TITLE: Sampling distribution of the mean confusion
QUESTION [1 upvotes]: Say we have some population and decide to randomly take a sample of size $n$ from this population.  What does it then mean to talk about the distribution of the sample mean?
In other words, what do we mean by the distribution of the sample mean here?

I’m quite confused about this as from what I understood from my textbook, it doesn’t make sense to talk about a distribution of a sample mean in this scenario (since we are looking at one sample).  Obviously, I’m wrong, but I’m not seeing why.

REPLY [0 votes]: As a relative newcomer to statistics, this confused me also. Similar wording is often used when comparing a single sample to the sampling distribution. When wording like this is used, they do not mean the sampling distribution was generated from that one sample, they are referring to the sampling distribution generated from all possible samples of the same size as the single sample.


text:
\section{Preliminaries}
\label{sec:model}

This section presents preliminaries on the spectral evolution model,
including common growth methods based on graph kernels and spectral
extrapolation used for link prediction. The overview is, mainly, based
on~\citep{kunegis-semodel-2013}.

\subsection{Spectral Representation}

A graph $G = (V, E)$ is specified by its \textit{vertex} set $V$ and
its \textit{edge} set $E$; throughout the rest of paper it is assumed
that any graph is finite (i.e., $|V|$ is a natural number),
undirected, and without self or multiple edges. Without loss of
generality, the vertex set $V$ is assumed to be ordered, e.g., $V =
\{1, 2, 3, \ldots, |V|\}$.

To any graph $G = (V, E)$ a matrix $\MAT{A}_G$ (or simply $\MAT{A}$)
is associated whose entries are given for any $u,v \in V$ by
$\MAT{A}(u, v) = 1$ if $(u, v) \in E$ and $\MAT{A}(u, v) = 0$ if $(u,
v) \notin E$. The matrix $\MAT{A}$ is called the \textit{adjacency
  matrix} of $G$. Under the above assumptions, any adjacency matrix is
symmetric.

A vector $\MAT{x}$ is an \textit{eigenvector} of a matrix $\MAT{A}$
with \textit{eigenvalue} $\lambda$ iff $\MAT{A}\MAT{x} =
\lambda\MAT{x}$. The spectral theorem states an important fact about
symmetric adjacency matrices.

\begin{theorem}\label{thm:eigendecom}
  Let $n \in \nats$ be positive. If $\MAT{A}$ is a $n$-by-$n$
  symmetric matrix, with real entries, then there exist real numbers
  $\lambda_1, \lambda_2, \ldots, \lambda_n$ and $n$ mutually
  orthogonal unit vectors $\MAT{x}_1, \MAT{x}_2, \ldots, \MAT{x}_n$
  such that $\MAT{x}_i$ is an eigenvector of $\MAT{A}$ with eigenvalue
  $\lambda_i$, for $1 \leq i \leq n$.
\end{theorem}

The pair $(\MAT{X}, \MAT{\Lambda})$, with $\MAT{X}$ being the
orthogonal matrix obtained from the unit vectors
$\MAT{x}_1,\ldots,\MAT{x}_n$ and $\MAT{\Lambda}$ the diagonal matrix
determined by $\lambda_1,\ldots, \lambda_n$ in
Theorem~\ref{thm:eigendecom}, is called the \emph{spectral
  decomposition} of $\MAT{A}$ and satisfies $\MAT{A} =
\MAT{X}\MAT{\Lambda}\MAT{X}^T$. It is important to note that
eigenvalues are uniquely determined, but can be repeated. However,
eigenvectors are not uniquely determined for a given eigenvalue. In
general, the eigenvectors of a given eigenvalue are determined modulo
orthogonal transformations.

\subsection{The Spectral Evolution Model}

The \textit{spectral evolution model}~\citep{kunegis-semodel-2013}
assumes that the evolution of a network can be described by the
evolution in the network's spectrum, while the networks eigenvectors
stay largely constant over time.

Formally, if $\MAT{A}_t$ is the adjacency matrix of a network at time
$t$ and $(\MAT{X}_t, \MAT{\Lambda}_t)$ its spectral decomposition,
then the spectral decomposition $(\MAT{X}_{t'}, \MAT{\Lambda}_{t'})$
of the adjacency matrix of the network at any time $t'$ satisfies
$\MAT{X}_{t} \approx \MAT{X}_{t'}$ (i.e., the eigenvectors of
$\MAT{A}_t$ and $\MAT{A}_{t'}$ are approximately the same over time).

There are, basically, two existing approaches to link prediction with
the spectral evolution model: spectral transformation functions (based on graph kernels) and
spectral extrapolation.

\subsubsection{Spectral Transformation}

Link prediction can be expressed as a spectral decomposition
$\MAT{X}F(\MAT{\Lambda})\MAT{X}^T$, where $F$ is a \textit{spectral
  transformation} that applies the same real function to each diagonal
element $\lambda_j$ of $\MAT{\Lambda}$.

\textbf{The Triangle Closing Kernel.}

The triangle closing kernel is expressed as
$\MAT{A}^2=\MAT{X}\MAT{\Lambda}^2\MAT{X}^T$. That is, the kernel
replaces the eigenvalues of
$\MAT{A}$ by their squared values. The  function associated to
the triangle closing kernel is $f(\lambda)=\lambda^2$.

\textbf{Exponential Kernel.}

The exponential kernel is expressed as

\[\exp{(\alpha\MAT{A})}=\sum_{k=0}^\infty\alpha^k\frac{1}{k!}\MAT{A}^k,\]

where $\alpha$ is a constant used to balance the weight of short and
long paths. It denotes the sum of every path between two vertices
weighted by the inverse factorial of its length. The function
associated to the exponential kernel is
$f(\lambda)=e^{\alpha\lambda}$.

\textbf{Neumann Kernel.}

The Neumann kernel is expressed as

\[(\MAT{I}-\alpha\MAT{A})^{-1}=\sum_{k=0}^\infty\alpha^k\MAT{A}^k,\]

where $\alpha^{-1}>|\lambda_1|$ and $\lambda_1$ is the largest
eigenvalue of $\MAT{A}$. The function associated to the Neumann kernel is
$f(\lambda)=1/(1-\alpha\lambda)$.

\subsubsection{Spectral Extrapolation}

Graph kernels can characterize a spectral transformation function for
the growth of the spectrum. However, when the evolution of the
spectrum is irregular, it is not always possible to find a simple
function that describes the trajectory for all eigenvalues. In this
situation, it is often convenient to use spectral extrapolation, which
can be viewed as a generalization of graph kernels.

Let $t_1$ be an intermediate time and $t_2$ the final time from which
link prediction is to be made, and let $\MAT{A}_1$ and $\MAT{A}_2$ be
the adjacency matrices of the network at times $t_1$ and $t_2$,
respectively. These matrices can be decomposed as $\MAT{A}_1 =
\MAT{X}_1\MAT{\Lambda}_1\MAT{X}_1^T$ and $\MAT{A}_2 =
\MAT{X}_2\MAT{\Lambda}_2\MAT{X}_2^T$, for some
$\MAT{X}_1,\MAT{\Lambda}_1,\MAT{X}_2,\MAT{\Lambda}_2$. If
$(\lambda_{2})_j$ is the $j$-th eigenvalue (say, under the ordering of
$\MAT{\Lambda}_2$'s diagonal) at $t_2$, its estimated previous value
$(\hat{\lambda}_1)_j$ at $t_1$ is computed as a diagonalization of
$\MAT{A}_1$ by $\MAT{X}_2$

\[(\hat{\lambda}_1)_j=\left(\sum_i(\MAT{X}_{1})_i^T(\MAT{X}_2)_j\right)^{-1}\sum_i(\MAT{X}_1^T)_i(\MAT{X}_2)_j(\lambda_1)_i,\]

where $(\MAT{X}_t)_k$ is the $k$-th eigenvector for $1 \leq k \leq n$,
and $(\lambda_t)_k$ is the $k$-th eigenvalue of $\MAT{A}_t$ ($t \in
\{1,2\}$), respectively (under the same ordering mentioned
above). Note that in this formula, the occurrences of $\MAT{X}_{1}$
could be replaced by $\MAT{X}_{2}$ because of the spectral evolution
model assumptions. However, if the possible small changes in the
eigenvectors are to be taken into account, the given version of the
formula should be used.

Linear extrapolation is used to predict link formation in the network
at a future time $t_3>t_2$. The $j$-th eigenvalue at $t_3$ is
approximated from the eigenvalues $\lambda_2$ at time $t_2$ and their
approximated counter-parts $\hat{\lambda}_1$ at time $t_1$, as
follows:

\[(\hat{\lambda}_3)_j=2(\lambda_{2})_j-(\hat{\lambda}_1)_j.\]

Finally, a link prediction real-valued matrix $\hat{\MAT{A}}_3$ is
computed as follows:

\[\hat{\MAT{A}}_3=\MAT{X}_2\hat{\MAT{\Lambda}}_3\MAT{X}_2^T.\]

Let $\hat{\MAT{A}}_3(u, v)$ denote the entry in row $u$ and column $v$
of $\hat{\MAT{A}}_3$.  For each $u,v \in V$, the value
$\hat{\MAT{A}}_3(u, v)$ is a link prediction score representing the
probability of having at time $t_3$ a link between $u$ and $v$. The
adjacency matrix $\MAT{A}_3$, which would represent the network at the
future time $t_3$, can be constructed from $\hat{\MAT{A}}_3$ in many
ways. For example, by setting a threshold $0 \leq \delta \leq 1$ such
that $\MAT{A}_3(u,v)=1$ if $\hat{\MAT{A}}_3(u,v) \geq \delta$ and
$\MAT{A}_3(u,v)=0$ if $\hat{\MAT{A}}_3(u,v) < \delta$, for each $u,v
\in V$.

config: arxiv


TITLE: Calculate $ \lim_{x \to 0} \frac{1}{[x]}$
QUESTION [2 upvotes]: Let $f = \dfrac{1}{[x]}$ where $[x]$ denotes greatest integer function.
(a) $\displaystyle\lim_{x \to 0} \dfrac{1}{[x]} = -1$
(b) $\displaystyle\lim_{x \to 0} \dfrac{1}{[x]}$ does Not Exist
(c) Domain of the function is $\mathbb{R}$
(d) Domain Is $\mathbb{R}-\{0\}$
Okay so, Domain will be set of points where $[x] \ne 0$
or, $x \ne [0,1)$
So,Domain is given by $\mathbb{R}-[0,1)$
Now, To solve for limit Only Left Hand Limit will exist as $(x \to 0+)$ does Not belong to the domain so I think only one sided limit will exist.
$\displaystyle \lim_{x \to 0} f(x)$ = $\displaystyle \lim_{x \to 0-} f(x) = -1$
So, only option (a) seems to be correct according to me .
Is my answer and solution correct? Can someone please check ?
Thank you.

