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\begin{abstract}
This paper analyzes a new regularized learning scheme for high dimensional partially linear support vector machine.
The proposed  approach consists of an empirical risk and the Lasso-type penalty for linear part, as well as the standard functional norm for nonlinear part. Here the linear kernel is used for model interpretation and feature selection, while the nonlinear kernel is adopted to enhance algorithmic flexibility. In this paper,
we develop a new technical analysis on the weighted empirical process, and establish the sharp learning rates for  the  semi-parametric estimator under the regularized conditions. Specially, our derived learning rates for semi-parametric SVM depend on not only the
sample size and the functional complexity, but also the sparsity and the margin parameters.

\end{abstract}

{\bf Key Words and Phrases:} partially linear models, high dimension, support vector machine, weighted
empirical process.

\section{Introduction}
Support vector machine (SVM), originally introduced by \cite{Vapnik1995}, is well known to be a popular and powerful technique mainly due to its successful practical performances and nice theoretical foundations in machine learning.
For supervised classification problems, SVM is based on the  margin-maximization principle endowed with a specified kernel, which is formulated by a nonlinear map from the input space to the feature space.

Denote $ \mathcal{X}$ and $\mathcal{Y}=\{-1,+1\}$ as the input space and
corresponding output space, respectively. Let $(X,Y)\in \mathcal{X}\times \mathcal{Y}$ be
a random vector drawn from an unknown joint distribution $\rho$ on $ \mathcal{X}\times \mathcal{Y}$.
Suppose that all the observations $\{(Y_i,X_i)\}_{i=1}^{n}$ are available from
$\rho$.
 In empirical risk minimization, the standard $L_2$-norm SVM has the widely-used hinge loss
plus $L_2$-norm penalty formulation. Recall that
  the empirical hinge loss function is defined by
 \begin{align*}
 \mathcal{R}_n(f)=\frac{1}{n}\sum_{i=1}^{n}  \phi_h(Y_if(X_i)),
 \end{align*}
where  the hinge loss is  $\phi_h(u)=(1-u )_+$,  with $u_+$ denoting the positive part of $u\in \mathbb{R}$.
The standard SVM can be expressed as the following regularization problem
\begin{equation*}
\min_{f\in\mathcal{H}_K}\big\{\mathcal{R}_n(f)+
\lambda \|f\|_{K}^2\big\},
\end{equation*}
where $\lambda $ is the regularized parameter for controlling the  functional complexity of $\mathcal{H}_K$. Note that
$\mathcal{H}_K$ is referred to a reproducing kernel Hilbert space (RKHS), often specified in advance. See Section 2 for more details on RKHS. The book by \citet*{Steinwart2008} contains a good overview of SVMs and the  particularly related
learning theory.

Among various kernel-based learning schemes including SVM, it is full of challenges how to select a suitable kernel  and there are not any perfect answers for such problem until now. See related work on kernel learning \citep*{Lancjriet2004, Micchelli2005, Wu2007, Kloft2011, Ying2009} for instances. In this paper, we consider a semi-parametric SVM problem of the linear kernel plus a general nonlinear kernel. Indeed, partial linear models in statistics have received a great attention in the last several decades, see \citep*{Muller2015, Hardle2007, Speckman1988}. Particularly, the linear part in the partial linear modelsaims at the model interpretation, and the nonlinear part is used to enhance the model flexibility. As a concrete example in stock market,   the future return of a stock $Y$ may depend on several company management indexes (e.g. shareholders structure) which are homogeneous for all the companies, and we allow linear relation with $Y$. However, the other features (e.g., from financial statements) should be nonlinear to the response, in that a company has a complex curve in terms of operation or profit pattern. In practice, load forecasting  using semi-parametric SVM gets a better prediction than the conventional way \citep*{ Abhisek2009}.  The   semi-parametric SVM are also successfully applied to analyze
pharmacokinetic and pharmacodynamic data \citep*{Seok2011}. However, to the best of our knowledge, the theoretical research on the semi-parametric support vector machines is still lacking, and this paper focuses on this topic in high dimensional setting.

High dimensional case refers to the setting where the ambient dimension $p$ of the  covariates  is very large (e.g. $p\gg n$), but only a small subset of the covariates are significantly relevant to the response. The high  dimensional estimation and inference for various models have been investigated in the last years, and the interested readers can refer to  two related book written by \cite{Buhlmann2019} and \cite{Giraud2014}. Specially, the high dimensional inference for the linear (or additive) SVM  has been wildly studied in recent years, see \citep*{Tarigan2006, Zhao2012, Zhang2016, Peng2016}.
Precisely, \cite{Tarigan2006} consider a $\ell_1$-penalized parametric estimation in high dimensions for SVM and prove the convergence rates of the excess risk term under regularity conditions. Similarly, \cite{Zhao2012} propose a group-Lasso
type regularized approachs for the nonparametric additive SVM, and provide the oracle properties of the estimator and develop an efficient numerical algorithm to compute it. For high dimensional linear SVM, \citet*{Zhang2016} and \cite{Peng2016}  explicitly investigate the statistical performance of the $\ell_1$-norm and non-convex-penalized SVM such as variable selection consistency. However, all the aforementioned works only consider a single kernel in high dimensions.
By contrast, a partially linear SVM has to consider the mutual correlation between these two kernels with different structures, and also considers the mutual effects between the sparsity and the nonlinear functional complexity.  So the non-asymptotic analysis of such semi-parametric models in high-dimensional SVM appears to be considerably more complicated than those  based on a single kernel.

Under our partial linear setting, the whole input feature  consists of two parts:
$X=(Z,T)'$, where $Z\in \mathbb{R}^p$ has a linear relation to the response, while the  sub-feature $T$ has a nonlinear effect to the response.
Given all the observations  $\{(Y_i,Z_i,T_i)\}_{i=1}^{n}$ with the  sample size $n$, we consider a two-fold regularized learning scheme for the high dimensional PLQR,  and the semi-parametric  estimation pair $(\hat \bbeta, \hat g)$ is the unique solution by minimizing the following unconstrained optimization
\begin{equation}\label{submethod}
\min_{(f=\bbeta'Z+ g)\in\mathcal{F}}\Big\{\mathcal{R}_n(f)+\lambda_n\|\bbeta\|_1+
\mu_n \|g\|_{K}^2\Big\},
\end{equation}
where $(\lambda_n,\mu_n)
$ are two regularized  hyper-parameters for controlling the coefficients of the sparsity and  functional complexity, respectively.
In the partial linear setting, the adopted hypothesis space $\mathcal{F}$ for SVM is a summation of the linear kernel and  the general nonlinear kernel. More precisely,
$$
\mathcal{F}:=\{f(X)=\bbeta'Z+g(T),\,\,\bbeta\in \mathbb{R}^p, g\in \mathcal{H}_K\}.
$$

To investigate the statistical performance of the proposed semi-parameter estimator \eqref{submethod}, we introduce a population target function for the partial linear SVM within $\mathcal{F}$. In this paper,
the target function we will focus on is  a global solution $f^*$ of the following population minimization  on $\mathcal{F}$,
\begin{align}
\min_{f\in\mathcal{F}}\mathcal{R}(f),\quad \hbox{where}\,\,\mathcal{R}(f):=\mathbb{E}_\rho[\phi_h(Yf(X))].
\end{align}
Under the partial linear framework, $f^*$ can be written as: $f^*(X)=(\bbeta^*)'Z+g^*(T)$, where
$g^*$ is the nonparametric component, belonging to a specific RKHS that will be defined in Section 2.
For the parametric part, one often assumes that the structure of $\bbeta^*$ is sparse  under high dimensional setting,
in sense that the cardinality of $S=\{j,\,\beta_j^*\neq 0,j=1,2,...,p\}$ is far less than the ambient dimension $p$.
Note that, the target function $f^*$ is quite different from the Bayes rule, and the latter  is an optimal decision function taken over all the measurable functions. We can  treat $f^*$ as a sparse approximation to the Bayes rule within $\mathcal{F}$, particularly when the true function is not sparse. In the current  literatures, we are not concerned with any approximation error induced by sparse approximation or kernel misspecification.

In this paper, we are primarily concerned with learning rates of the excess risk $\mathcal{R}(\hat f)-\mathcal{R}(f^*)$
and the estimation errors of the parametric estimator and the nonparametric estimator for the high dimensional SVM. Interestingly, the  theoretical results reveal that our derived rate of the parametric estimator depends on not only
the sample size and the sparsity parameter, but also the functional complexity generated by the non-parametric component
and vice versa. As a byproduct, we develop a new weighted empirical process  to refine our analysis. This is one of the  key theoretical tools in the high dimensional literatures of the semi-parametric  estimation.

The rest of this article is organized as follows. In Section 2, we introduce some basic notations on RKHS that is used  to characterize the functional complexity. Then we impose some regular assumptions required to establish the convergence rates. In the end of Section 2, we explicitly propose our main theoretical results in terms of the excess risk and estimation errors. Section 3 is devoted to a detailed proof for the main theorems, and also proves some useful lemmas associated with the weighted empirical process. Section 4 concludes this paper with discussions and future possible researches.

{\bf Notations}.
We use $[p]$ to denote the set $\{1,2,...,p\}$. For a vector $\ba=(a_1,a_2,...,a_p)\in \mathbb{R}^p$,  the $\ell_q$-norm is defined as $\|\ba\|_q=\big(\sum_{i\in[p]}|a_i|^q\big)^{1/q}$.   For two sequences of numbers $a_n$  and  $b_n$, we use an $a_n=O(b_n)$ to denote that $a_n\leq Cb_n$ for some finite positive constant $C$ for all $n$. If
both $a_n=O(b_n)$ and $b_n=O(a_n)$, we use the notation $a_n\simeq b_n$. We also use an $a_n=\Omega(b_n)$ for $a_n\geq Cb_n$.
For a function, we
denote the $L_2$-norm of $f$ by $\|f\|_2=\big(\int_X f(x)^2d\rho_X(x)\big)^{1/2}$ with some distribution $\rho_X$.

