-\alpha_1)},\\
\lim_{q\to1}\frac{\theta_q\big((-1)^{1-k}q^{\alpha_1}\lambda\big)}{\theta_q\big((-1)^{1-k}\lambda\big)}=\big\{(-1)^{1-k}\lambda\big\}^{-\alpha_1},\\
\lim_{q\to1}\frac{\theta_{p}\big(\frac{p^{\alpha_1+1}(-1)^kx}{\lambda(1-q)^k}\big)}{\theta_{p}\big(\frac{p(-1)^kx}{\lambda(1-q)^k}\big)}(1-q)^{-k\alpha_1}=\left\{(-1)^k\frac{x}{\lambda}\right\}^{-\alpha_1},\\
\lim_{q\to1}{}_r\varphi_{r-1}\left(\begin{matrix}q^{\alpha_1},q^{1+\alpha_1-\beta_1},\ldots,q^{1+\alpha_1-\beta_s},\bm{0}_k\\q^{1+\alpha_1-\alpha_2},\ldots,q^{1+\alpha_1-\alpha_r}
\end{matrix} ;q;q^{\gamma}\frac{(1-q)^k}{(-1)^kx}\right)\\
\qquad{} ={}_{s+1}F_{r-1}\left(\begin{matrix}\alpha_1,1+\alpha_1-\bm{\beta}\\1+\alpha_1-\widehat{\bm{\alpha}_1}
\end{matrix} ;\frac{(-1)^k}{x}\right)
\end{gather*}
from \eqref{qgamma_reduction}, \eqref{theta_reduction1}, \eqref{theta_reduction2} and
\begin{gather*}
\lim_{q\to1}\frac{(q^\alpha;q)_n}{(1-q)^n}=(\alpha)_n
\end{gather*}
respectively, we obtain \eqref{Main2}.

\subsection*{Acknowledgements}

The author would like to thank the referees for their helpful suggestions and variable comments. Additionally, the author is grateful for that one of the referee let him know the existence of the papers \cite{Dreyfus2015_2} and \cite{Sauloy2000}.

\pdfbookmark[1]{References}{ref}


TITLE: Most probably value of velocity does not coincide with most probable energy for a 2D ideal gas
QUESTION [2 upvotes]: I'm having trouble understanding the physical concept of the following statistical mechanical problem.
I'm given the function
$$ \Phi(v_x) = (\frac{m}{2\pi \kappa T})^{1/2} \cdot e^{-mv_x^2/2\kappa T}.$$
To solve the density of a a speed $v = [v_x, v_y]$
$ \Phi(v_x,v_y) = \Phi(v_x)\Phi(v_y) = (\frac{m}{2\pi \kappa T}) \cdot e^{-m(v_x^2+v_y^2)/2\kappa T}$
A coordinate transfer taking $dxdy=r\ dr\ d\theta$
$f(v)dv = \int_0^{2\pi} d\theta \ v(\frac{m}{2\pi \kappa T}) \cdot e^{-m(v_x^2+v_y^2)/2\kappa T}$
Which gives
$f(v) = 2\pi v(\frac{m}{2\pi \kappa T}) \cdot e^{-mv^2/2\kappa T}$
The most probable speed is then just
$v_{mpv} = \sqrt{\kappa T/m}$
I would have expected that the most probable kinetic energy would then be
$\epsilon_{mpv} = \frac{mv_{mpv}^2}{2}$
However by doing the correct transformations the distribution of $\epsilon$ looks instead like
$f(\epsilon) = \frac{e^{-\epsilon/\kappa T}}{\kappa T}$
which has a most probable value of
$\epsilon_{mpv} = 0$
I think I missed something conceptually as to why the most probable values of $\epsilon$ and $v$ don't coincide. Any help is greatly appreciated.
Edit: I would like to add that this is not a homework problem, but an example problem. I took a stab at it without looking at the answer and am confused by the disconnect with my logic and the answer.

REPLY [1 votes]: I stumbled also into this problem and took me a while to realize why the most probable speed and the most probable energy do not have to correspond.
I want to preface this by saying that it is always the integral of the distribution function that matters, not the value of the distribution function itself. Therefore, asking for the most probable velocity is not really physically meaningful -- there are zero particles with exactly  the most probable velocity! (There are zero particles with any exact velocity value, you can only ask "How many particles are there with velocities in a given interval")
The problem of why the maxima of the distribution functions do not match is because the transformation of variables here is not linear ($E \propto v^2$). The transformation of the distribution function preserves area under the distribution function at each interval:
$$
   f(v) dv = g(E) dE
$$
Now, if you transform your x-axis of the distribution function with $E \propto v^2$, you will have to spread out the number of particles such that the area-preserving relation holds at every portion of the distribution. This will modify the shape of the function and displace your maxima.
Let me show this by an animation. In the following I start with a histogram (approximation of the distribution function) for the speed variable (2D case), then modify the variable shown to be $\xi = v^p / 2$ where I change $p$ to be from $1$ to $3$. Notice that at $p = 2$ this corresponds to the energy distribution.
(The plotting script is here.)


TITLE: Finding a polynomial of degree $n$ when value of $f(k)$ is equal to some value
QUESTION [0 upvotes]: Problem :
If $f(x)$ is a polynomial of degree $n$ and if $f(k) = \frac{k}{k+1}$ where $k =0,1,2,\ldots,n$, find $f(x)$.
Can we go like this : Let the polynomial be $a(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)......(x-\alpha_n)$ where $\alpha_1, \alpha_2....\alpha_n$ are the roots of the $f(x)$ and $a$ is the leading coefficient.
Please guide how to proceed in such question.

REPLY [1 votes]: Hint: Consider the polynomial $g(x) = (x+1) f(x) - x$. What is the degree of $g(x)$? What are the zeros of $g(x)$? What is the value of $g(-1)$?

More details added.
The degree of $g(x)$ is equal to the degree of $f(x)$ plus 1 (since we multiplied by $x+1$), which is equal to $n+1$.
The zeros of $g(x)$ are (at least) $0, 1, 2, \ldots n$, so by the Remainder-Factor Theorem,
$$g(x) = A(x) \prod_{i=0}^n (x-i)$$
Since $g$ has degree $n+1$, it follows that $A(x)$ is a constant polynomial, which we will denote by $A$.
The value of $g(-1)$ is $g(-1) = (-1 + 1) f(-1) - (-1) = 1 $. Hence
$$1 = g(-1) = A \prod_{i=0}^n (-1-i)$$
This allows us to determine that $A = \frac{1}{(n+1)!}$.
Hint: $\frac{ g(x) + x) } { x+1}$ is indeed a polynomial, and is our $f(x)$.