REPLY [0 votes]: Since $x=0$ is an accumulation point for the domain $D$, the right definition of limit at $x=0$ is this:
$$
\forall \varepsilon  > 0\,\,\,\exists \delta  = \delta \left( \varepsilon  \right) > 0:\forall x \in \left( { - \delta ,\delta } \right) \cap D\setminus\left\{ 0 \right\} \Rightarrow \left| {f\left( x \right) - l} \right| < \varepsilon
$$
Now
$$
\left( { - \delta ,\delta } \right) \cap D = \left( { - \delta ,0} \right)
$$
and you have that
$$
\forall \varepsilon  > 0\,\,\,\exists \delta  = \delta \left( \varepsilon  \right) > 0:\forall x \in \left( { - \delta ,0} \right) \Rightarrow \left| {f\left( x \right) + 1} \right| < \varepsilon
$$
and this means that
$$
\mathop {\lim }\limits_{x \to 0} f(x) =  - 1
$$
This is to think as a full limit even though it could seems a left limit. In fact it is not so due to the definition of limit and in particular due to the condition
$$
\forall x \in \left( { - \delta ,\delta } \right) \cap D\setminus\left\{ 0 \right\}
$$


text:
\begin{document}

\begin{frontmatter}

\title{An extended framework of continuous-stage Runge-Kutta methods}

\author[a,b]{Wensheng Tang\corref{cor1}}
\ead{tangws@lsec.cc.ac.cn} \cortext[cor1]{Corresponding author.}
\address[a]{College of Mathematics and Statistics,\\
    Changsha University of Science and Technology,\\ Changsha 410114, China}
\address[b]{Hunan Provincial Key Laboratory of \\
   Mathematical Modeling and Analysis in Engineering,\\ Changsha 410114, China}
\author[]{}

\begin{abstract}
We propose an extended framework for continuous-stage Runge-Kutta
methods which enables us to treat more complicated cases especially
for the case weighting on infinite intervals. By doing this, various
types of weighted orthogonal polynomials (e.g., Jacobi polynomials,
Laguerre polynomials, Hermite polynomials etc.) can be used in the
construction of Runge-Kutta-type methods. Particularly, families of
Runge-Kutta-type methods with geometric properties can be
constructed in this new framework. As examples, some new symmetric
and symplectic integrators by using Legendre polynomials, Laguerre
polynomials and Hermite polynomials are constructed.

\end{abstract}

\begin{keyword}

Ordinary differential equations; Continuous-stage Runge-Kutta
methods; Legendre  polynomials; Laguerre polynomials; Hermite
polynomials; Hamiltonian systems; Symplectic methods.

\end{keyword}

\end{frontmatter}

\section{Introduction}
\label{}

It is widely believed that numerical methods are a vital component
in the discovery and study of natural phenomena and law, especially
for solving various differential equations arising in the field of
natural science, primarily due to the fact that in most cases we can
not find the analytical or precise solutions. If we put aside those
intricate and complicated mathematical theories, then it is
conceived that a ``good" numerical method should be simple, easily
understood and conveniently implemented in the practical
application. As is well known, polynomials are the most simple
mathematical object for computers to treat, and thus they ought to
be strongly suggested for use in the construction of numerical
methods --- indeed, they have attracted much attention and been
extensively used in many fields, including finite element methods
\cite{Brenners08tmt}, spectral methods
\cite{Gottliebo77nao,Guo98sma},  numerical differentiation and
integration (quadrature) \cite{sulim03ait}, interpolation theory and
numerical approximation \cite{sulim03ait,Szeg85op} and so forth.
Amongst multitudinous types of polynomials, orthogonal polynomials
are of significant importance mainly owing to that they are more
convenient to use and sometimes it may lead to surprising
applications \cite{butcher92tro}.

The well-known Runge-Kutta (RK) method, as one of the most popular
classes of methods for solving initial value problems, has been a
central topic in numerical ordinary differential equations (ODEs)
since the pioneering work of Runge (1895) \cite{runge95udn}. They
have been highly developed for over a hundred and twenty years (see
\cite{butcher96aho,butcher64ipa,butcher87tna,hairernw93sod,hairerw96sod}
and references therein). One of the most important points we have to
speak out is that RK methods possess a close relationship with
polynomials. A basic evidence for supporting such view is that, the
RK order conditions corresponding to bushy trees (by B-series theory
\cite{butcher87tna,hairerlw06gni}) are equivalent to those
polynomial-based quadrature rules \cite{hairernw93sod}. Another good
case in evidence is the $W$-transformation \cite{hairerw96sod} which
is defined on the basis of Legendre orthogonal polynomials. By using
$W$-transformation many special-purpose methods can be established,
e.g., symplectic RK methods, symmetric RK methods, algebraically
stable RK methods, stiffly accurate \& $L$-stable RK methods etc
\cite{chan90osr,dingt09irk,hairernw93sod,hairerw96sod,lius05irk,sun93coh,sun00asw}.

A novel RK approach reflecting stronger relationship between RK-type
methods and polynomials are developed in recent years. It is clear
and easy to understand it by introducing the concept of
``continuous-stage Runge-Kutta (csRK) methods". Actually, the theory
of csRK methods was initially launched by Butcher in 1972
\cite{butcher72ato} (see also \cite{butcher87tna}) and subsequently
developed by Hairer \cite{hairer10epv,hairerz13oco},  Tang \& Sun
\cite{Tangs12tfe,tangs12ana,Tangs14cor,Tangsc17dgm}, and Miyatake \&
Butcher \cite{miyatake14aep,miyatakeb16aco}. The coefficients of
csRK methods are assumed to be ``continuous" functions and thus it
is a simple and natural choice to let them be polynomials. The first
example of such methods was given by Hairer using Lagrangian
interpolatory polynomials for deriving energy-preserving collocation
(EPC) methods \cite{hairer10epv}, whereas a low-order version of EPC
methods was proposed earlier by Quispel \& McLaren
\cite{quispelm08anc} with the name ``averaged vector field (AVF)
methods" but without being interpreted as csRK methods. A
closely-related type of methods which can also be transformed into
csRK methods attributes to Brugnano et al \cite{brugnanoit10hbv},
called infinity Hamiltonian Boundary Value Methods (denoted by
$\infty$-HBVMs), but also had not been explained within the
framework of csRK methods at that time. It was firstly pointed out
in \cite{Tangs12tfe} (and subsequently in \cite{tangs12ana}) that
four kinds of Galerkin time finite element methods can be recast as
csRK methods, and particularly, EPC methods, AVF methods as well as
$\infty$-HBVMs can be explained as continuous finite element
methods, and all of them can be unified in the context of csRK
methods. Instead of using Lagrangian polynomials \cite{hairer10epv},
Tang et al \cite{Tangs12tfe,tangs12ana,Tangs14cor,Tang18ano} resort
to Legendre orthogonal polynomials for constructing csRK methods,
and some special-purpose methods including symplectic csRK methods
\cite{tangs12ana,Tangs14cor,Tang18ano}, conjugate-symplectic (up to
a finite order) csRK methods \cite{Tangs14cor}, symmetric csRK
methods \cite{tangs12ana,Tangs14cor,Tang18ano}, energy-preserving
csRK methods \cite{tangs12ana,Tangs14cor,Tang18ano} are developed in
use of Legendre polynomials. Besides, Some extensions of csRK
methods are also obtained by Tang et al
\cite{Tangsz15hos,Tanglx16cos,Tangz18spc}. Another interesting
related work are given by Li \& Wu \cite{liw16ffe}, showing that
their energy-preserving methods can be explained as a class of csRK
methods.

More recently, new ideas have been introduced for construction of
csRK methods in \cite{Tang18ano,Tang18csr,Tang18csm,Tang18siw},
where the Butcher coefficients are partially assumed to be
polynomial functions by leading weight functions (not necessarily
polynomial) into the formalism of csRK methods.  These new
developments have greatly extended the previous studies from the
simplest case with coefficient $B_\tau=1$ to a much more general
case with $B_\tau\neq1$.  As is shown in
\cite{Tang18ano,Tang18csr,Tang18csm,Tang18siw}, by using different
weighted orthogonal polynomials including Legendre polynomials,
Chebyshev polynomials and their superset ``Jacobi polynomials", new
families of symplectic methods and symmetric methods can be derived.
However, these investigations are following Hairer's original
formalism \cite{hairer10epv} with integrals restricted on a finite
interval $[0,1]$. In this paper, we are going to break through such
limitation and further enlarge the primitive framework of csRK
methods to a super new one which admits us to treat the case for
infinite intervals.