\section{Conditions and Main Theorems}
We begin with the background and notation required for the main statements of our problem.
First of all, we introduce the notation of RKHS. RKHS can be defined by any symmetric and positive semidefinite kernel function $K:\,\mathcal{T} \times \mathcal{T}\rightarrow \mathbb{R}$. For each $t \in \mathcal{T}$, the function $t'\rightarrow K(t',t)$ is contained with the Hilbert space $\mathcal{H}_K$; moreover, the Hilbert space is endowed with an inner product $\langle\cdot,\cdot\rangle_K$ such that $K(\cdot,t)$ acts as the representer of the evaluation. Especially,
the reproducing property of RKHS plays an important role in the theoretical analysis and numerical optimization for any kernel-based method,
\begin{align}
\label{C1}
f(t)=\langle f,K(\cdot,t)\rangle_K,\quad \forall\, t \in \mathcal{T}.
\end{align}
This property also implies that $\|f\|_\infty\leq \kappa \|f\|_K$ with $\kappa:=\max_{t\in \mathcal{T}}|K(t,t)|<\infty$.  Moreover, by Mercer's theorem, a kernel $K$ defined on a compact subset of $\mathcal{T}$ admits the following eigen-decomposition,
\begin{align}
\label{C2}
K(t,t')=\sum_{\ell=1}^\infty \mu_\ell \phi_\ell(t) \phi_\ell(t'),\,\,t,t'\in \mathcal{T},
\end{align}
where $\mu_1\geq \mu_2\geq \cdots>0$ are the eigenvalues and $\{\phi_\ell\}_{\ell=1}^\infty$ is an orthonormal basis in $L_2(\rho_T)$. The decay rate of $\mu_\ell$ completely characterizes the complexity of RKHS induced by a kernel $K$, and generally it has equivalent relationships with various entropy numbers, see \citet*{Steinwart2008}
for details. With these preparations, we define the quantity,
\begin{align}
\label{C3}
\mathcal{Q}_n(r)=\frac{1}{\sqrt{n}}\Big[\sum_{\ell=1}^{\infty}\min\{r^2,\mu_\ell\}\Big]^{1/2},\quad \forall \,r>0.
\end{align}
Let $\nu_n$ be the smallest positive solution to the inequality,
$
40\nu_n^2\geq\mathcal{Q}_n(\nu_n),
$
where $40$ is only a technical constant.

Then, due to the mutual effects between the high dimensional parametric component and the nonparametric one, we  introduce the following quantity related to the convergence rates of the semi-parametric estimate, as illustrated in (\ref{C4}),
\begin{align}
\label{C4}
\gamma_n:=\max\Big\{\nu_n,\sqrt{\frac{\log p}{n}} \Big\}.
\end{align}

We now describe our main assumptions. Our first assumption deals with the tail behavior of the covariate of the linear part.

{\bf Assumption A}. (i) For simplicity,  we assume that $\|Z\|_\infty\leq C_0<\infty$ with some positive constant $C_0$;
(ii) The largest eigenvalue of $\mathbb{E}[ZZ']$ is finite, denoted by $\Lambda_{\max}>0$.

It appears that a bound on the $Z$-values is a restrictive assumption, ruling out the standard sub-gaussian covariates.
However, we can usually approximate a non-bounded distribution with its truncated version. Imposing such assumption is only for technical simplicity and may be relaxed to general thin-tail random variables. Assumption A(ii) is fairly standard in the literature to identify the coefficients associated to $Z$.

{\bf Assumption B}. There exist the constants $C_1>0$ and $\zeta\geq 2$ such that, for all $f\in \mathcal{F}$, the equation (\ref{C5}) holds,
\begin{align}
\label{C5}
\mathcal{R}( \bbeta,  g)-\mathcal{R}(\bbeta^*, g^*)\geq C_1\|f-f^*\|_2^{\zeta}.
\end{align}
The parameter $\zeta$ is called the Bernstein parameter introduced by \citet*{Bartlett2006,Pierre2019}.  Fast rates will usually be derived when $\zeta=2$.  This condition is essentially a qualification of the identifiability condition of the objective function at its minimum $f^*$.
Note that, the Bernstein parameter is slightly different from the classical margin parameter adopted by \cite{Tarigan2006,Chen2004}.

To estimate the parametric and nonparametric parts respectively,
 some conditions concerning correlations between $Z$ and $T$ are required.
For each $j\in[p]$, let $\Pi^{(j)}_T$ be the projection of $Z^{(j)}$ onto $\mathcal{H}_K$. To be precise, $\Pi^{(j)}_T=g_j^*(T)$ with (\ref{C6}),
\begin{align}
\label{C6}
g_j^*=\arg\min_{g\in\mathcal{H}_K}\mathbb{E}_{Z^{(j)},T}[(Z^{(j)}-g(T))^2].
\end{align}
Let $\Pi_{Z|T}:=(\Pi^{(1)}_T,...,\Pi^{(p)}_T)'$ and $Z_T=Z-\Pi_{Z|T}$. Each function $g^*_j$ can be viewed as the best approximation  of $\mathbb{E}[Z^{(j)}|T]$ within  $\mathcal{H}_K$.
In the extreme case ($Z$ is uncorrelated with $T$),
$\Pi_{Z|T}=0$. The following condition is quite common in the semi-parametric estimation \citep*{Muller2015}, ensuring that there is enough information in the data to identify the parametric coefficients.

{\bf Assumption C}.   The smallest eigenvalue  of
$\mathbb{E}[Z_TZ_T']$ is  bounded below by a constant $\Lambda_{\min}>0$.

Note that, the equation (\ref{projection}) always holds with the definition of  projection on the  $\|\cdot\|_2$-norm,
\begin{align}\label{projection}
\|\bbeta'Z+g(T)\|_2^2=\|\bbeta'Z_T\|_2^2+|\bbeta'\Pi_{Z|T}+g(T)\|_2^2,\quad \forall \,f\in \mathcal{F}.
\end{align}
This equality ensures that the parametric estimation can be separated from the total estimation, which is very useful in our proof.

We are in a position to derive the learning rate of the estimator $(\hat \bbeta,\hat g)$ defined by minimization \eqref{submethod}. We  allow that the number of dimension $p$ and the number of active covariates $s:=|S|$ which are increasing with respect to the sample size $n$, while $s\ll p$ and the dimension of $T$  is fixed.

\begin{thm}
Let  $(\hat \bbeta,\hat g)$ be the proposed semi-parametric estimator for SVM defined in \eqref{submethod}, with the regularization parameters  $\lambda_n=\sqrt{\log p/n}$ and $\mu_n\simeq \gamma_n^2$. If Assumptions A, B, and C hold, the equation \eqref{C7} holds with the probability at least $1-2p^{-A/2}-16p^{3-A}$ with some $A>3$,
\begin{align}
\label{C7}
\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)=O\big((\gamma_n+\sqrt{s\log p/ n})^{\frac{\zeta}{\zeta-1}}\big),
\end{align}
and at the meantime the estimation error has the form \eqref{C8},
\begin{align}
\label{C8}
\|\hat \bbeta-\bbeta^*\|_2=O\big((\gamma_n+\sqrt{s\log p/ n})^{\frac{1}{\zeta-1}}\big),\quad
\|\hat g-g^*\|_2
=O\big((\gamma_n+\sqrt{s\log p/ n})^{\frac{1}{\zeta-1}}\big).
\end{align}
\end{thm}

Remark that, this rate may be interpreted as the sum of a subset selection term ($\sqrt{s\log p/ n}$) for the linear part and a fixed dimensional non-parametric estimation term ($\nu_n$). Depending on the scaling of the triple $(n,p,s)$ and the smoothness of the  RKHS $\mathcal{H}_K$, either the subset selection term or the non-parametric estimation term may dominate the estimation. In general, if $s \log p/ n = o(\nu_n^2)$, the $s$-dimensional parametric term can dominate the estimation, so can the vice versa otherwise. At the boundary, the scalings of the two terms are equivalent. In the best situation ($\zeta=2$), our derived rate of the excess risk is the same as the optimal rate achieved by those least square approaches, see \citep*{Koltchinskii2010,Muller2015}
for details.

Note also that, it is easy to check that Theorems 1 still holds if $p$ in the confidence probability is replaced by an arbitrary $\tilde{p}\geq p$ such that $\log \tilde{p} \geq 2\log\log n$. In this case, the divergence of $p$ is not needed and the probability bounds in the theorem becomes $1-2\tilde{p}^{-A/2}-16\tilde{p}^{3-A}$.

A number of corollaries of Theorem 1 can be obtained with particular choices of different kernels. First of all, we present finite-dimensional  $m$-rank operators, i.e., the kernel function $K$ can be expressed in terms of $m$ eigenfunctions. These eigenfunctions include the linear functions, polynomial functions, as well as the function class based on finite dictionary expansions.

\begin{cor}
Under the same conditions as Theorem 1, consider a nonlinear kernel with finite rank $m$. Then the semi-parametric estimator  for SVM defined in \eqref{submethod} with $\lambda_n=\sqrt{\log p/n}$ and $\mu_n\simeq \gamma_n^2$ satisfies the condition \eqref{C9},
\begin{align}
\label{C9}
\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)=O_p\Big(\Big(\frac{s\log p}{n}+\frac{m}{n}\Big)^{\frac{\zeta}{2(\zeta-1)}}\Big),
\end{align}
where
$$
\|\hat \bbeta-\bbeta^*\|_2=O_p\Big(\Big(\frac{s\log p}{n}+\frac{m}{n}\Big)^{\frac{1}{2(\zeta-1)}}\Big),\quad
\|\hat g-g^*\|_2
=O_p\Big(\Big(\frac{s\log p}{n}+\frac{m}{n}\Big)^{\frac{1}{2(\zeta-1)}}\Big).
$$
\end{cor}
For a finite rank kernel and for any $r>0$, we have
$\mathcal{Q}_n(r)\leq r\sqrt{\frac{m}{n}}$, which follows by the result of Theorem 1. Corollary 1
corresponds to the linear case for SVM when $s\simeq m$.
The existing theory in the literatures on the linear SVM has paid constant attention to the analysis of the generalization error and
variable selection consistency.
\cite{Zhang2016} considers the non-convex penalized SVM in terms of the variable selection consistency and oracle property in high dimension, however, their results are based on a restrictive condition in case of $p\ll n^{1/2}$. So the ultra-high dimensional cases ($p=O(e^{n^r})$ with $r<1$) are excluded. Under the constrained eigenvalues constant condition, \cite{Peng2016} provides a tight upper bound of the linear SVM estimator in the $\ell_2$ norm, with an order $\sqrt{s\log p/n}$, which is the same as our rate in Corollary 1 when $\zeta=2$.