This paper will be organized as follows. Section 2 is devoted to
give some preliminaries about orthogonal polynomials and weighted
interpolatory quadrature. This is followed by Section 3,  in which
we will speak out our new idea for extending the previous framework
of csRK methods. After that, some discussions about geometric
integration by csRK methods will be given in Section 4. Section 5 is
devoted to showing a few numerical experiments for the sake of
verifying our theoretical results. At last, we conclude this paper.

\section{Orthogonal polynomials and weighted interpolatory quadrature}

Assume $I$ is an interval, and we do not restrict it to be a finite
interval. Particularly, it can be an infinite interval in the
following three types: (1) $(-\infty, b]$ with $b$ a finite real
number; (2) $[a, +\infty)$ with $a$ a finite real number; (3)
$(-\infty, +\infty)$.

\begin{defi}\label{weight_func}
A non-negative (or positive almost everywhere) function $w(x)$ is
called a \emph{weight function} on $I$, if it satisfies the
following two conditions:
\begin{itemize}
 \item[(a)] The $k$-th moment $\int_I x^k w(x)\,\dif x, \;k\in\mathbb{N}$ exists;
 \item[(b)] For any non-negative function $u(x)$, $\int_Iu(x)w(x)\,\dif
 x=0$ implies $u(x)\equiv0$.
 \end{itemize}
\end{defi}

It is known that for a given weight function $w(x)$, there exists a
sequence of orthogonal polynomials in the \emph{weighted function
space} (Hilbert space) \cite{Szeg85op}
\begin{equation*}
L^2_w(I)=\{u \text{ is measurable on}\, I:\;\int_I|u(x)|^2w(x)\,\dif
x<+\infty\}
\end{equation*}
which is linked with the inner product
\begin{equation*}
(u,v)_w=\int_Iu(x)v(x)w(x)\,\dif x.
\end{equation*}

In this paper, we denote a sequence of weighted orthogonal
polynomials by $\{P_n(x)\}_{n=0}^\infty$, which consists of a
complete set in the Hilbert space $L^2_w(I)$. It is known that
$P_n(x)$ has exactly $n$ real simple zeros in the interval $I$. For
convenience, in what follows we always assume the orthogonal
polynomials are normalized, i.e., satisfying
\begin{equation*}
(P_i,P_j)_w=\delta_{ij},\;\;i, j=0,1,2,\cdots.
\end{equation*}

It is known that orthogonal polynomials have important applications
in numerical integration. An $s$-point \emph{weighted interpolatory
quadrature formula} is in the form
\begin{equation}\label{wquad}
\int_I\Phi(\tau)w(\tau)\,\dif \tau\approx\sum\limits_{i=1}^sb_i
\Phi(c_i),\;\;c_i\in I,
\end{equation}
where
\begin{equation*}
b_i=\int_I\ell_i(\tau)w(\tau)\,\dif
\tau,\;\;\ell_i(\tau)=\prod\limits_{j=1,j\neq
i}^s\frac{\tau-c_j}{c_i-c_j},\;\;i=1,\cdots,s.
\end{equation*}
Here, remark that for the simplest case $s=1$, the number of
Lagrangian basis functions in interpolation is just one, i.e.,
$\ell_1(\tau)=\tau/c_1$.

As far as we know, the optimal quadrature technique is the
well-known Gauss-Christoffel's, which can be stated as follows (see,
for example, \cite{Abramowitzs65hom,sulim03ait}).
\begin{thm}\label{GCqua}
Consider $I$ is a finite interval. If $c_1, c_2,\cdots, c_s$ are
chosen as the $s$ distinct zeros of the normalized orthogonal
polynomial $P_s(x)$ of degree $s$ in $L^2_w(I)$, then the
interpolatory quadrature formula \eqref{wquad} is exact for
polynomials of degree $2s-1$, i.e., of the optimal order $p=2s$. If
$\Phi\in C^{2s}$, then it has the following remainder
\begin{equation*}
\int_I\Phi(\tau)w(\tau)\,\dif \tau-\sum\limits_{i=1}^sb_i
\Phi(c_i)=\frac{\Phi^{(2s)}(\xi)}{(2s)!\mu^2_s},
\end{equation*}
for some $\xi\in I$, where $\mu_s$ is the leading coefficient of
$P_s(x)$, and $b_i=-\frac{\mu_{s+1}}{\mu_sP'_s(c_i)P_{s+1}(c_i)}>0$
\cite{judd12qmp}.
\end{thm}
\begin{rem}\label{GassChriRem}
If the weight functions are defined on an infinite interval, then we
have similar results. For instance, if we take the weight functions
as
\begin{equation*}
e^{-x},\;x\in[0,+\infty)\;\;\text{and}\;\;e^{-x^2},\;x\in(-\infty,+\infty),
\end{equation*}
respectively, then we have the well-known Gauss-Christoffel-Laguerre
and Gauss-Christoffel-Hermite quadrature rules of order $p=2s$ and
the remainders of them are dependent on $\Phi^{(2s)}(\xi)$ (for more
details, see \cite{Abramowitzs65hom}).
\end{rem}

Other suboptimal quadrature rules, e.g., Gauss-Christoffel type
quadrature with some fixed nodes (including Gauss-Radau,
Gauss-Lobatto quadrature etc.) can be found in many literatures
(see, for example, \cite{sulim03ait}). We find that numerical
solution of ordinary differential equations can be closely related
to quadrature techniques associated with suitable orthogonal
polynomials. This will be clearly shown later by studying csRK
methods.

\section{Continuous-stage Runge-Kutta methods}

Consider the following initial value problem
\begin{equation}\label{eq:ode}
\dot{\vec{z}}=\vec{f}(t,\vec{z}),\quad \vec{z}(t_0)=\vec{z}_0\in
\mathbb{R}^d,
\end{equation}
where
$\vec{f}:\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is
assumed to be sufficiently differentiable. For such problem, we
propose the following concept of csRK methods \emph{weighted} on
$I$, the initial version of which was proposed by Tang in
\cite{Tang18csr}.

\begin{defi}\label{defi1}
Let $w(x)$ be a weight function defined on $I$, $A_{\tau,\, \sigma}$
be a function of two variables $\tau$, $\sigma$, and $B_\tau$,
$C_\tau$ be functions of $\tau$. The one-step method $\Psi_h:
\vec{z}_0 \mapsto \vec{z}_{1}$ given by
\begin{equation}\label{crk}
\begin{split}
&\vec{Z}_\tau=\vec{z}_0+h\int_IA_{\tau,\,\sigma}w(\sigma)\vec{f}(t_0+C_\sigma
h,\vec{Z}_\sigma)\,\dif
\sigma,\quad\tau\in I,\\
&\vec{z}_{1}=\vec{z}_0+h\int_IB_\tau w(\tau)\vec{f}(t_0+C_\tau
h,\vec{Z}_\tau)\,\dif\tau,
\end{split}
\end{equation}
is called a \emph{continuous-stage Runge-Kutta} (csRK) method, where
$\vec{Z}_\tau\approx \vec{z}(t_0+C_\tau h).$ Here we often assume
\begin{equation}\label{consis}
C_\tau=\int_IA_{\tau,\,\sigma}w(\sigma)\,\dif\sigma,
\end{equation}
and denote a csRK method by the triple $(A_{\tau,\,
\sigma}w(\sigma),\,B_\tau w(\tau),\,C_\tau)$.
\end{defi}
\begin{rem}
For the case when $I$ is an infinite interval, we assume that the
improper integrals of \eqref{crk} satisfy some conditions (in terms
of uniform convergence) such that differentiation under the integral
sign with respect to parameter $h$ (step size) is legal.
Consequently, the Taylor expansion of the numerical solution can be
written as a B-series \cite{hairerlw06gni} and the standard order
theory of B-series integrators can be applied.
\end{rem}

\begin{rem}
If take $I$ as the standard interval $[0,1]$, then we regain the
methods developed in \cite{Tang18csr}. However, remark that the
primitive framework of csRK methods given in \cite{Tang18csr} can
not cover the case for weighting on an infinite interval and many
classical orthogonal polynomials such as Laguerre polynomials,
Hermite polynomials etc can not be applied in that framework.
\end{rem}