Secondly, we state a result for the RKHS with  countable eigenvalues,  decaying at a rate $\mu_\ell \simeq(1/\ell)^{2\alpha}$ for some smooth parameter $\alpha > 1/2$. In fact, this type of scaling covers the Sobolev spaces,  consisting of derivative functions with $\alpha$.

\begin{cor}
Under the same conditions as Theorem 1, consider a kernel with the eigenvalue decay $\mu_\ell \simeq(1/\ell)^{2\alpha}$ for some  $\alpha > 1/2$. Then the semi-parametric estimator defined in \eqref{submethod} with $\lambda_n=\sqrt{\log p/n}$ and $\mu_n\simeq \gamma_n^2$ satisfies the equation \eqref{C10},
\begin{align}
\label{C10}
\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)=O_p\Big(\Big(\frac{s\log p}{n}+n^{-\frac{2\alpha}{2\alpha+1}}\Big)^{\frac{\zeta}{2(\zeta-1)}}\Big),
\end{align}
where
$$
\|\hat \bbeta-\bbeta^*\|_2=O_p\Big(\Big(\frac{s\log p}{n}+n^{-\frac{2\alpha}{2\alpha+1}}\Big)^{\frac{1}{2(\zeta-1)}}\Big),\quad
\|\hat g-g^*\|_2
=O_p\Big(\Big(\frac{s\log p}{n}+n^{-\frac{2\alpha}{2\alpha+1}}\Big)^{\frac{1}{2(\zeta-1)}}\Big).
$$
\end{cor}
In the previous corollary, we need to compute the critical univariate rate $\nu_n$. Given the assumption of polynomial eigenvalue decay, a truncation argument shows that $\mathcal{Q}_n(r)=O(\frac{r^{\frac{2\alpha}{2\alpha-1}}}{\sqrt{n}})$, i.e., $\nu_n^2\simeq n^{-\frac{2\alpha}{2\alpha+1}}$. As opposed to Corollary 1,
we now discuss the special case where the functional complexity dominates the esimation, that is, the rate of the excess risk is an order $O\big(n^{-\frac{\alpha}{2\alpha+1}}\big)^{\frac{\zeta}{\zeta-1}}$. This is a better rate campared with those in \cite{Chen2004} and  \cite{Wu2007}. The learning rate in \cite{Chen2004} is
derived as $O_p(n^{\frac{\theta}{\theta+\eta+\theta \eta}})$, where $\theta$ is a separation parameter corresponding to $\zeta=\frac{1+2\theta}{\theta}$, and $\eta$ is a power appearing in the covering number, satisfying $\eta\simeq \frac{1}{\alpha}$.  Our rate can be proved to be better than that of Chen in the best case with $\zeta=2$. The similar arguments also hold when considering the result in \cite{Wu2007}.

\section{Proofs}
In this section, we provide the proofs of our  main theorem (Theorem 1). At a high-level, Theorem 1 is based on an appropriate adaptation to the semi-parametric settings of various techniques, developed for sparse linear regression or additive non-parametric estimation in high dimensions \citep*{Buhlmann2019}. In contrast to the parametric setting or additive setting, it involves structural deals from the empirical process theory to control the error terms in the semi-parametric case . In particular, we make use of several concentration theorems for the empirical processes \citep*{Geer2000}, as well as the results on the Rademacher complexity of kernel classes \citep*{Bartlett2005}.

\subsection{Proof for Theorem 1}
We write the total empirical objective as the equation \eqref{P1},
\begin{align}
\label{P1}
\mathcal{L}(\bbeta,g)=\mathcal{R}_n(\bbeta,g)+\lambda_n\|\bbeta\|_1
+\mu_n
\|g\|_{K}^2.
\end{align}

The population risk for partial linear SVM is defined by \eqref{P2},
\begin{align}
\label{P2}
\mathcal{R}(\bbeta,g)=\mathbb{E}\big[ \phi_h(Y[\bbeta'Z+g(T)])\big].
\end{align}

According to the definition of $(\hat \bbeta, \hat g)$, it holds that
	$\mathcal{L}(\hat \bbeta, \hat g)\leq \mathcal{L}(\bbeta^*, g^*)$. That means,
	\begin{align}
\label{P3}
\mathcal{R}_n(\hat \bbeta, \hat g)+\lambda_n\|\hat \bbeta\|_1+\mu_n
	\|\hat g\|_{K}^2\leq \mathcal{R}_n(\bbeta^*, g^*)+\lambda_n\|\bbeta^*\|_1+\mu_n
	\| g^*\|_{K}^2.
	\end{align}
The inequality \eqref{P3} can be rewritten into the form \eqref{firststep},
\begin{align}\label{firststep}
&\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)+\lambda_n\|\hat \bbeta-\bbeta^*\|_1+\mu_n/2	\|\hat g-g^*\|_{K}^2\nonumber\\
&\leq \mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}_n(\hat \bbeta, \hat g)+\mathcal{R}_n(\bbeta^*, g^*)-\mathcal{R}(\bbeta^*, g^*)
+2\lambda_n\|(\hat \bbeta-\bbeta^*)_S\|_1+2\mu_n
\| g^*\|_{K}^2.
\end{align}
For simplicity, we denote,
\begin{align}
\label{P4}
\nu_n(f):=\nu_n(\bbeta,g)=(\mathcal{R}_n( \bbeta,  g)-\mathcal{R}( \bbeta, g)),\quad \forall \,f(X)=\bbeta'Z+g(T).
\end{align}

In order to derive the upper bound of $|\nu_n(\hat f)-\nu_n(f^*)|$ in  \eqref{P4}, a new weighted empirical process is proposed in our semi-parametric high dimensional setting. The process is relevant to the uniform law of large number in a mixed function space. The \textbf{Lemma \ref{wieghtineq}} can be derived via the peeling device which is often used in probabilistic theory.

\begin{lem}\label{wieghtineq}
	Let $\mathcal{E}$ be the event
	\begin{align}
\label{P5}
\mathcal{E}:=\Big\{|\nu_n(f)-\nu_n(f^*)|\leq D_0\Big(\sqrt{\frac{\log p}{n}}\|\bbeta-\bbeta^*\|_1+\gamma_n\| g-g^*\|_{2}+	\gamma_n^2\| g-g^*\|_{K}+e^{-p}\Big)\Big\},
\end{align}
where $D_0$ is a constant in the proof of the \textbf{Lemma \ref{wieghtineq}}.
Because $p=\Omega(\log n)$ and $p=o(e^n)$, the inequality \eqref{P6} holds for  some universal constant $A>3$,
\begin{align}
\label{P6}
\mathbb{P}(\mathcal{E})\geq 1-2p^{-A/2}-16p^{3-A}.
\end{align}
\end{lem}

We continue our proof  \eqref{firststep} along with the results established in Lemma  \ref{wieghtineq}.
Apply the weighted empirical process and we can obtain \eqref{secdstep} from \eqref{firststep},
\begin{align}\label{secdstep}
&\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)+\lambda_n\|\hat \bbeta-\bbeta^*\|_1+\frac{\mu_n}{2}	\|\hat g-g^*\|_{K}^2\nonumber\\
&\leq  D_0\big(\sqrt{\log p/n}\|\bbeta-\bbeta^*\|_1+\gamma_n\| g-g^*\|_{2}+	\gamma_n^2\| g-g^*\|_{K}+e^{-p}\big)\nonumber\\
&+2\lambda_n\|(\hat \bbeta-\bbeta^*)_S\|_1+2\mu_n
\| g^*\|_{K}^2.
\end{align}
Therefore, when the conditions $\lambda_n\geq 2D_0\sqrt{\log p/n}$ and $\mu_n\geq 2D_0\gamma_n^2$ are both satisfied, \eqref{thirdstep} holds after the inequality \eqref{secdstep},
\begin{align}\label{thirdstep}
&\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)+\frac{\lambda_n}{2}\|\hat \bbeta-\bbeta^*\|_1+\frac{\mu_n}{4}	\|\hat g-g^*\|_{K}^2\nonumber\\
&\leq  D_0\big(\gamma_n\| g-g^*\|_{2}+	\gamma_n^2/2\big)
+2\lambda_n\|(\hat \bbeta-\bbeta^*)_S\|_1+2\mu_n
\| g^*\|_{K}^2,
\end{align}
where we use the basic inequality $2uv\leq u^2+v^2$. Since $p=\Omega(\log n)$ implies that
$e^{-p}=O(n^{-1})=O(\gamma_n^2)$, \eqref{storng} can be derived with Assumption B, C and the  equality \eqref{projection},
\begin{align}\label{storng}
\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)\geq C_1\|\hat f-f^*\|_2^{\zeta}
\geq C_1\|(\hat \bbeta-\bbeta^*)'Z_T\|_2^{\zeta}\geq C_1\Lambda_{\min}^{\zeta/2}\|\hat \bbeta-\bbeta^*\|_2^{\zeta}.
\end{align}
Moreover, \eqref{P7} follows by Assumption A after some simple computations,
\begin{align}
\label{P7}
\left\{
\begin{array}{l}
\| g-g^*\|_{2}\leq \| f-f^*\|_{2}+\Lambda_{\max}\|\hat \bbeta-\bbeta^*\|_2,\\
\lambda_n\|(\hat \bbeta-\bbeta^*)_S\|_1\leq \lambda_n\sqrt{s}\|(\hat \bbeta-\bbeta^*)_S\|_2\leq \lambda_n\sqrt{s}\|\hat \bbeta-\bbeta^*\|_2,
\end{array}
\right.
\end{align}
where the last inequality follows from the Cauchy-Schwartz inequality.
Substitute \eqref{P7} into \eqref{secdstep} and we can obtain the inequality \eqref{imdestep},
\begin{align}\label{imdestep}
&C_1\|\hat f-f^*\|_2^{\zeta}+\lambda_n\|\hat \bbeta-\bbeta^*\|_1+\mu_n/2	\|\hat g-g^*\|_{K}^2\nonumber\\
&\leq  D_0\gamma_n\| f-f^*\|_{2}+ D_0\gamma_n\Lambda_{\max}\|\hat \bbeta-\bbeta^*\|_2+	D_0\gamma_n^2/2
+2\sqrt{s}\lambda_n\|\hat \bbeta-\bbeta^*\|_2+2\mu_n
\| g^*\|_{K}^2.
\end{align}
 For any $\theta>0$, we can then derive the inequality \eqref{yongy} with the Young inequality ($uv\leq u^{q}/q+v^p/p$ with $1/q+1/p=1$),
\begin{align}\label{yongy}
&\gamma_n\| f-f^*\|_{2}\leq \big(\frac{\zeta-1}{\zeta}\big)\theta^{\frac{\zeta}{1-\zeta}}\gamma_n^{\frac{\zeta}{\zeta-1}}+\frac{\theta^\zeta}{\zeta}\|\hat f-f^*\|_2^{\zeta},\\
&  c_n\|\hat \bbeta-\bbeta^*\|_2\leq \Big(\frac{\zeta-1}{\zeta}\Big)\theta^{\frac{\zeta}{1-\zeta}}c_n^{\frac{\zeta}{\zeta-1}}+\frac{\theta^\zeta}{\zeta}\|\hat \bbeta-\bbeta^*\|_2^{\zeta},
\end{align}

where $c_n:=D_0\gamma_n\Lambda_{\max}+2\sqrt{s}\lambda_n$.  \eqref{iiimdestep} holds if $\theta$ is small enough to satisfy $\theta^{\zeta}\leq \frac{C_1\zeta}{2}$,
\begin{align}\label{iiimdestep}
&\frac{C_1}{2}\|\hat f-f^*\|_2^{\zeta}+\lambda_n\|\hat \bbeta-\bbeta^*\|_1+\frac{\mu_n}{2}	\|\hat g-g^*\|_{K}^2\nonumber\\
&\leq D_0 \big(\frac{\zeta-1}{\zeta}\big)\theta^{\frac{\zeta}{1-\zeta}}\gamma_n^{\frac{\zeta}{\zeta-1}}+ 	\frac{D_0\gamma_n^2}{2}
+c_n\|\hat \bbeta-\bbeta^*\|_2+2\mu_n
\| g^*\|_{K}^2.
\end{align}
Furthermore, combine  \eqref{storng}, \eqref{yongy} with \eqref{iiimdestep} and we can conclude the inequality \eqref{P8},
\begin{align}
\label{P8}
&\frac{C_1\Lambda_{\min}^{\frac{\zeta}{2}}}{2}\|\hat \bbeta-\bbeta^*\|_2^{\zeta}+\frac{\lambda_n}{2}\|\hat \bbeta-\bbeta^*\|_1+\frac{\mu_n}{4}	\|\hat g-g^*\|_{K}^2\nonumber\\
&\leq D_0 \big(\frac{\zeta-1}{\zeta}\big)\theta^{\frac{\zeta}{1-\zeta}}\gamma_n^{\frac{\zeta}{\zeta-1}}+ 	\frac{D_0\gamma_n^2}{2}+2\mu_n
\| g^*\|_{K}^2+\big(\frac{\zeta-1}{\zeta}\big)\theta^{\frac{\zeta}{1-\zeta}}c_n^{\frac{\zeta}{\zeta-1}},
\end{align}
where the condition $\theta^{\zeta}\leq \frac{C_1\zeta\Lambda^{\zeta/2}_{\min}}{4}$ is additionnally required so that
$\frac{\theta^\zeta}{\zeta}$ is ignorable. In this case, we can derive \eqref{P9},
\begin{align}
\label{P9}
&\|\hat \bbeta-\bbeta^*\|_2^{\zeta}=O_p\big((\gamma_n+\sqrt{s}\lambda_n)^{\frac{\zeta}{\zeta-1}}+\mu_n\big),\\
&\lambda_n\|\hat \bbeta-\bbeta^*\|_1=O_p\big((\gamma_n+\sqrt{s}\lambda_n)^{\frac{\zeta}{\zeta-1}}+\mu_n\big).
\end{align}
Moreover, we will obtain \eqref{P10} based on \eqref{P9} and \eqref{iiimdestep},
\begin{align}
\label{P10}
\|\hat f-f^*\|_2^{\zeta}=O_p\big((\gamma_n+\sqrt{s}\lambda_n)^{\frac{\zeta}{\zeta-1}}+\mu_n+\gamma_n\mu_n^{\frac{1}{\zeta}}\big)
=O_p\big(\gamma_n^{\frac{\zeta}{\zeta-1}}+\gamma_n^{\frac{\zeta+2}{\zeta}}\big),
\end{align}
where we choose $\mu_n\simeq\gamma_n^2$ and $\gamma_n=\Omega(\lambda_n)$. Therefore, it is concluded that \eqref{P11} holds by the triangle inequality and Assumption C,
\begin{align}
\label{P11}
\|\hat g-g^*\|_2
=O_p\big((\gamma_n+\sqrt{s}\lambda_n)^{\frac{1}{\zeta-1}}\big).
\end{align}
Finally, plugging the  derived upper bounds into \eqref{thirdstep},
we obtain the desired upper bound of the excess risk $\mathcal{R}(\hat \bbeta, \hat g)-\mathcal{R}(\bbeta^*, g^*)$.
This completes the proof. $\Box$

\subsection{The semi-parametric weighted emprical process}
In order to prove Lemma 1, some auxiliary results is required which is on uniform law of large number or concerntation inequalities, stated as Lemmas \ref{consentration} \citep*{Massart2000}.

\begin{lem}\label{consentration}
	Let $U_1,...,U_n$ be independent and identically distributed copies of a random variable $U\in \mathcal{U}$.
	Let $\Gamma$ be a class of real-valued functions on $\mathcal{U}$ satisfying $\sup_{u}|\gamma(u)|\leq D$ for all
	$\gamma\in \Gamma$. Define
	$$
	\mathbf{Z}:=\sup_{\gamma\in \Gamma}\Big|\frac{1}{n}\sum_{i=1}^n\{\gamma(U_i)-\mathbb{E}[\gamma(U_i)]\}\Big|,
	$$
	and
	$$
B^2:=\sup_{\gamma\in \Gamma}var(\gamma(U)).
	$$
	Then there exists a universal constant $N_0$ such that
	$$
	\mathbb{P}\Big(\mathbf{Z}\geq N_0 \big[ \mathbb{E}[\mathbf{Z}]+B\sqrt{r/n}+D r/n\big]\Big)\leq \exp(-r),\quad \forall\,r>0.
	$$
\end{lem}

{\bf Proof for Lemma 1.}
For any $f(x)= \bbeta'z+g(t)$, we define \eqref{P12} to apply Lemma \ref{consentration},
\begin{align}
\label{P12}
\gamma(u)=\phi_h(y[\bbeta'z+g(t)])-\phi_h(y[(\bbeta^*)'z+g^*(t)]),\,\,\bbeta\in \mathbb{R}^p \,\hbox{and} \,g\in \mathcal{H}_K.
\end{align}
Based on \eqref{P12}, a bounded set of functions is introduced,
$$
\Gamma_\Delta:=\big\{\gamma,\,\,\sqrt{\log p/n}\|\bbeta-\bbeta^*\|_1\leq \Delta_\beta, \gamma_n\| g-g^*\|_{2}\leq \Delta_{-}, \gamma_n^2\| g-g^*\|_{K}\leq \Delta_{+}\big\},
$$
where we write the triplet $\Delta=( \Delta_\beta,\Delta_{-},\Delta_{+})$ .

Since $\phi_h(\cdot)$ in \eqref{P12} is Lipschitz with constant $1$, the inequality \eqref{P13} holds for any $u$,
\begin{align}
\label{P13}
\begin{array}{rl}
|\gamma(u)|&\leq |(\bbeta-\bbeta^*)'z|+|g(t)-g^*(t)|\leq C_0\|\bbeta-\bbeta^*\|_1+\kappa \| g-g^*\|_{K}\\
&\leq C_0\sqrt{\frac{n}{\log p}}\Delta_\beta+\kappa\frac{n}{\log p}\Delta_{+}.
\end{array}
\end{align}
\eqref{P13} implies that  if we take \eqref{dbound} in Lemma \ref{consentration},
\begin{align}\label{dbound}
D:=C_0\sqrt{\frac{n}{\log p}}\Delta_\beta+\kappa\frac{n}{\log p}\Delta_{+},
\end{align}
and \eqref{P14} is also derived by the Lipschitz property,
\begin{align}
\label{P14}
\begin{array}{rl}
B^2&\leq 2\mathbb{E}[((\bbeta-\bbeta^*)'Z)^2]+2\mathbb{E}[(g(T)-g^*(T))^2]\\
&\leq \frac{2n}{\log p}\big(C_0^2\Delta_{\beta}^2+\Delta_{-}^2\big),
\end{array}
\end{align}
we can plug \eqref{dbound} and \eqref{boubund} into Lemma \ref{consentration} to yield \eqref{immderuslt},
\begin{align}\label{boubund}
B:=\sqrt{\frac{2n}{\log p}}\big(C_0\Delta_{\beta}+\Delta_{-}\big),
\end{align}
\begin{align}\label{immderuslt}
\mathbb{P}\Big(\mathbf{Z}\geq N_0 \Big[ \mathbb{E}[\mathbf{Z}]+\sqrt{\frac{2r}{\log p}}\big(C_0\Delta_{\beta}+\Delta_{-}\big)+\frac{\kappa r}{\log p}\Delta_{+}\Big]\Big)\leq \exp(-r),\quad \forall\,r\in(0,2n).
\end{align}

It remains to provide the upper bound of the term $\mathbb{E}[\mathbf{Z}]$.
Let $\sigma_1,...,\sigma_n$ be a Rademacher sequence independent of $(Y_1,X_1),...,(Y_n,X_n)$.  The inequality \eqref{P15} can be obtained by symmetrization and
the contraction inequality,
\begin{align}
\label{P15}
\begin{array}{rl}
\mathbb{E}[\mathbf{Z}]&\leq 4\mathbb{E}\Big(\sup_{\gamma\in \Gamma_\Delta}\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i(f(X_i)-f^*(X_i))\Big|\Big)\\
&\leq 4\mathbb{E}\Big(\sup_{\gamma\in \Gamma_\Delta }\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i((\bbeta-\bbeta^*)'Z_i)\Big|\Big)\\
&+4\mathbb{E}\Big(\sup_{\gamma\in \Gamma_\Delta}\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i(g(T_i)-g^*(T_i))\Big|\Big).
\end{array}
\end{align}
By Bernstein inequality and the union bound, we can get the inequality \eqref{uuperexp},
\begin{align}\label{uuperexp}
\mathbb{E}\Big(\sup_{\gamma\in \Gamma_\Delta }\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i((\bbeta-\bbeta^*)'Z_i)\Big|\Big)
&\leq \mathbb{E}\Big(\sup_{j\in[p]}\Big|\frac{1}{n}\sum_{i=1}^n\sigma_iZ_{ij}\Big|\Big)
\sup_{\gamma\in \Gamma_\Delta }\|\bbeta-\bbeta^*\|_1\nonumber\\
&\leq C_0\Delta_{\beta}.
\end{align}
Moreover, applying Talagrand's concentration inequality once again, we get \eqref{P16} with the probability at least $1-e^{-r}$
\begin{align}
\label{P16}
\begin{array}{rl}
\mathbb{E}\Big(\sup_{\gamma\in \Gamma_\Delta}\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i(g(T_i)-g^*(T_i))\Big|\Big)
&\leq
N_0\Big(\sup_{\gamma\in \Gamma_\Delta}\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i(g(T_i)-g^*(T_i))\Big|\\
&+\sqrt{\frac{2r}{\log p}}\Delta_{-}+\frac{\kappa r}{\log p}\Delta_{+}\Big).
\end{array}
\end{align}
Besides the result in \citet*{Koltchinskii2010} that,
$$
\sup_{\gamma\in \Gamma_\Delta}\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i(g(T_i)-g^*(T_i))\Big|\leq \nu_n\| g-g^*\|_{2}+
\nu_n^2\| g-g^*\|_{K},\quad \forall\,g\in \mathcal{H}_K,
$$
the inequality \eqref{nonlexp} holds with the probability at least $1-2e^{-r}-p^{-A/2}$,
\begin{align}\label{nonlexp}
\mathbb{E}\Big(\sup_{\gamma\in \Gamma_\Delta}\Big|\frac{1}{n}\sum_{i=1}^n\sigma_i(g(T_i)-g^*(T_i))\Big|\Big)
&\leq
N_0\Big(\Delta_{-}+\Delta_{+}+\sqrt{\frac{2r}{\log p}}\Delta_{-}+\frac{\kappa r}{\log p}\Delta_{+}\Big),
\end{align}
where $A>3$ is some constant in Section 5 in \citet*{Koltchinskii2010}.
Thus, combining \eqref{uuperexp}, \eqref{nonlexp} with \eqref{immderuslt}, we get \eqref{finderuslt} on an event $E$ of  the probability at least $1-2e^{-r}-p^{-A/2}$,
\begin{align}\label{finderuslt}
\mathbf{Z}\leq L_0 \Big[ \Delta_{\beta}+\Delta_{-}+\Delta_{+}+\sqrt{\frac{r}{\log p}}\big(\Delta_{\beta}+\Delta_{-}\big)+\frac{ r}{\log p}\Delta_{+}\Big],\,\, \forall\,r\in(0, 2n),
\end{align}
where $L_0$ is some constant depending on $N_0$, $C_0$ and $\kappa$.