Following Hairer's idea \cite{hairer10epv}, we introduce the
following variant of \emph{simplifying assumptions}\footnote{It
should be noticed that in $\breve{D}(\zeta)$ we have removed
``$w(\sigma)$" from both sides of the formula.}
\begin{equation}\label{csRK_sim_assum}
\begin{split}
&\breve{B}(\xi):\quad \int_IB_\tau w(\tau) C_\tau^{\kappa-1}\,\dif
\tau=\frac{1}{\kappa},\quad \kappa=1,\ldots,\xi,\\
&\breve{C}(\eta):\quad
\int_IA_{\tau,\,\sigma}w(\sigma)C_\sigma^{\kappa-1}\,\dif
\sigma=\frac{1}{\kappa}C_\tau^{\kappa},\quad \kappa=1,\ldots,\eta,\\
&\breve{D}(\zeta):\quad \int_IB_\tau w(\tau) C_\tau^{\kappa-1}
A_{\tau,\,\sigma}\,\dif \tau=\frac{1}{\kappa}B_\sigma
(1-C_\sigma^{\kappa}),\quad \kappa=1,\ldots,\zeta.
\end{split}
\end{equation}
Remark that the range of integration on the left-hand side is $I$
(finite or infinite) which has extended the original simplifying
assumptions (with $I=[0,1]$) given by Hairer \cite{hairer10epv}. The
following result is similar to the classical result by Butcher in
1964 \cite{butcher64ipa}.
\begin{thm}\label{crk:order}
If the coefficients $(A_{\tau,\,\sigma}w(\sigma),\,B_\tau
w(\tau),\,C_\tau)$ of method \eqref{crk} satisfy $\breve{B}(\xi)$,
$\breve{C}(\eta)$ and $\breve{D}(\zeta)$, then the method is of
order at least $\min (\xi,2\eta+2,\,\eta+\zeta+1)$.
\end{thm}
In this article, we will hold on a simple assumption abidingly as
done in \cite{Tangs14cor,Tang18ano,Tang18csr}
\begin{equation}\label{C_assum}
C_\tau\equiv\tau, \quad\tau\in I.
\end{equation}
The following results are presented for generalizing those available
results given in \cite{Tang18csr}.
\begin{lem}\label{lem_assum}
Under the assumption \eqref{C_assum}, $\breve{B}(\xi),
\breve{C}(\eta)$ and $\breve{D}(\zeta)$ are equivalent to
respectively,
\begin{align}\label{eq:cd1}
&\breve{B}(\xi):\quad \int_IB_\tau w(\tau) \phi(\tau)\,\dif
\tau=\int_0^1\phi(x)\,\dif x,\quad \text{for}\; \forall\,  \phi\;
\text{with}\;\emph{deg}(\phi)\leq\xi-1,\\\label{eq:cd2}
&\breve{C}(\eta):\quad \int_IA_{\tau,\,\sigma} w(\sigma)
\phi(\sigma)\,\dif \sigma=\int_0^\tau \phi(x)\,\dif x,\quad
\text{for}\; \forall\,  \phi\; \text{with}\;
\emph{deg}(\phi)\leq\eta-1,\\\label{eq:cd3} &\breve{D}(\zeta):\quad
\int_IB_\tau  A_{\tau,\,\sigma}w(\tau)\phi(\tau)\,\dif
\tau=B_\sigma\int_\sigma^1\phi(x)\,\dif x,\quad  \text{for}\;
\forall\, \phi\; \text{with}\; \emph{deg}(\phi)\leq\zeta-1,
\end{align}
where $\emph{deg}(\phi)$ stands for the degree of polynomial
function $\phi$.
\end{lem}
\begin{proof}
Please refer to Lemma 2.1 of \cite{Tang18csr} to obtain a
straightforward proof.
\end{proof}

\begin{thm}\label{ordcon_var}
Suppose\footnote{The notation $A_{\ast,\,\sigma}$ stands for the
one-variable function in terms of $\sigma$, and $A_{\tau,\,\ast}$
can be understood likewise.}
$B_{\tau},\,\,A_{\ast,\,\sigma},\,\,(B_\tau\,A_{\tau,\,\ast})\in
L^2_w(I)$, then, under the assumption \eqref{C_assum} we have
\begin{itemize}
\item[\emph{(a)}] $\breve{B}(\xi)$ holds $\Longleftrightarrow$ $B_\tau$ has
the following form in terms of the normalized orthogonal polynomials
in $L^2_w(I)$:
\begin{equation}\label{Bt}
B_\tau=\sum\limits_{j=0}^{\xi-1}\int_0^1P_j(x)\,\dif x
P_j(\tau)+\sum\limits_{j\geq\xi}\lambda_j P_j(\tau),
\end{equation}
where $\lambda_j$ are any real parameters;
\item[\emph{(b)}] $\breve{C}(\eta)$ holds $\Longleftrightarrow$ $A_{\tau,\,\sigma}$ has
the following form in terms of the normalized orthogonal polynomials
in $L^2_w(I)$:
\begin{equation}\label{Ats}
A_{\tau,\,\sigma}=\sum\limits_{j=0}^{\eta-1}\int_0^{\tau}P_j(x)
\,\dif x P_j(\sigma)+\sum\limits_{j\geq\eta}\phi_j(\tau)
P_j(\sigma),
\end{equation}
where $\phi_j(\tau)$ are any real functions;
\item[\emph{(c)}] $\breve{D}(\zeta)$ holds $\Longleftrightarrow$ $B_\tau A_{\tau,\,\sigma}$ has
the following form in terms of the normalized orthogonal polynomials
in $L^2_w(I)$:
\begin{equation}\label{BtAts}
B_\tau\,A_{\tau,\,\sigma}=\sum\limits_{j=0}^{\zeta-1}B_\sigma\int_\sigma^1P_j(x)
\,\dif x P_j(\tau)+\sum\limits_{j\geq\zeta}\psi_j(\sigma) P_j(\tau),
\end{equation}
where $\psi_j(\sigma)$ are any real functions.
\end{itemize}
\end{thm}
\begin{proof}
This theorem can be proved in the same manner as Theorem 2.3 of
\cite{Tang18csr}.
\end{proof}

In general, we must truncate the series \eqref{Bt} and \eqref{Ats}
suitably for practical use, and the integrals of \eqref{crk} need to
be approximated by using a \emph{weighted interpolatory quadrature
formula} \eqref{wquad}. Thus, after applying the numerical
quadrature, it results in an $s$-stage conventional RK method
\begin{equation}\label{crk:quad}
\begin{split}
&\widehat{\vec{Z}}_i=\vec{z}_0+h\sum_{j=1}^sb_j
A_{c_i,\,c_j}\vec{f}(t_0+c_{j}h,
\widehat{\vec{Z}}_j),\quad i=1,\cdots,s,\\
&\vec{z}_{1}=\vec{z}_0+h\sum_{i=1}^sb_{i}B_{c_i}
\vec{f}(t_0+c_{i}h,\widehat{\vec{Z}}_i),
\end{split}
\end{equation}
where $\widehat{\vec{Z}}_i\approx\vec{Z}_{c_i}$.  The following
result is also an extension of the previous result by Tang et al
\cite{Tangs14cor,Tanglx16cos}.
\begin{thm}\label{qua:csRK}
Assume the underlying quadrature formula \eqref{wquad} is of
order\footnote{That is, \eqref{wquad} exactly holds for any
polynomial $\Phi(x)$ of degree less than $p$.} $p$, and
$A_{\tau,\,\sigma}$ is of degree $\pi_A^\tau$ with respect to $\tau$
and of degree $\pi_A^{\sigma}$ with respect to $\sigma$, and
$B_{\tau}$ is of degree $\pi_B^\tau$. If all the simplifying
assumptions $\breve{B}(\xi)$, $\breve{C}(\eta)$ and
$\breve{D}(\zeta)$  in \eqref{csRK_sim_assum} are fulfilled, then
the standard RK method \eqref{crk:quad} is at least of order
$$\min(\rho, 2\alpha+2, \alpha+\beta+1),$$
where $\rho=\min(\xi, p-\pi_B^{\tau})$, $\alpha=\min(\eta,
p-\pi_A^{\sigma})$ and $\beta=\min(\zeta,
p-\pi_A^{\tau}-\pi_B^{\tau})$.
\end{thm}
\begin{proof}
Please refer to Theorem 2.4 of \cite{Tang18csr}, the proof of which
can be adapted to the present case.
\end{proof}

\section{Geometric integration by csRK  methods}

In this section, we discuss the geometric integration by csRK
methods within the new framework of csRK methods. It is well-known
that symmetric methods and symplectic methods are of significant
importance in numerical integration of time-reversible systems and
Hamiltonian systems respectively, owing to that they have excellent
discrete dynamic behaviors in long-time numerical simulation
\cite{hairerlw06gni}. The definitions and relevant knowledge can be
found in many literatures (see, for example,
\cite{Feng95kfc,Fengqq10sga,hairernw93sod,hairerlw06gni,
Leimkuhlerr04shd,sanzc94nhp} and references therein).

\subsection{Symmetric methods}

\begin{defi}\cite{hairerlw06gni}
A one-step method $\phi_h$ is called symmetric if it satisfies
$$\phi^*_h=\phi_h,$$
where $\phi^*_h=\phi^{-1}_{-h}$ is referred to as the adjoint method
of $\phi_h$.
\end{defi}
By the definition, a method $z_1=\phi_h(z_0; t_0,t_1)$ is symmetric
if exchanging $h\leftrightarrow -h$, $z_0\leftrightarrow z_1$ and
$t_0\leftrightarrow t_1$ leaves the original method unaltered. For
the sake of deriving symmetric integrators, we now assume the
interval $I$ to be the following two cases:
\begin{itemize}
  \item[(i)]  $I=[a,b]$ (finite interval) with $a+b=1$;
  \item[(ii)] $I=(-\infty,+\infty)$ (infinite interval).
\end{itemize}

\begin{thm}\label{symmcon}
Under the assumption \eqref{consis} and we suppose $\breve{B}(\xi)$
holds with $\xi\geq1$ (which means the method is of order at least
1), then when $I=[a,b],\,a+b=1$ or $I=(-\infty,+\infty)$ the csRK
method \eqref{crk} is symmetric, if
\begin{equation}\label{symm}
A_{\tau,\,\sigma}w(\sigma)+A_{1-\tau,\,1-\sigma}w(1-\sigma)\equiv
B_{\sigma}w(\sigma),\;\;\tau,\,\sigma\in I.
\end{equation}
Particularly, if $w(\sigma)=w(1-\sigma)$, then the symmetric
condition \eqref{symm} becomes
\begin{equation}\label{symmred}
A_{\tau,\,\sigma}+A_{1-\tau,\,1-\sigma}\equiv
B_{\sigma},\;\;\tau,\,\sigma\in I.
\end{equation}
\end{thm}
\begin{proof}
For the proof, we have referred to the technique given in
\cite{Tang18ano}. Obviously, \eqref{symm} implies
$B_{\sigma}w(\sigma)\equiv B_{1-\sigma}w(1-\sigma)$ in $I$.
Furthermore, by taking an integral on both sides of \eqref{symm}
with respect to $\sigma$ (in use of \eqref{consis}), we get
$C_\tau+C_{1-\tau}\equiv1,\;\;\tau\in I$.