We will now choose $r=A\log p$ so as to obtain a weighted empirical process that holds uniformly over the constrains \eqref{splitnum},
\begin{align}\label{splitnum}
e^{-p}\leq \Delta_{\beta}\leq e^p,\,e^{-p}\leq \Delta_{-}\leq e^p,\,e^{-p}\leq \Delta_{+}\leq e^p.
\end{align}
To achieve this purpose, if we choose
$$
\Delta_{\beta}^k=\Delta_{-}^k=\Delta_{+}^k:=2^{-k},\quad k=-p,-p+1...,p-1,p,
$$
and
\begin{align}\label{splisec}
\Gamma_\Delta^{k,l,h}:=\Big\{\gamma,\,\,&\frac{1}{2}\Delta_\beta^k\leq \sqrt{\frac{\log p}{n}}\|\bbeta-\bbeta^*\|_1\leq \Delta_\beta^k, \frac{1}{2}\Delta_{-}^l\leq \gamma_n\| g-g^*\|_{2}\leq \Delta_{-}^l,\nonumber\\
& \frac{1}{2}\Delta_{+}^h\leq \gamma_n^2\| g-g^*\|_{K}\leq \Delta_{+}^h\Big\},
\end{align}
based on \eqref{finderuslt}, \eqref{weigheruslt} holds over an event $F(\Delta_{\beta}^k,\Delta^l_{-},\Delta^h_{+})$ of the probability $\mathbb{P}\big(F(\Delta_{\beta}^k,\Delta^l_{-},\Delta^h_{+})\big)\geq 1-p^{A/2}$ for any triplet $(\Delta_{\beta}^{k},\Delta^l_{-},\Delta^h_{+})$ satisfying \eqref{splitnum} and \eqref{splisec},
\begin{align}\label{weigheruslt}
\mathbf{Z}\leq L_0 \Big[ \Delta_{\beta}^k+\Delta_{-}^l+\Delta_{+}^h+\sqrt{A}\big(\Delta^k_{\beta}+\Delta^l_{-}\big)+A\Delta^h_{+}\Big],\,\, \forall\,r\in(0, 2n).
\end{align}
 On the event
$E':=E\cap \big(\bigcap_{k,l,h}F(\Delta_{\beta}^k,\Delta^l_{-},\Delta^h_{+})\big)$, the intersection $\bigcap_{k,l,h}F(\Delta_{\beta}^k,\Delta^l_{-},\Delta^h_{+})$ is bounded by $(2p+1)^3$. Therefore, the lower bound of the probability $E'$ can be formulated as \eqref{P17},
\begin{align}
\label{P17}
\mathbb{P}(E')\geq 1-\mathbb{P}(E^c)- (2p+1)^3\exp(-A\log p)\geq 1-2p^{-A/2}-16p^{3-A}.
\end{align}
Thus, for any $k,l,h$, \eqref{P18} holds on the event  $E'$ with the construction of the function sets  $\Gamma_\Delta^{k,l,h}$ and \eqref{weigheruslt},
\begin{align}
\label{P18}
\mathbf{Z}\leq2L_0(1+\sqrt{A})\Big(\sqrt{\frac{\log p}{n}}\|\bbeta-\bbeta^*\|_1+\gamma_n\| g-g^*\|_{2}\Big)+
2L_0(1+A)\gamma_n^2\| g-g^*\|_{K}.
\end{align}
If it is true for either of the conditions $\Delta_{\beta}\leq e^{-p}$, or $\Delta_{-}\leq e^{-p}$ or $\Delta_{+}\leq e^{-p}$, it follows that \eqref{P18} with almost the same probability by monotonnicity of the left-hand side,
$$
\mathbf{Z}\leq2L_0(1+\sqrt{A})\Big(\sqrt{\frac{\log p}{n}}\|\bbeta-\bbeta^*\|_1+\gamma_n\| g-g^*\|_{2}\Big)+
2L_0(1+A)\gamma_n^2\| g-g^*\|_{K}+3e^{-p}.
$$
This completes the proof of Lemma \ref{wieghtineq}. \hspace*{10cm}$\Box$

\section{Discussion and Future Work}
In this paper, we have studied the estimation in the partially linear sparse  models for support vector machine, where the covariates split the linear component and the nonlinear component   within a reproducing kernel Hilbert space.
An important feature of our analysis is that we develop a new weighted empirical process in the high dimensional semi-parametric setting, so that our derived rates are sharp  in comparison with the existing related results, even are comparable to  the least square estimations in the high dimensional setting.

There are some further research topics of this work. It is known that the parametric estimation for the partially linear mean regression does not depend on the functional complexity under some additional conditions.
This paper has not achieved such a better rate for the partially linear SVM. Therefore, how to improve the parametric estimation error is an interesting research problem. Besides, the lower bound for the partially linear SVM in the high dimensional setting have not been established under the margin information, which is an important complementary to our upper bounds.

\bigskip

\noindent\textbf{\Large Acknowledgement}\\
Yifan Xia's research is partially supported by
the National Social Science Fund of China (NSSFC-16BTJ013)(NSSFC-16ZDA010) and the Sichuan Project of Science \& Technology (2016JY0273). Shaogao Lv's research is partially supported by NSFC-11871277 and NSFC-11829101.

\bigskip
\bigskip


TITLE: Guessing game where opponent can lie
QUESTION [2 upvotes]: I encountered the following problem:

Your opponent chooses a number $n \in \{1, \dots, m\}$ (where $m$ is a power of $2$) and your task is to guess it.
You are allowed to ask only $m$ questions of type: "Is the number in the set X?" for $X \subset \{1, \dots, m\}$.
However, the opponent need not to always answer truthfully and may lie less than $m/4$ times (i.e. in at most $m/4 - 1$ answers).

The source suggest that there is a winning strategy (I assume that for us) and that it is related to error correction codes.
I tried to use induction of $m$ ($2^{k-1} \to 2^k$) but failed.
It is easily achievable to detect the half of $\{1, \dots, m\}$ in which $n$ lies, but not to force the opponent to spend a half of opportunities to lie.
Since it has something to do with error correction codes, I also tried to get some new insights by constructing a code which translates the words from $\mathbb{Z}_2^{\log_2 m}$ ($m$ in binary) to $\mathbb{Z}_2^m$ (sequence of answers, some of them are lies).
The code needs to have distance at least $m/2$ to detect $m/4$ "errors" (code distance = minimal Hamming distance between the code words).
However, just having the code is not sufficient.
It is also necessary to have the proper set of questions which makes the translation (i.e. the coding function).
Then it is possible to ask the questions and reconstruct the number $n$ from the answers (i.e. correct the errors and deduce the $n$).
But now, it is an idea which is very far for anything concrete.

REPLY [2 votes]: Here's a start.
Let $m = 2^k$. Consider the Hadamard code $[2^k,k, 2^{k - 1}]_2$. Every number $n \in {1, \ldots, m}$ corresponds to a codeword $Had(n)$ of length $m$, i.e. a sequence of $m$ bits. For each $i \in {1, \ldots, m}$, let $X_i$ be the set $$X_i = \{n \in \{1, \ldots, m\} | \text{the }i\text{-th bit of Had}(n)\text{ is }1 \}.$$
There's the obvious $m$ questions you can ask, and since your opponent only lies $m/4 - 1$ times at most, you will be able to decode the codeword to the correct number.


text:
\begin{document}

\title{Painlev\'{e} V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight}
\author{Chao Min and Yang Chen}

\date{\today}

\maketitle
\begin{abstract}
We study the Hankel determinant generated by a singularly perturbed Jacobi weight
$$
w(x,t):=(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;\al>0,\;\;t\geq 0.
$$
If $t=0$, it is reduced to the classical symmetric Jacobi weight. For $t>0$, the factor $\mathrm{e}^{-\frac{t}{x^{2}}}$ induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities.

In the finite $n$ dimensional case, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of $R_n(t)$, where $R_n(t)$ is closely related to a particular Painlev\'{e} V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto $\s$-function of a particular Painlev\'{e} V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=2n^2 t$ is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large $s$ and small $s$ asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.
\end{abstract}

$\mathbf{Keywords}$: Random matrix theory; Hankel determinant; Singularly perturbed Jacobi weight;

Ladder operators; Painlev\'{e} V; Double scaling.

\section{Introduction and Preliminaries}
Random matrices were introduced in nuclear physics by Wigner in the 1950s to describe the statistics of the energy levels of quantum systems. The theory of random matrices makes the hypothesis that the local statistical behavior of the energy levels is identical with that of the eigenvalues of a random matrix.

In random matrix theory (RMT), it is well known that the joint probability density of the eigenvalues $\{x_j\}_{j=1}^{n}$ of $n\times n$ Hermitian matrices in the unitary ensemble is \cite{Mehta}
$$
p(x_1, x_2,\ldots, x_n)dx_1 dx_2\cdots dx_n=\frac{1}{D_n[w]}\frac{1}{n!}\prod_{1\leq j<k\leq n}(x_j-x_k)^2\prod_{l=1}^{n}w(x_l)dx_l,
$$
where $w(x)$ is a weight or probability density supported on the interval $I\subseteq \mathbb{R}$, and we suppose the moments of all orders,
$$
\mu_{j}:=\int_{I}x^{j}w(x)dx,\qquad j=0,1,2,\ldots
$$
exist. In addition, $D_n[w]$ is the normalization constant or the partition function,
$$
D_n[w]=\frac{1}{n!}\int_{I^n}\prod_{1\leq j<k\leq n}(x_j-x_k)^2\prod_{l=1}^{n}w(x_l)dx_l.
$$
As a matter of fact, $D_n[w]$ can be expressed as the determinant of the Hankel or moment matrix \cite[(2.2.11)]{Szego}:
$$
D_{n}[w]=\det\left(\mu_{j+k}\right)_{j,k=0}^{n-1}.
$$

Hankel determinants have been studied extensively over the past few years in part due to connections with RMT \cite{BWW,Charlier,CG,CC,LCF,MLC,Wu,Xu2015,Xu2016}. This is because Hankel determinants compute the most fundamental objects in RMT, such as the partition function of a random matrix ensemble, the probability distribution function of the largest eigenvalue of Hermitian matrices and the moment generating function of linear statistics associated with the ensemble.

In this paper, we consider the Hankel determinant generated by a singularly perturbed Jacobi weight, namely,
$$
D_{n}(t):=\det\left(\int_{-1}^{1}x^{j+k}w(x,t)dx\right)_{j,k=0}^{n-1},
$$
where
\be\label{wei}
w(x,t):=(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;\al>0,\;\;t\geq 0.
\ee
The study of this Hankel determinant is motivated in part by the Wigner time-delay distribution in chaotic cavities \cite{Texier}. In recent years, the asymptotic analysis of orthogonal polynomials and Hankel determinants for the singularly perturbed weights have attracted a lot of interests; see, e.g., \cite{BMM,CCF2019,MLC,Wang,Xu2015,Xu2016}.

Noting that the weight (\ref{wei}) is even, an evaluation of the moments shows that
\bea
\mu_{j}(t):&=&\int_{-1}^{1}x^{j}w(x,t)dx\nonumber\\[5pt]
&=&\left\{
\begin{aligned}
&0,&j=1,3,5,\ldots,\\
&(-1)^{\frac{j}{2}}\pi\left(\frac{\Gamma(1+\al)\Phi\left(-\frac{j+1}{2}-\al,\frac{1-j}{2};-t\right)}{\Gamma\left(\frac{1-j}{2}\right)\Gamma\left(\frac{j+3}{2}+\al\right)}
-\frac{t^{\frac{j+1}{2}}\Phi\left(-\al,\frac{j+3}{2};-t\right)}{\Gamma\left(\frac{j+3}{2}\right)}\right),&j=0,2,4,\ldots.
\end{aligned}
\right.\nonumber
\eea
where $\Phi(\cdot, \cdot; \cdot)$ is the confluent hypergeometric function \cite{Lebedev}.

In addition, for the unperturbed symmetric Jacobi weight, i.e. $t=0$, we have the explicit formula of $D_n(0)$ \cite[(17.6.2)]{Mehta}:
\bea\label{bg}
D_n(0)&=&\frac{2^{n(n+2\al)}}{n!}\prod_{j=1}^{n}\frac{\Gamma(j+1)\Gamma^2(j+\al)}{\Gamma(j+n+2\al)}\nonumber\\
&=&2^{n(n+2\al)}\frac{G(n+1)G(n+1+2\al)G^2(n+\al+1)}{G(2n+2\al+1)G^2(\al+1)},
\eea
where $G(\cdot)$ is the Barnes $G$-function which satisfies the relation \cite{Barnes}
$$
G(z+1)=\Gamma(z)G(z),\qquad\qquad G(1):=1.
$$
See also \cite{Voros} for more properties of this function.

It is well known that $D_n(t)$ can be expressed as the product of the square of the $L^2$ norms of the monic orthogonal polynomials \cite[(2.1.6)]{Ismail}, namely
\be\label{hankel}
D_{n}(t)=\prod_{j=0}^{n-1}h_{j}(t),
\ee
where
\be\label{or}
h_{j}(t)\delta_{jk}=\int_{-1}^{1}P_{j}(x,t)P_{k}(x,t)w(x,t)dx,\qquad j, k=0,1,2,\ldots.
\ee
and $P_{j}(x,t),\; j=0,1,2,\ldots,$ are the monic polynomials of degree $j$ orthogonal with respect to the weight function $w(x,t)$.

The main method of this paper is the ladder operator approach associated with the orthogonal polynomials. This approach has been widely applied to solve the problems on Hankel determinants generated by various weight functions; see \cite{ChenDai,ChenIts,Dai,Min2020,MLC} for reference. So we introduce some elementary facts about the orthogonal polynomials.

In the following discussions, $n$ denotes any nonnegative integers instead of the dimension of the Hermitian matrices. Since the weight $w(x,t)$ is even, $P_{n}(x,t)$ comprises only even or odd powers of $x$ when $n$ is even or odd respectively \cite{Chihara}. That is,
\be\label{expan}
P_{n}(x,t)=x^{n}+\mathrm{p}(n,t)x^{n-2}+\cdots,\qquad n=0,1,2,\ldots,
\ee
and we shall see that $\mathrm{p}(n,t)$, the coefficient of $x^{n-2}$, will play an important role. We set $\mathrm{p}(0,t)=0, \mathrm{p}(1,t)=0$.

The orthogonal polynomials $\{P_{n}(x,t)\}_{n=0}^{\infty}$ satisfy the three-term recurrence relation \cite{Szego}
\be\label{rr}
xP_{n}(x,t)=P_{n+1}(x,t)+\beta_{n}(t)P_{n-1}(x,t)
\ee
with the initial conditions
$$
P_{0}(x,t)=1,\;\;\;\;\beta_{0}(t)P_{-1}(x,t)=0.
$$
From (\ref{or}), (\ref{expan}) and (\ref{rr}), it is easy to see that $\bt_n(t)$ has two alternative representations:
\bea
\beta_{n}(t)&=&\mathrm{p}(n,t)-\mathrm{p}(n+1,t)\label{be1}\\
&=&\frac{h_{n}(t)}{h_{n-1}(t)},\label{be2}
\eea
and a telescopic sum gives
\be\label{sum}
\sum_{j=0}^{n-1}\beta_{j}(t)=-\mathrm{p}(n,t).
\ee
Note that from (\ref{be1}) we have $\bt_0(t)=0$.

The rest of this paper is arranged as follows. In Sec. 2, we apply the ladder operators and its supplementary conditions to the singularly perturbed Jacobi weight, and we obtain the second-order nonlinear difference equations satisfied by the auxiliary quantities $R_n(t)$ and $r_n(t)$, respectively. In Sec. 3, we show that $R_n(t)$ and $r_n(t)$ satisfy the coupled Riccati equations, from which we obtain the second-order differential equations satisfied by $R_n(t)$ and $r_n(t)$, respectively. We find that the equation for $R_n(t)$ can be transformed to a particular Painlev\'{e} V. In Sec. 4, we derive the second-order differential equation and also the difference equation satisfied by the logarithmic derivative of the Hankel determinant. We show that this quantity can be expressed as an integral in terms of $R_n(t)$. In Sec. 5, we study the asymptotic behavior of the Hankel determinant under a suitable double scaling, and the large $s$ and small $s$ asymptotic expansions of the scaled Hankel determinant are established.

\section{Ladder Operators and Second-Order Nonlinear Difference Equations}
In this section, we introduce a pair of ladder operators and the compatibility conditions at first. The information below can be found in Chen and Its \cite{ChenIts}.

Suppose that $w(x)$ is a continuous even weight function defined on $[-1,1]$, and $w(-1)=w(1)=0$. The monic orthogonal polynomials with respect to $w(x)$ satisfy the lowering operator equation
\be\label{lowering}
\left(\frac{d}{dz}+B_{n}(z)\right)P_{n}(z)=\beta_{n}A_{n}(z)P_{n-1}(z)
\ee
and the raising operator equation
\be\label{raising}
\left(\frac{d}{dz}-B_{n}(z)-\mathrm{v}'(z)\right)P_{n-1}(z)=-A_{n-1}(z)P_{n}(z),
\ee
where $\mathrm{v}(z):=-\ln w(z)$, and
\be\label{an}
A_{n}(z):=\frac{1}{h_{n}}\int_{-1}^{1}\frac{\mathrm{v}'(z)-\mathrm{v}'(y)}{z-y}P_{n}^{2}(y)w(y)dy,
\ee
\be\label{bn}
B_{n}(z):=\frac{1}{h_{n-1}}\int_{-1}^{1}\frac{\mathrm{v}'(z)-\mathrm{v}'(y)}{z-y}P_{n}(y)P_{n-1}(y)w(y)dy.
\ee

The functions $A_n(z)$ and $B_n(z)$ are not independent but must satisfy the following supplementary conditions valid for $z\in \mathbb{C}\cup\{\infty\}$:
\be
B_{n+1}(z)+B_{n}(z)=z A_{n}(z)-\mathrm{v}'(z), \tag{$S_{1}$}
\ee
\be
1+z(B_{n+1}(z)-B_{n}(z))=\beta_{n+1}A_{n+1}(z)-\beta_{n}A_{n-1}(z), \tag{$S_{2}$}
\ee
\be
B_{n}^{2}(z)+\mathrm{v}'(z)B_{n}(z)+\sum_{j=0}^{n-1}A_{j}(z)=\beta_{n}A_{n}(z)A_{n-1}(z). \tag{$S_{2}'$}
\ee

Solving for $P_{n-1}(z)$ from (\ref{lowering}) and substituting it into (\ref{raising}), we find that
$y(z):=P_{n}(z)$ satisfies the following second-order linear ordinary differential equation:
$$
y''(z)-\left(\mathrm{v}'(z)+\frac{A_{n}'(z)}{A_{n}(z)}\right)y'(z)+\left(B_{n}'(z)-B_{n}(z)\frac{A_{n}'(z)}{A_{n}(z)}
+\sum_{j=0}^{n-1}A_{j}(z)\right)y(z)=0,
$$
where use has been made of ($S_{2}'$).

In our problem, the weight and associated quantities are
$$
w(x,t)=(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}},
$$
$$
\mathrm{v}(z)=\frac{t}{z^{2}}-\al\ln(1-z^2),\qquad\mathrm{v}'(z)=-\frac{2t}{z^3}+\frac{2\al z}{1-z^2},
$$
$$
\frac{\mathrm{v}'(z)-\mathrm{v}'(y)}{z-y}=\frac{2t(z^2+zy+y^2)}{z^3y^3}+\frac{2\al(1+zy)}{(1-z^2)(1-y^2)}.
$$
In the following discussions, we suppress the $t$ dependence in $w(x), h_n$ and $\bt_n$ for brevity.
\begin{proposition}
For our problem, we have
\be\label{anz}
A_{n}(z)=\frac{R_{n}(t)}{z^2}+\frac{2n+1+2\al+R_{n}(t)}{1-z^2},
\ee
\be\label{bnz}
B_{n}(z)=\frac{\left[1-(-1)^n\right]t}{z^3}+\frac{r_{n}(t)}{z}+\frac{z(n+r_n(t))}{1-z^2},
\ee
where
\be\label{Rnt}
R_{n}(t):=\frac{2t}{h_{n}}\int_{-1}^{1}\frac{P_{n}^{2}(y)w(y)}{y^2} dy,
\ee
\be\label{rnt}
r_{n}(t):=\frac{2t}{h_{n-1}}\int_{-1}^{1}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y^3} dy.
\ee
\end{proposition}

\begin{proof}
According to the definition of $A_{n}(z)$ in (\ref{an}), we get
\bea
A_{n}(z)
&=&\frac{1}{h_{n}}\int_{-1}^{1}\left[\frac{2t(z^2+zy+y^2)}{z^3y^3}+\frac{2\al(1+zy)}{(1-z^2)(1-y^2)}\right]P_{n}^{2}(y)w(y)dy\nonumber\\[5pt]
&=&\frac{2t}{z^2h_{n}}\int_{-1}^{1}\frac{P_{n}^{2}(y)w(y)}{y^2} dy+\frac{2\al}{(1-z^2)h_{n}}\int_{-1}^{1}\frac{P_{n}^{2}(y)w(y)}{1-y^2} dy,\nonumber
\eea
where we have used the parity of the integrand to simplify the result. The above two integrals have a simple relation through integration by parts,
\bea
\frac{2t}{h_{n}}\int_{-1}^{1}\frac{P_{n}^{2}(y)w(y)}{y^2} dy&=&\frac{1}{h_{n}}\int_{-1}^{1}y P_{n}^{2}(y)(1-y^2)^\al d\mathrm{e}^{-\frac{t}{y^{2}}}\nonumber\\
&=&-2n-1+\frac{2\al}{h_{n}}\int_{-1}^{1}y^2P_{n}^{2}(y)(1-y^2)^{\al-1}\mathrm{e}^{-\frac{t}{y^{2}}}dy\nonumber\\
&=&-2n-1-2\al+\frac{2\al}{h_{n}}\int_{-1}^{1}\frac{P_{n}^{2}(y)w(y)}{1-y^2} dy.\nonumber
\eea
That is,
$$
\frac{2\al}{h_{n}}\int_{-1}^{1}\frac{P_{n}^{2}(y)w(y)}{1-y^2} dy=2n+1+2\al+R_n(t).
$$
Then formula (\ref{anz}) follows.