Next, let us establish the adjoint method of a given csRK method.
From \eqref{crk}, by interchanging $t_0, \vec{z}_0, h$ with $t_1,
\vec{z}_1, -h$ respectively, we have
\begin{equation*}
\begin{split}
&\vec{Z}_\tau=\vec{z}_1-h\int_IA_{\tau,\,\sigma}w(\sigma)\vec{f}(t_1-C_\sigma
h,\vec{Z}_\sigma)\,\dif
\sigma,\;\;\tau\in I,\\
&\vec{z}_{0}=\vec{z}_1-h\int_IB_\tau w(\tau)\vec{f}(t_1-C_\tau
h,\vec{Z}_\tau)\,\dif\tau,
\end{split}
\end{equation*}
Note that $t_1-C_\tau h=t_0+(1-C_\tau)h=t_0+C_{1-\tau}h$, thus the
second formula can be recast as
\begin{equation*}
\vec{z}_{1}=\vec{z}_0+h\int_I B_\tau w(\tau)\vec{f}(t_0+C_{1-\tau}
h,\vec{Z}_\tau)\,\dif\tau.
\end{equation*}
By plugging it into the first formula, it ends up with
\begin{equation*}
\begin{split}
&\vec{Z}_\tau=\vec{z}_0+h\int_I(B_\sigma-A_{\tau,\,\sigma})w(\sigma)\vec{f}(t_0+C_{1-\sigma}
h,\vec{Z}_\sigma)\,\dif
\sigma,\;\;\tau\in I,\\
&\vec{z}_{1}=\vec{z}_0+h\int_IB_\tau w(\tau)\vec{f}(t_0+C_{1-\tau}
h,\vec{Z}_\tau)\,\dif\tau,
\end{split}
\end{equation*}
By change of integral variables,  we obtain an equivalent
scheme\footnote{Note that according to the assumption about $I$ the
ranges of integration are invariant.}
\begin{equation*}
\begin{split}
&\vec{Z}^*_\tau=\vec{z}_0+h\int_I
(B_{1-\sigma}-A_{1-\tau,\,1-\sigma})w(1-\sigma)\vec{f}(t_0+C_\sigma
h,\vec{Z}^*_\sigma)\,\dif
\sigma,\;\;\tau\in I,\\
&\vec{z}_{1}=\vec{z}_0+h\int_I B_{1-\tau}w(1-\tau)\vec{f}(t_0+C_\tau
h,\vec{Z}^*_\tau)\,\dif\tau,
\end{split}
\end{equation*}
which is the adjoint method of the original method, where
$\vec{Z}^*_\tau=\vec{Z}_{1-\tau}$. Take into account that
\begin{equation*}
\begin{split}
&(B_{1-\sigma}-A_{1-\tau,\,1-\sigma})w(1-\sigma)
\equiv B_{\sigma}w(\sigma)-A_{1-\tau,\,1-\sigma}w(1-\sigma),\\
&B_{1-\tau}w(1-\tau)\equiv B_\tau w(\tau),
\end{split}
\end{equation*}
and let the coefficients of the adjoint method match with that of
the original method, i.e., imposing the condition \eqref{symm}, then
by definition the original method is symmetric. The formula
\eqref{symmred} is straightforward by removing
$w(\sigma)\,(=w(1-\sigma))$ from both sides of \eqref{symm}.
\end{proof}
\begin{thm}\label{symm_quad}
If the coefficients of the underlying csRK method \eqref{crk}
satisfy \eqref{symmred}, then the RK scheme \eqref{crk:quad} derived
by numerical quadrature is symmetric, provided that the quadrature
weights and abscissae  (see \eqref{GCqua}) satisfy $b_{s+1-i}=b_i$
and $c_{s+1-i}=1-c_i$ for all $i$.
\end{thm}
\begin{proof}
Please refer to the proof of Theorem 3.5 in \cite{Tang18ano}.
\end{proof}

\begin{thm}\label{symmcon3}
Suppose that $w(\sigma)\equiv w(1-\sigma)$ and
$A_{\tau,\sigma}/B_\sigma\in L_w^2(I\times I)$, then symmetric
condition \eqref{symm} is equivalent to the fact that
$A_{\tau,\sigma}$ has the following form in terms of the orthogonal
polynomials $P_n(x)$ in $L_w^2(I)$
\begin{equation}\label{symmconvari}
A_{\tau,\sigma}=B_\sigma\Big(\frac{1}{2}+\sum_{i+j\,\text{is}\,\text{odd}\atop
0\leq i,j\in \mathbb{Z}}\alpha_{(i,j)}
P_i(\tau)P_j(\sigma)\Big),\quad\alpha_{(i,j)}\in \mathbb{R},
\end{equation}
with $B_{\sigma}\equiv B_{1-\sigma}$, where $\alpha_{(i,j)}$ can be
any real parameters for odd $i+j$, provided that the orthogonal
polynomials $P_n(x)$ satisfy
\begin{equation}\label{symm_relation}
P_n(1-x)=(-1)^nP_n(x),\;n\in \mathbb{Z}.
\end{equation}
\end{thm}
\begin{proof}
Please refer to the proof of Theorem 4.6 of \cite{Tang18csr}.
\end{proof}

The following results can be easily verified by using a change of
variables.
\begin{thm}\label{symm_wf}
If $w(x)$ is an even function, i.e., satisfying $w(-x)\equiv w(x)$,
then the shifted function defined by $\widehat{w}(x)=w(2\mu x-\mu)$
satisfies the symmetry relation: $\widehat{w}(x)\equiv
\widehat{w}(1-x)$. Here $\mu$ is a non-zero constant and usually we
take it as $\mu=1$.
\end{thm}
\begin{thm}\label{symm_pol}
If a sequence of polynomials $\{P_n(x)\}_{n=0}^\infty$ satisfy the
symmetry relation
\begin{equation}\label{symm_relat}
P_n(-x)=(-1)^nP_n(x),\;n\in \mathbb{Z},
\end{equation}
then the shifted polynomials defined by $\widehat{P}_n(x)=P_n(2\mu
x-\mu)$  are bound to satisfy the property \eqref{symm_relation}.
Here $\mu$ is a non-zero constant and usually we take it as $\mu=1$.
\end{thm}
We find that many classical (standard) orthogonal polynomials
including Hermite polynomials, Legendre polynomials, Chebyshev
polynomials of the first and second kind, and any other general
Gegenbauer polynomials etc., do not satisfy \eqref{symm_relation},
but they possess the symmetry relation \eqref{symm_relat}. Theorem
\ref{symm_pol} helps us to find suitable orthogonal polynomials to
satisfy \eqref{symm_relation}.

\begin{thm}\label{symm_Bs}
If a sequence of polynomials $\{P_n(x)\}_{n=0}^\infty$ satisfy
\eqref{symm_relation}, then we have
\begin{equation*}
\int_0^1P_j(x)\,\dif
x=0,\;\;\text{for}\;\;j\;\;\text{is}\;\;\text{odd},
\end{equation*}
and the following function
\begin{equation}
B_\sigma=\sum\limits_{j=0}^{\xi-1}\int_0^1P_j(x)\,\dif x
P_j(\sigma)+\sum\limits_{\text{even}\;j\geq \xi}\lambda_j
P_j(\sigma),\;\; \xi\geq1,
\end{equation}
always satisfies $B_{\sigma}\equiv B_{1-\sigma}$.
\end{thm}

With the help of these results, we can conveniently construct
symmetric methods by Theorem \ref{symmcon3}.

\subsection{Symplectic methods}

We can directly apply some available results of
\cite{Tang18ano,Tang18csr} to csRK method \eqref{crk}, it then gives
the following results.

\begin{thm}\label{symcond1}
If the coefficients of a csRK method \eqref{crk} satisfy
\begin{equation}\label{symcond}
B_\tau A_{\tau,\sigma}+B_\sigma A_{\sigma,\tau}\equiv B_\tau
B_\sigma,\;\; \tau,\,\sigma \in I,
\end{equation}
then it is symplectic.
\end{thm}
\begin{proof}
This theorem can be proved in a similar manner as Theorem 3.1 of
\cite{Tang18ano}. In fact, after conducting the same arguments as
shown in \cite{Tang18ano}, we find that if
\begin{equation}\label{symproof}
\Big(B_\tau
w(\tau)\Big)\Big(A_{\tau,\sigma}w(\sigma)\Big)+\Big(B_\sigma
w(\sigma)\Big) \Big(A_{\sigma,\tau}w(\tau)\Big)\equiv \Big(B_\tau
w(\tau)\Big)\Big(B_\sigma w(\sigma)\Big),
\end{equation}
then the method \eqref{crk} is symplectic. By removing the factor
``$w(\tau)w(\sigma)$" from both sides of \eqref{symproof}, it gives
\eqref{symcond}. And this in turn implies that if \eqref{symcond} is
fulfilled, then it gives \eqref{symproof} which verifies the
symplecticity of the method.
\end{proof}
\begin{rem}
Theorem \ref{symcond1} implies that the symplecticity of the csRK
methods \eqref{crk} is independent of its weight function.
\end{rem}
\begin{thm}\label{symRKquad}
The RK scheme \eqref{crk:quad} based on the underlying csRK method
with coefficients satisfying \eqref{symcond} is always symplectic.
\end{thm}
\begin{proof}
Please see \cite{Tang18ano} for getting a similar proof.
\end{proof}
\begin{thm}\label{sym_design}
Under the assumption \eqref{C_assum}, for a symplectic csRK method
with coefficients satisfying \eqref{symcond}, we have the following
statements:
\begin{itemize}
\item[\emph{(a)}] $\breve{B}(\xi)$ and $\breve{C}(\eta)$
$\Longrightarrow$ $\breve{D}(\zeta)$, where
$\zeta=\min\{\xi,\,\eta\}$;
\item[\emph{(b)}] $\breve{B}(\xi)$ and $\breve{D}(\zeta)$
$\Longrightarrow$ $\breve{C}(\eta)$, where
$\eta=\min\{\xi,\,\zeta\}$.
\end{itemize}
\end{thm}
\begin{proof}
Please refer to the proof of Theorem 4.1 in \cite{Tang18csr}.
\end{proof}
\begin{thm}\label{sympcond3}
Suppose that $A_{\tau,\sigma}/B_\sigma\in L_w^2(I\times I)$, then
symplectic condition \eqref{symcond} is equivalent to the fact that
$A_{\tau,\sigma}$ has the following form in terms of the normalized
orthogonal polynomials $P_n(x)$ in $L_w^2(I)$
\begin{equation}\label{sympconvari}
A_{\tau,\sigma}=B_\sigma\Big(\frac{1}{2}+\sum_{0<i+j\in
\mathbb{Z}}\alpha_{(i,j)}
P_i(\tau)P_j(\sigma)\Big),\quad\alpha_{(i,j)}\in \mathbb{R},
\end{equation}
where $\alpha_{(i,j)}$ is skew-symmetric, i.e.,
$\alpha_{(i,j)}=-\alpha_{(j,i)},\,i+j>0$.
\end{thm}
\begin{proof}
Please refer to the proof of Theorem 4.3 in \cite{Tang18csr}.
\end{proof}

On the basis of these preliminaries, we could introduce an
operational \emph{procedure} for establishing symplectic RK-type
methods --- actually the similar technique has been developed in
\cite{Tang18csr,Tang18csm,Tang18siw}. The \emph{procedure} is as
follows\footnote{For more details, we refer the readers to \cite{Tang18siw}.}:\\
\textbf{Step 1.} Make an ansatz for $B_\tau$ which satisfies
$\breve{B}(\xi)$ with $\xi\geq1$ according to \eqref{Bt}, and a
finite number of $\lambda_\iota$ could be kept as parameters;\\
\textbf{Step 2.} Suppose $A_{\tau,\,\sigma}$ is in the form (by
Theorem \ref{sympcond3})
\begin{equation}\label{Ats_proc}
A_{\tau,\sigma}=B_\sigma\Big(\frac{1}{2}+\sum_{0<i+j\in
\mathbb{Z}}\alpha_{(i,j)}
P_i(\tau)P_j(\sigma)\Big),\quad\alpha_{(i,j)}=-\alpha_{(j,i)},
\end{equation}
where $\alpha_{(i,j)}$ are kept as parameters with a finite number,
and then substitute $A_{\tau,\,\sigma}$ into\footnote{An alternative
technique is to consider using $\breve{D}(\zeta)$.}
$\breve{C}(\eta)$ (see \eqref{eq:cd2}, usually we let $\eta<\xi$)
for determining $\alpha_{(i,j)}$:
\begin{equation*}
\int_IA_{\tau,\,\sigma}w(\sigma) \phi_k(\sigma)\,\dif
\sigma=\int_0^\tau \phi_k(x)\,\dif x,\;\;k=0,1,\cdots,\eta-1,
\end{equation*}
Here, $\phi_k(x)$ stands for any polynomial of degree $k$,  which
performs very similarly as the
``test function" used in general finite element analysis; \\
\textbf{Step 3.} Write down $B_\tau w(\tau)$ and $A_{\tau,\,\sigma}
w(\sigma)$ (satisfy $\breve{B}(\xi)$ and $\breve{C}(\eta)$
automatically), which results in a symplectic csRK method of order
at least $\min\{\xi,\,2\eta+2,\,\eta+\zeta+1\}$ with
$\zeta=\min\{\xi,\,\eta\}$ by Theorem \ref{crk:order} and
\ref{sym_design}. If needed, we then get symplectic RK methods by
using quadrature rules (see Theorem \ref{symRKquad}).

Remark that the procedure above provides a general approach for
establishing symplectic integrators. In view of Theorem
\ref{qua:csRK} and \ref{sym_design}, it is suggested to design RK
coefficients with low-degree $A_{\tau,\,\sigma}$ and $B_\tau$. For
the sake of easily calculating those integrals of $\breve{C}(\eta)$
in the second step, the following ansatz may be advisable (with
$C_\tau$ given by \eqref{C_assum} and let $\rho\geq\eta$ and
$\xi\geq2\eta$)
\begin{equation}\label{symBA}
B_\tau=\sum\limits_{j=0}^{\xi-1}\int_0^1P_j(x)\,\dif x
P_j(\tau),\qquad
A_{\tau,\sigma}=B_\sigma\Big(\frac{1}{2}+\sum_{0<i+j\in
\mathbb{Z}\atop i\leq\rho,\,j\leq \xi-\eta}\alpha_{(i,j)}
P_i(\tau)P_j(\sigma)\Big),
\end{equation}
where $\alpha_{(i,j)}=-\alpha_{(j,i)}$. Because of the index $j$
restricted by $j\leq \xi-\eta$ in the second formula of
\eqref{symBA}, we can use $\breve{B}(\xi)$ to arrive at (please c.f.
\eqref{eq:cd1})
\begin{equation*}
\begin{split}
&\int_IA_{\tau,\,\sigma}w(\sigma) \phi_k(\sigma)\,\dif
\sigma=\int_IB_\sigma\Big(\frac{1}{2}+\sum_{0<i+j\in \mathbb{Z}\atop
i\leq\rho,\,j\leq \xi-\eta}\alpha_{(i,j)}
P_i(\tau)P_j(\sigma)\Big)w(\sigma)\phi_k(\sigma)\,\dif \sigma\\
&=\frac{1}{2}\int_0^1 \phi_k(x)\,\dif x+\sum_{0<i+j\in
\mathbb{Z}\atop i\leq\rho,\,j\leq \xi-\eta}\alpha_{(i,j)}
P_i(\tau)\int_0^1P_j(\sigma)\phi_k(\sigma)\,\dif \sigma,\;\;0\leq
k\leq\eta-1.
\end{split}
\end{equation*}
Therefore, $\breve{C}(\eta)$ implies that
\begin{equation}\label{symp_eqs}
\frac{1}{2}\int_0^1 \phi_k(x)\,\dif x+\sum_{0<i+j\in \mathbb{Z}\atop
i\leq\rho,\,j\leq \xi-\eta}\alpha_{(i,j)}
P_i(\tau)\int_0^1P_j(\sigma)\phi_k(\sigma)\,\dif \sigma=\int_0^\tau
\phi_k(x)\,\dif x,\;\;0\leq k\leq\eta-1.
\end{equation}
Finally, it needs to settle $\alpha_{(i,j)}$ by transposing,
comparing or merging similar items of \eqref{symp_eqs} after the
polynomial on right-hand side being represented by the basis
$\{P_j(x)\}_{j=0}^\infty$. In view of the skew-symmetry of
$\alpha_{(i,j)}$, if we let $r=\min\{\rho,\xi-\eta\}$, then actually
the degrees of freedom of these parameters is $r(r+1)/2$, by
noticing that
\begin{equation*}
\alpha_{(i,i)}=0,\;i\geq1\;\;\text{and}\;\;\alpha_{(i,j)}=0,
\;\;\text{for}\;i>r\;\text{or}\;j>r.
\end{equation*}

\subsection{Some examples}

In this part, we give some examples for illustrating the application
of our theoretical results given in the preceding section. In view
of the skew-symmetry of $\alpha_{(i,j)}$, we only provide the values
of $\alpha_{(i,j)}$ with $i<j$, and the Gauss-Christoffel's
quadrature rule with optimal order (see Theorem \ref{GCqua} and
Remark \ref{GassChriRem}) will be used in these examples.

\begin{exa}\label{Legend}
Consider using the normalized Legendre polynomials which are
orthogonal with respect to the weight function $w(x)=1$ on $[-1,1]$.
These Legendre polynomials $P_n(x)$ can be defined by Rodrigues'
formula
\begin{equation*}
P_0(x)=\frac{1}{\sqrt{2}},\;P_n(x)=\frac{\sqrt{2n+1}}{2^{n+\frac{1}{2}}n!}
\frac{\dif^n}{dx^n}\big(x^2-1\big)^n,\; \;n=1,2,\cdots.
\end{equation*}
\end{exa}
Let $\xi=3,\,\eta=1,\,\rho=2$, after some elementary calculations,
it gives
\begin{equation*}
\alpha_{(0,1)}=-\frac{2\sqrt{3}}{3},\;\;\alpha_{(1,2)}
=-\frac{2\sqrt{3}}{3}\alpha_{(0,2)}.
\end{equation*}
Let $\mu=\alpha_{(0,2)}$ be a free parameter, then we get a family
of $\mu$-parameter symplectic csRK methods of order $3$. By using
Gauss-Christoffel's quadrature rules with $2$ nodes and $3$ nodes
respectively, we get symplectic RK methods of order $3$ which are
shown in Tab. \ref{exa1:symp01} and \ref{exa1:symp02}.