Similarly, from the definition of $B_n(z)$ in (\ref{bn}), we have
\bea
B_{n}(z)&=&\frac{1}{h_{n-1}}\int_{-1}^{1}\left[\frac{2t(z^2+zy+y^2)}{z^3y^3}+\frac{2\al(1+zy)}{(1-z^2)(1-y^2)}\right]P_{n}(y)P_{n-1}(y)w(y)dy\nonumber\\[5pt]
&=&\frac{2t}{z^3h_{n-1}}\int_{-1}^{1}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y} dy+\frac{2t}{zh_{n-1}}\int_{-1}^{1}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y^3} dy\nonumber\\[5pt]
&+&\frac{2\al z}{(1-z^2)h_{n-1}}\int_{-1}^{1}\frac{yP_{n}(y)P_{n-1}(y)w(y)}{1-y^2} dy.\nonumber
\eea
In view of (\ref{or}) and (\ref{expan}), it follows that
\bea\label{gf}
\frac{1}{h_{n-1}}\int_{-1}^{1}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y} dy&=&\left\{
\begin{aligned}
&0,&n=0,2,4,\ldots,\\
&1,&n=1,3,5,\ldots
\end{aligned}
\right.\nonumber\\[5pt]
&=&\frac{1-(-1)^n}{2}.
\eea
By integration by parts, we have
\bea
\frac{2\al}{h_{n-1}}\int_{-1}^{1}\frac{yP_{n}(y)P_{n-1}(y)w(y)}{1-y^2} dy&=&-\frac{1}{h_{n-1}}\int_{-1}^{1}P_{n}(y)P_{n-1}(y)\mathrm{e}^{-\frac{t}{y^2}} d(1-y^2)^\al\nonumber\\[5pt]
&=&\frac{1}{h_{n-1}}\int_{-1}^{1}(1-y^2)^\al \left(P_{n}(y)P_{n-1}(y)\mathrm{e}^{-\frac{t}{y^2}}\right)' dy\nonumber\\[5pt]
&=&n+\frac{2t}{h_{n-1}}\int_{-1}^{1}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y^3} dy\nonumber\\[5pt]
&=&n+r_n(t).\nonumber
\eea
Then we arrive at formula (\ref{bnz}). This completes the proof.
\end{proof}

Substituting (\ref{anz}) and (\ref{bnz}) into ($S_{1}$), we find
\be\label{s1}
R_{n}(t)=r_{n+1}(t)+r_{n}(t).
\ee
Similarly, substituting (\ref{anz}) and (\ref{bnz}) into ($S_{2}$) gives the following two equations:
\be\label{s21}
\bt_{n+1}R_{n+1}(t)-\bt_{n}R_{n-1}(t)=2(-1)^nt,
\ee
\be\label{s22}
\bt_{n+1}(2n+3+2\al+R_{n+1}(t))-\bt_{n}(2n-1+2\al+R_{n-1}(t))=1+r_{n+1}(t)-r_n(t).
\ee
The combination of (\ref{s21}) and (\ref{s22}) shows
$$
2(-1)^nt+(2n+3+2\al)\bt_{n+1}-(2n-1+2\al)\bt_{n}=1+r_{n+1}(t)-r_n(t),
$$
i.e.,
$$
2(-1)^nt+\left[(2n+1+2\al)\bt_{n+1}-(2n-1+2\al)\bt_{n}\right]+2\bt_{n+1}=1+r_{n+1}(t)-r_n(t).
$$
A telescopic sum gives
\be\label{s2}
\left[1-(-1)^n\right]t+(2n+1+2\al)\bt_n-2\mathrm{p}(n,t)=n+r_n(t),
\ee
where we have used (\ref{sum}) and the initial conditions $\bt_0=0, r_0(t)=0$.

Finally, from ($S_{2}'$), we obtain the following three equations:
\be\label{s2p1}
-2(-1)^n t\: r_{n}(t)=\beta_{n}R_{n}(t)R_{n-1}(t),
\ee
\be\label{s2p2}
(n+r_{n}(t))^2+2\al(n+r_{n}(t))=\beta_{n}\left(2n+1+2\al+R_{n}(t)\right)\left(2n-1+2\al+R_{n-1}(t)\right),
\ee
\bea\label{s2p3}
&&r_{n}^2(t)-2(-1)^n t(n+r_{n}(t))+2\al\left[1-(-1)^n\right]t+\sum_{j=0}^{n-1}R_{j}(t)\nonumber\\
&=&\beta_{n}\left(2n+1+2\al+R_{n}(t)\right)R_{n-1}(t)+\beta_{n}R_{n}(t)\left(2n-1+2\al+R_{n-1}(t)\right).
\eea
The combination of (\ref{s2p1}) and (\ref{s2p2}) gives an important formula:
\bea\label{imp}
&&(2n+1+2\al)\bt_nR_{n-1}(t)+(2n-1+2\al)\bt_nR_{n}(t)\nonumber\\
&=&(n+r_{n}(t))^2+2\al(n+r_{n}(t))+2(-1)^n t\: r_n(t)-(2n+1+2\al)(2n-1+2\al)\bt_n.
\eea
Furthermore, substituting $\bt_nR_{n-1}(t)=-2(-1)^n t\: r_{n}(t)/R_n(t)$ into (\ref{imp}), we obtain the expression of $\bt_n$ in terms of $R_n(t)$ and $r_n(t)$:
\be\label{be3}
\bt_n=\frac{(n+r_{n}(t))^2+2\al(n+r_{n}(t))}{(2n-1+2\al)(2n+1+2\al+R_{n}(t))}+\frac{2(-1)^n t\: r_{n}(t)}{(2n-1+2\al)R_{n}(t)}.
\ee
\begin{theorem}
The auxiliary quantities $r_n(t)$ and $R_n(t)$ satisfy the following second-order difference equations:
\bea\label{rn}
&&(n+r_{n})(n+2\al+r_{n})(r_{n+1}+r_{n})(r_n+r_{n-1})\nonumber\\
&+&2(-1)^n t\: r_{n}(2n-1+2\al+r_n+r_{n-1})(2n+1+2\al+r_{n+1}+r_{n})=0,
\eea\\[-40pt]
\begin{small}
\bea\label{Rn}
&&\Big\{\big[R_{n+1}R_n^4+ 2\big((2 n+2 \alpha+1) R_{n+1}- (-1)^nt(2 n+2 \alpha+3)\big)R_n^3+\big( (4 n^2+4n(2 \alpha +1)\nonumber\\
&-&2 t^2+2 (-1)^n t+2 \alpha ^2+4 \alpha+1)R_{n+1}-2 t (2 n+2 \alpha+3) (t+(-1)^n(3n+3 \alpha +1))\big)R_n^2\nonumber\\
&+&2 t (2 n+2 \alpha+1)\big( ((-1)^n-2 t)R_{n+1}-(2 n+2 \alpha+3) (2 t+(-1)^n (n+\al))\big) R_n\nonumber\\
&-&2 t^2 (2 n+2 \alpha+1)^2 (2n+2\al+3+R_{n+1})\big]R_{n-1}+2 t (2 n+2 \alpha-1) (2n+2 \alpha+1 +R_n)\nonumber\\
&\times&\big[(-1)^n R_{n+1}R_n^2+ \big( ((-1)^n (n+\al+1)-t)R_{n+1}-t (2 n+2 \alpha+3)\big)R_n-t (2 n+2 \alpha+1)\nonumber\\
&\times&(2n+2 \alpha+3 +R_{n+1})\big]\Big\}^2=4\Big\{\big[(-1)^n t (2 n+2 \alpha-1) (2n+2 \alpha+1 +R_n)\nonumber\\
&+& \big( (n+\alpha +(-1)^n t)R_n+(-1)^n t (2 n+2 \alpha+1)\big)R_{n-1}\big]^2- n (n+2 \alpha) R_{n-1}^2 R_n^2\Big\}\nonumber\\
&\times&\Big\{\big[(-1)^n t (2 n+2 \alpha+1) (2n+2 \alpha+3 +R_{n+1})- \big( (n+\alpha+1 -(-1)^n t)R_{n+1}\nonumber\\
&-&(-1)^n t (2 n+2 \alpha+3)\big)R_{n}\big]^2-(n+1) (n+2 \alpha+1) R_n^2 R_{n+1}^2\Big\}.
\eea
\end{small}
The initial conditions are given by
$$
r_0(t)=0,\\[10pt]
$$
\be\label{ic1}
r_1(t)=R_0(t)=\frac{2\sqrt{t}\left[\Gamma\left(\frac{3}{2}+\al\right)\Phi\left(-\al,\frac{1}{2};-t\right)-(1+2\al)\sqrt{t}\:\Gamma(1+\al)\Phi\left(\frac{1}{2}-\al,\frac{3}{2};-t\right)
\right]}{\Gamma(1+\al)\Phi\left(-\frac{1}{2}-\al,\frac{1}{2};-t\right)-2\sqrt{t}\:\Gamma\left(\frac{3}{2}+\al\right)\Phi\left(-\al,\frac{3}{2};-t\right)},\\[10pt]
\ee
\be\label{ic2}
R_1(t)=\frac{6t\left[(3+2\al)\Gamma(1+\al)\Phi\left(-\frac{1}{2}-\al,\frac{1}{2};-t\right)-4\sqrt{t}\:\Gamma\left(\frac{5}{2}+\al\right)\Phi\left(-\al,\frac{3}{2};-t\right)
\right]}{3\Gamma(1+\al)\Phi\left(-\frac{3}{2}-\al,-\frac{1}{2};-t\right)+8t\sqrt{t}\:\Gamma\left(\frac{5}{2}+\al\right)\Phi\left(-\al,\frac{5}{2};-t\right)},
\ee
where $\Phi(\cdot, \cdot; \cdot)$ is the confluent hypergeometric function.
\end{theorem}
\begin{proof}
Substituting (\ref{s1}) into (\ref{s2p1}), we have
\be\label{d1}
\bt_n=-\frac{2(-1)^n t\: r_{n}(t)}{(r_{n+1}(t)+r_{n}(t))(r_{n}(t)+r_{n-1}(t))}.
\ee
The combination of (\ref{d1}) and (\ref{be3}) gives (\ref{rn}) with the aid of (\ref{s1}). On the other hand, eliminating $\bt_n$ from (\ref{s2p1}) and (\ref{s2p2}), we find
\bea
&&\left[(n+r_{n}(t))^2+2\al(n+r_{n}(t))\right]R_{n}(t)R_{n-1}(t)\nonumber\\
&+&2(-1)^n t\: r_n(t)\left(2n+1+2\al+R_{n}(t)\right)\left(2n-1+2\al+R_{n-1}(t)\right)=0.\nonumber
\eea
This equation can be viewed as the qradratic equation for $r_n(t)$. Substituting either solution into (\ref{s1}) and after clearing the square root, we obtain (\ref{Rn}). The initial conditions follow from straightforward computations of (\ref{Rnt}) and (\ref{rnt}) when $n=0$ and 1.
\end{proof}