\begin{rem}
If we shift the Legendre polynomials from $[-1,1]$ to the standard
interval $[0,1]$, then we can use Theorem \ref{symmcon3} and
\ref{sympcond3} to derive many classical high-order symmetric and
symplectic RK methods (e.g., Gauss-Legendre RK methods). The
construction of symplectic RK methods with shifted Legendre
polynomials has been investigated in the literature
\cite{tangs12ana,Tangs14cor,Tanglx16cos,Tang18ano}.
\end{rem}
\begin{table}
\[\ba{c|cc}
-\frac{\sqrt{3}}{3} &\frac{2-\sqrt{3}}{8} &  \frac{-6-5\sqrt{3}}{24}\\[2pt]
\frac{\sqrt{3}}{3} & \frac{-6+5\sqrt{3}}{24} & \frac{2+\sqrt{3}}{8}\\[2pt]
\hline &\frac{2-\sqrt{3}}{4} & \frac{2+\sqrt{3}}{4} \ea\] \caption{A
$2$-stage $3$-order symplectic RK method, based on Legendre
polynomials defined on $[-1,1]$.}\label{exa1:symp01}
\end{table}
\begin{table}
\small{\[\ba{c|ccc} -\frac{\sqrt{15}}{5} & \frac{10-3\sqrt{15}}{72}
& \frac{10-4\sqrt{15}-(10\sqrt{3}+9\sqrt{5})\mu}{45} &
-\frac{26+5\sqrt{15}}{72}
+\frac{(10\sqrt{3}+9\sqrt{5})\mu}{45}\\[2pt]
0 & \frac{-16+2\sqrt{15}-7\sqrt{3}\mu}{144}& \frac{2}{9}&
\frac{-16-2\sqrt{15}+7\sqrt{3}\mu}{144}\\[2pt]
\frac{\sqrt{15}}{5}& \frac{-26+5\sqrt{15}}{72}
+\frac{(-10\sqrt{3}+9\sqrt{5})\mu}{45} & \frac{10+4\sqrt{15}+
(10\sqrt{3}-9\sqrt{5})\mu}{45}&\frac{10+3\sqrt{15}}{72}\\[2pt]
\hline & \frac{10-3\sqrt{15}}{36} &
\frac{4}{9}&\frac{10+3\sqrt{15}}{36}\ea\]} \caption{A family of
$3$-stage $3$-order symplectic RK methods, based on Legendre
polynomials defined on $[-1,1]$.}\label{exa1:symp02}
\end{table}

\begin{exa}\label{Lague}
Consider using the normalized Laguerre polynomials which are
orthogonal with respect to the weight function $w(x)=e^{-x}$ on
$[0,+\infty)$. These Laguerre polynomials $L_n(x)$ can be defined by
Rodrigues' formula
\begin{equation*}
L_0(x)=1,\;L_n(x)=\frac{e^x}{n!}\frac{\dif^n}{dx^n}\big(x^ne^{-x}\big),
\;\;n=1,2,\cdots.
\end{equation*}
\end{exa}
Let $\xi=3,\,\eta=1,\,\rho=2$, after some elementary calculations,
it gives
\begin{equation*}
\alpha_{(0,1)}=1-\frac{1}{3}\alpha_{(0,2)},\;\;\alpha_{(1,2)}
=-2\alpha_{(0,2)}.
\end{equation*}
Let $\mu=\alpha_{(0,2)}$ be a free parameter, then we get a family
of $\mu$-parameter symplectic csRK methods of order $3$. A family of
symplectic RK methods of order $2$ are shown in Tab.
\ref{exa2:symp01}. If we use $3$-point Gauss-Christoffel's
quadrature rule, then we get the family of $3$-stage $3$-order
symplectic RK methods (we do not show it here since the exact zeros
of $L_3(x)$ is too complicated to be exhibited.).

\begin{table}
\[\ba{c|cc}
2-\sqrt{2} &\frac{4+3\sqrt{2}}{16} &  \frac{28-19\sqrt{2}}{16}+\frac{-3+2\sqrt{2}}{6}\mu\\[2pt]
2+\sqrt{2} & \frac{28+19\sqrt{2}}{16}-\frac{3+2\sqrt{2}}{6}\mu & \frac{4-3\sqrt{2}}{16}\\[2pt]
\hline &\frac{4+3\sqrt{2}}{8} & \frac{4-3\sqrt{2}}{8} \ea\]
\caption{A family of $2$-stage $2$-order symplectic RK methods,
based on Laguerre polynomials defined on
$[0,+\infty)$.}\label{exa2:symp01}
\end{table}
\begin{exa}\label{Hermi}
Consider using the normalized Hermite polynomials which are
orthogonal with respect to the weight function $w(x)=e^{-x^2}$ on
$(-\infty,+\infty)$. These Hermite polynomials $H_n(x)$ can be
defined by Rodrigues' formula
\begin{equation*}
H_0(x)=\frac{1}{\pi^{\frac{1}{4}}},\;H_n(x)=\frac{(-1)^ne^{x^2}}{\sqrt{2^nn!}\pi^{\frac{1}{4}}}
\frac{\dif^n}{dx^n}\big(e^{-x^2}\big),\; \;n=1,2,\cdots.
\end{equation*}
\end{exa}
Let $\xi=3,\,\eta=1,\,\rho=2$, after some elementary calculations,
it gives
\begin{equation*}
\alpha_{(0,1)}=\frac{1}{3}\alpha_{(0,2)}-\frac{\sqrt{2\pi}}{2},\;\;\alpha_{(1,2)}
=-\sqrt{2}\alpha_{(0,2)}.
\end{equation*}
Let $\mu=\alpha_{(0,2)}$ be a free parameter, then we get a family
of $\mu$-parameter symplectic csRK methods of order $3$. Tab.
\ref{exa3:symp01} gives a family of symplectic RK methods by using
$3$-point Gauss-Christoffel's quadrature.

Next, let us consider using the shifted Hermite polynomials defined
by
\begin{equation*}
\widehat{H}_n(x)=\sqrt{2}H_n(2x-1),\;\;x\in(-\infty,+\infty),\;\;n=0,1,\cdots,
\end{equation*}
where the factor $\sqrt{2}$ is for the sake of guaranteeing the
normalization of the new orthogonal polynomials on
$(-\infty,+\infty)$.  We still take $\xi=3,\,\eta=1,\,\rho=2$ but
impose $\alpha_{(0,2)}=-\alpha_{(2,0)}=0$ (by Theorem
\ref{symmcon3}), then it enables us to get symmetric and symplectic
methods. Actually, after some elementary calculations, it yields
\begin{equation*}
\alpha_{(1,2)}=-3\sqrt{2}\alpha_{(0,1)}-\frac{3\sqrt{\pi}}{4},\;\;\alpha_{(0,2)}=0.
\end{equation*}
Let $\mu=\alpha_{(0,1)}$ be a free parameter, then we get a family
of $\mu$-parameter symmetric and symplectic csRK methods of order
$4$.  We show the family of $3$-stage $4$-order symmetric and
symplectic RK methods in Tab. \ref{exa3:symp02}.

\begin{table}
\small{\[\ba{c|ccc} -\frac{\sqrt{6}}{2} & \frac{4-3\sqrt{6}}{72} &
\frac{7-7\sqrt{6}}{18}-\frac{28\sqrt{3}+63\sqrt{2}}{54\sqrt{\pi}}\mu
& -\frac{32+5\sqrt{6}}{72}+\frac{28\sqrt{3}+63\sqrt{2}}{54\sqrt{\pi}}\mu\\[2pt]
0 & \frac{-14+\sqrt{6}}{72}-\frac{19\sqrt{3}}{108\sqrt{\pi}}\mu&
\frac{7}{18}&-\frac{14+\sqrt{6}}{72}+\frac{19\sqrt{3}}{108\sqrt{\pi}}\mu\\[2pt]
\frac{\sqrt{6}}{2}&-\frac{32-5\sqrt{6}}{72}+\frac{-28\sqrt{3}+63\sqrt{2}}{54\sqrt{\pi}}\mu&
\frac{7+7\sqrt{6}}{18}+\frac{28\sqrt{3}-63\sqrt{2}}{54\sqrt{\pi}}\mu & \frac{4+3\sqrt{6}}{72}\\[2pt]
\hline & \frac{4-3\sqrt{6}}{36} & \frac{7}{9} &
\frac{4+3\sqrt{6}}{36}\ea\]} \caption{A family of $3$-stage
$3$-order symplectic RK methods, based on Hermite polynomials
defined on $(-\infty,+\infty)$.}\label{exa3:symp01}
\end{table}
\begin{table}
\small{\[\ba{c|ccc} \frac{2-\sqrt{6}}{4} & \frac{1}{18} &
\frac{14-21\sqrt{6}-112\sqrt{3/\pi}\mu}{36} &
\frac{2+12\sqrt{6}+112\sqrt{3/\pi}\mu}{36}\\[2pt]
\frac{1}{2} & \frac{2+3\sqrt{6}+16\sqrt{3/\pi}\mu}{36} &
\frac{7}{18} & \frac{2-3\sqrt{6}-16\sqrt{3/\pi}\mu}{36}\\[2pt]
\frac{2+\sqrt{6}}{4} & \frac{2-12\sqrt{6}-112\sqrt{3/\pi}\mu}{36} &
\frac{14+21\sqrt{6}+112\sqrt{3/\pi}\mu}{36} & \frac{1}{18}\\[2pt]
\hline & \frac{1}{9} & \frac{7}{9} & \frac{1}{9}\ea\]} \caption{A
family of $3$-stage $4$-order symmetric and symplectic RK methods,
based on Hermite polynomials defined on
$(-\infty,+\infty)$.}\label{exa3:symp02}
\end{table}

\begin{rem}
From these examples, it is seen that the same type of orthogonal
polynomials can be defined on different intervals (by shifting
variables), and when we use their different versions, it may result
in different symplectic schemes with different order accuracy.
\end{rem}

\section{Numerical experiments}

For the sake of verifying the efficiency of our new integrators, we
perform some numerical experiments with symplectic integrators of
order 2, 3 and 4 respectively (see Tab. \ref{exa1:symp01},
\ref{exa2:symp01} and \ref{exa3:symp02}). We consider the well-known
Kepler's problem determined by the Hamiltonian function
\cite{hairerlw06gni,Calvofmr09sos}
\begin{equation*}
H(p, q)=\frac{p_1^2+p_2^2}{2}-\frac{1}{\sqrt{q_1^2+q_2^2}}
\end{equation*}
with initial value conditions
$(p_1(0),\,p_2(0),\,q_1(0),\,q_2(0))=(0,1,1,0)$. The exact solution
is
\begin{equation*}
p_1(t)=-\textrm{sin}(t),\;\;p_2(t)=\textrm{cos}(t),\;\;q_1(t)=\textrm{cos}(t),\;\;
q_2(t)=\textrm{sin}(t).
\end{equation*}
The numerical results are shown in Fig.
\ref{fig-2order01}-\ref{fig-4order02}. From the plots of energy
error and solution error, we observe that all the symplectic methods
nearly preserve the energy with a bounded error and they basically
possess a linear growth of solution error. Besides, the exact
trajectory of the Kepler's problem is well simulated by our
numerical methods. The symplecticity of our methods is therefore
verified by these numerical behaviors.

\begin{figure}
\begin{center}
\scalebox{0.8}[0.60]{\includegraphics{Laguerre2HSE.pdf}}
\caption{Energy error and solution error by $2$-order Laguerre
symplectic method ($\mu=0$) given in Tab. \ref{exa2:symp01}, step
size $h=0.1$.}\label{fig-2order01}
\end{center}
\begin{center}
\scalebox{0.8}[0.60]{\includegraphics{Laguerre2Trajectory}}
\caption{Trajectories of exact and numerical solution, using
$2$-order Laguerre symplectic method ($\mu=0$) given in Tab.
\ref{exa2:symp01}, step size $h=0.1$.}\label{fig-2order02}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\scalebox{0.8}[0.60]{\includegraphics{Legendre3HSE.pdf}}
\caption{Energy error and solution error by $3$-order Legendre
symplectic method given in Tab. \ref{exa1:symp01}, step size
$h=0.1$.}\label{fig-3order01}
\end{center}
\begin{center}
\scalebox{0.8}[0.60]{\includegraphics{Legendre3Trajectory}}
\caption{Trajectories of exact and numerical solution, using
$3$-order Legendre symplectic method given in Tab.
\ref{exa1:symp01}, step size $h=0.1$.}\label{fig-3order02}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\scalebox{0.8}[0.60]{\includegraphics{Hermite4HSE.pdf}}
\caption{Energy error and solution error by $4$-order Hermite
symplectic method ($\mu=\sqrt{2\pi}/14$) given in Tab.
\ref{exa3:symp02}, step size $h=0.1$.}\label{fig-4order01}
\end{center}
\begin{center}
\scalebox{0.8}[0.60]{\includegraphics{Hermite4Trajectory}}
\caption{Trajectories of exact and numerical solution, using
$4$-order Hermite symplectic method ($\mu=\sqrt{2\pi}/14$) given in
Tab. \ref{exa3:symp02}, step size $h=0.1$.}\label{fig-4order02}
\end{center}
\end{figure}

\section{Conclusions}

This paper shows how to extend the primitive framework of continuous
stage Runge-Kutta methods to a new larger one such that it enables
us to treat more complicated cases. By using the new theoretical
framework, we can not only obtain more general-purpose RK-type
methods for practical use, but also it allows us to discover new
geometric integrators for special purpose e.g., symplectic-structure
preserving integrators for Hamiltonian systems. There is a very
interesting fact in the construction of symplectic methods, that is,
by using the same type of orthogonal polynomials we can get
different symplectic RK schemes if the polynomials are shifted to
different intervals. It is shown that the orthogonal range $I$ of
the polynomials could heavily impact the formalism of the resulting
methods.

Although we only present three examples for deriving symmetric and
symplectic integrators with order up to 4, essentially the same
technique can be directly used to get more higher order methods.
Moreover, other types of orthogonal polynomials can also be
considered for use, regardless of weighting on a finite or infinite
interval.

At last but not the least, we emphasize that our new symplectic
methods may not be necessarily implemented via RK schemes. Recognize
that for all our practical csRK methods  (after truncating the
series \eqref{Bt} and \eqref{Ats}), the Butcher coefficient
$A_{\tau,\sigma}w(\sigma)$ appeared in \eqref{crk} is a polynomial
with respect to $\tau$ (but not to $\sigma$ in general, since
$w(\sigma)$ may be not a polynomial), this implies that the stage
value $\vec{Z}_\tau$ of the csRK methods is also a (vector-valued)
polynomial. This fact suggests that we can use a polynomial
expansion of $\vec{Z}_\tau$ in terms of unknown coefficients
$\vec{\gamma}_j$, say, let
\begin{equation}\label{Z_expan}
\vec{Z}_\tau=\sum_{j=1}^{k+1}\vec{\gamma}_j \phi_j(\tau),
\end{equation}
where $\vec{Z}_\tau$ is assumed to be of degree $k$ and
$\{\phi_j(\tau)\}$ is a suitable basis in the polynomial function
space of degree $k$ at most. Therefore, by substituting the
expansion \eqref{Z_expan} into \eqref{crk} it results in an
algebraic system in terms of $\vec{\gamma}_j$ which can be solved by
iteration. Compared with the classical RK scheme \eqref{crk:quad},
such a technique may bring us more cost savings in computation,
especially when we consider using a high-order quadrature formula
with many nodes for approximating the integrals of \eqref{crk}. In a
word, conventional RK scheme (see \eqref{crk:quad}) may be
implemented in other ways and we can understand the same numerical
method from a different perspective.

\section*{Acknowledgments}

This work was supported by the National Natural Science Foundation
of China (11401055), China Scholarship Council and Scientific
Research Fund of Hunan Provincial Education Department (15C0028).


TITLE: Prove that the conective $\beta(a,b,c,d)=(a\implies b) \implies (c\implies d)$ is not adequate.
QUESTION [1 upvotes]: A set C of conectives is called adequate if you every formula written with the conectives $\{\text{implies, and, not, or}\}$ has an equivalent formula written only with conectives from $C$.
Prove that the conective $\beta(a,b,c,d)=(a\implies b) \implies (c\implies d)$ is / is not not adequate.
I believe this conective is not adequate: I've come to think you can't form contradictions using it alone. Also, I could form the $\implies$ conective using $\beta$, thus $\{\beta, \implies\} \equiv\{\beta\}$ in terms of "formula making power".
E: I should add I haven't yet studied boolean algebras, so I really don't understand many terms.
Is it possible to make a (perhaps sillier) argument by contradiction, assuming there exists a contradiction expressible only with $\beta$ and reaching and absurdity?

REPLY [2 votes]: Post's characterisation of functional completeness gives necessary and sufficient conditions for the adequacy of a set of propositional connectives. $\{\beta\}$ fails Post's test because $\beta(\top, \top, \top, \top) = \top$.


TITLE: Roots in Banach algebras.
QUESTION [1 upvotes]: I'm studying positive functionals on normed algebras and I got stuck in the following problem:

Let $A$ be a unital Banach algebra, and $x\in A$ be such that $\Vert x\Vert <1$. Then the series
  $$\sum_{k=0}^\infty\dfrac{1}{k!}\left(\dfrac{1}{2}-0\right)\cdots\left(\dfrac{1}{2}-k+1\right)x^k$$
  converges (absolutely) to and element $y\in A$ such that $y^2=1_A+x$.

The series above is just the Taylor series of $\sqrt{1+x}$ centered at $0$. That the series converges absolutely is simple (one can use, for example, the root test). The problem is to show that $y^2=1+x$.
Let $a_k=\dfrac{1}{k!}\left(\dfrac{1}{2}-0\right)\cdots\left(\dfrac{1}{2}-k+1\right)$. It can be shown by induction on $k$ that
$a_k=\dfrac{(-1)^k(2k)!}{(1-2k)(k!)^2(4^k)}$ (Wikipedia)
Since the series $y=\sum_k a_kx^k$ converges absolutely, then
$$y^2=\sum_{n=0}^\infty\sum_{k=0}^n a_k a_{n-k}x^n$$
If we calculate the initial terms, we get $y=1+x+0x^2+0x^3+0x^4+0x^5+\cdots$, but I'm having trouble showing that $\sum_{k=0}^na_ka_{n-k}=0$ for every $n\geq 2$. I've tried induction, but it didn't work for me.

REPLY [3 votes]: Since $\Vert x\Vert<1$ then its spectrum $\sigma(x)\subset r\mathbb{D}$ for some $r<1$, so we have well defined holomorphic calculus of element $x$
$$
\gamma_x:\mathcal{O}(r\mathbb{D})\to A, f\mapsto \gamma_x(f):=f(x)
$$
which in fact is unital involutive $*$-homomorphism. Note that $h(z)=\sqrt{1+z}\in\mathcal{O}(r\mathbb{D})$ and we get some $y=h(x)\in A$. Finally
$$
y^2=h(x)h(x)=\gamma_x(h)\gamma_x(h)=\gamma_x(h\cdot h)=\gamma_x(1+z)=1+x
$$