\section{Coupled Riccati Equations, Painlev\'{e} V}
In this section, we derive the coupled Riccati equations satisfied by the auxiliary quantities $R_n(t)$ and $r_n(t)$ from taking derivatives in the orthogonality conditions. Then we obtain the second-order differential equations for $R_n(t)$ and $r_n(t)$, respectively. We find that $R_n(t)$ is closely related to a particular Painlev\'{e} V transcendent.

We start from taking a derivative with respect to $t$ in the equality
$$
\int_{-1}^{1}P_{n}^2(x,t)(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}}dx=h_{n}(t),\;\;n=0,1,2,\ldots,
$$
and have
$$
h_n'(t)=-\int_{-1}^{1}\frac{P_{n}^2(x,t)w(x)}{x^2}dx.
$$
It follows that
\be\label{eq1}
2t \frac{d}{dt}\ln h_{n}(t)=-R_{n}(t).
\ee
In view of (\ref{be2}), we find
$$
2t\frac{d}{dt}\ln\beta_{n}(t)=R_{n-1}(t)-R_{n}(t).
$$
Then
\be\label{eq2}
2t\beta_{n}'(t)=\beta_{n}R_{n-1}(t)-\beta_{n}R_{n}(t).
\ee

On the other hand, taking a derivative in the equation
$$
\int_{-1}^{1}P_{n}(x,t)P_{n-2}(x,t)(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}}dx=0,\;\;n=1,2,\ldots,
$$
we have
\be\label{pn}
\frac{d}{dt}\mathrm{p}(n,t)=\frac{1}{h_{n-2}}\int_{-1}^{1}\frac{P_{n}(x,t)P_{n-2}(x,t)w(x)}{x^2}dx.
\ee
From the three-term recurrence relation (\ref{rr}) and exploiting (\ref{be2}), we get
\be\label{rr1}
\frac{P_{n-2}(x,t)}{h_{n-2}}=\frac{xP_{n-1}(x,t)}{h_{n-1}}-\frac{P_{n}(x,t)}{h_{n-1}}.
\ee
Substituting (\ref{rr1}) into (\ref{pn}), we obtain
\be\label{pnt}
2t\frac{d}{dt}\mathrm{p}(n,t)=\left[1-(-1)^n\right]t-\beta_{n}R_{n}(t),
\ee
where we have used formula (\ref{gf}).

\begin{proposition} The auxiliary quantities $r_n(t)$ and $R_n(t)$ satisfy the coupled Riccati equations:
\be\label{ricca1}
2t\:r_{n}'(t)=-\frac{2(-1)^nt\:r_n(t) (2n+1+2\al+R_n(t))}{R_{n}(t)}-\frac{(n+r_{n}(t))(n+2\al+r_{n}(t))R_{n}(t)}{2n+1+2\al+R_{n}(t)},
\ee
\be\label{ricca2}
2tR_{n}'(t)=R_{n}^2(t)+\left[1-2(-1)^n t-2r_{n}(t)\right]R_{n}(t)-2(-1)^n(2n+2\al+1) t.
\ee
\end{proposition}

\begin{proof}
By taking a derivative in (\ref{s2}) and with the aid of (\ref{pnt}), we have
\be\label{for}
2t\: r_n'(t)=2(2n+1+2\al)t\bt_n'(t)+2\bt_nR_n(t).
\ee
Replacing $2t\bt_n'(t)$ by (\ref{eq2}), equation (\ref{for}) becomes
\bea
2t\:r_n'(t)&=&(2n+1+2\al)\bt_nR_{n-1}(t)-(2n-1+2\al)\bt_nR_{n}(t)\nonumber\\
&=&-\frac{2(-1)^n (2n+1+2\al)t\:r_n(t)}{R_{n}(t)}-(2n-1+2\al)\bt_nR_{n}(t).\nonumber
\eea
where we have used (\ref{s2p1}) in the second step. Substituting the expression of $\bt_n$ in (\ref{be3}) into the above equality, we obtain (\ref{ricca1}).

On the other hand, substituting (\ref{be3}) into (\ref{for}) to eliminate $\bt_n$ and $\bt_n'(t)$, and then replacing $2t\: r_n'(t)$ by (\ref{ricca1}), we find
\bea
&&\Big\{2tR_{n}'(t)-R_{n}^2(t)-\left[1-2(-1)^n t-2r_{n}(t)\right]R_{n}(t)+2(-1)^n(2n+2\al+1) t\Big\}\nonumber\\
&\times&\Big\{ \left[r_n^2(t)+2  \left(n+\alpha +(-1)^n t\right)r_n(t)+n (n+2 \alpha)\right]R_n^2(t)+4 (-1)^n t (2 n+2 \alpha +1) r_n(t) R_n(t)\nonumber\\
&&+2 (-1)^n t (2 n+2 \alpha +1)^2 r_n(t)\Big\}=0.\nonumber
\eea
Obviously, the above formula yields two equations. The latter algebraic equation does not hold and should be discarded, which can be verified by taking special values such as $n=1$. Hence we obtain (\ref{ricca2}). The proof is complete.
\end{proof}

\begin{theorem}
The auxiliary quantity $R_n(t)$ satisfies the following second-order nonlinear ordinary differential equation,
\bea\label{ode}
&&8 t^2 R_n(t)  (2 n+2 \alpha+1+R_n(t))R_n''(t)-4 t^2  (4 n+4 \alpha +2+3 R_n(t))(R_n'(t))^2\nonumber\\
&+&8 t  (2 n+2 \alpha +1+R_n(t))R_n(t) R_n'(t)-R_n^5(t)-2 (2 n+2 \alpha+1) R_n^4(t)\nonumber\\
&-& \left[4 n^2+4(2 \alpha +1) n+4 \alpha +1-4 t^2-4 (-1)^n t\right]R_n^3(t)+8 t (2 n+2 \alpha +1) \left[(-1)^n+2 t\right] R_n^2(t)\nonumber\\
&+&4 t (2 n+2 \alpha +1)^2 \left[(-1)^n+5 t\right] R_n(t)+8 t^2 (2 n+2 \alpha +1)^3=0.
\eea
Let $S_n(t):=1+\frac{R_{n}(t)}{2n+2\al+1}$. Then $S_n(t)$ satisfies a particular Painlev\'{e} V equation,
\bea\label{pv}
S_{n}''(t)&=&\frac{(3 S_n(t)-1) (S_n'(t))^2}{2S_n(t) (S_n(t)-1) }-\frac{S_n'(t)}{t}+\frac{(S_n(t)-1)^2 }{t^2}\left(\frac{(2n+2 \alpha+1)^2 S_n(t)}{8} -\frac{\al^2}{2S_n(t)}\right)\nonumber\\[10pt]
&-&\frac{(-1)^n S_n(t)}{2t}-\frac{S_n(t)(S_n(t)+1) }{2(S_n(t)-1)},
\eea
i.e. $P_{V}\left(\frac{(2n+2 \alpha+1)^2}{8}, -\frac{\al^2}{2}, -\frac{(-1)^n}{2}, -\frac{1}{2}\right)$, following the convention of \cite{Gromak}.
\end{theorem}
\begin{proof}
Note that equation (\ref{ricca2}) can be viewed as a linear equation for $r_n(t)$. Solving $r_{n}(t)$ from (\ref{ricca2}) we have
\be\label{rexp}
r_n(t)=\frac{1-2 (-1)^n t}{2}+\frac{R_n(t)}{2}-\frac{t \left[(-1)^n (2 n+1+2 \alpha)+R_n'(t)\right]}{R_n(t)}.
\ee
Substituting it into (\ref{ricca1}), we obtain (\ref{ode}). With the linear transformation $S_n(t)=1+\frac{R_{n}(t)}{2n+2\al+1}$, we obtain the Painlev\'{e} V equation (\ref{pv}).
\end{proof}

\begin{theorem}
The quantity $r_n(t)$ satisfies the second-order nonlinear differential equation:
\begin{small}
\bea\label{ode2}
&&4 t^3 (r_n''(t))^2+4t^2\big[ r_n'(t)-2(-1)^n \left(3r_n^2(t)+4(n+\alpha) r_n(t)+n  (n+2 \alpha)\right)\big]r_n''(t)\nonumber\\
&-&t\big[4r_n^2(t)-8 (-1)^n t r_n(t)+4 t^2-1\big] (r_n'(t))^2-4(-1)^nt\big[3r_n^2(t)+4   (n+\alpha)r_n(t)+ n  (n+2 \alpha)\big]r_n'(t)\nonumber\\
&+&8 (-1)^n r_n^5(t)+4  \big[4 (-1)^n (n+\al)+5 t\big]r_n^4(t)+8\big[(-1)^n (n^2+2n\alpha+t^2)+8 t (n+\alpha)\big]r_n^3(t)\nonumber\\
&+&8t\big[2(-1)^nt(n+\al)+(3n+2\al)(3n+4\al)\big]r_n^2(t)+8 n (n+2 \alpha)t \big[4 n+4 \alpha +(-1)^n t\big]r_n(t)\nonumber\\
&+&4 n^2 (n+2 \alpha)^2t=0.
\eea
\end{small}
\end{theorem}
\begin{proof}
Solving $R_n(t)$ from (\ref{ricca1}), which is actually a quadratic equation for $R_n(t)$, we get two solutions. Substituting either solution into (\ref{ricca2}), and clearing the square root, we obtain (\ref{ode2}) after making some simplifications.
\end{proof}

\begin{remark}
The second-order differential equation (\ref{ode2}) for $r_n(t)$ may be transformed to a Chazy type equation \cite{Chazy1909,Chazy1911,Cosgrove}. See \cite{Min2019,Lyu2017} on Chazy equations satisfied by $r_n(t)$ in other problems.
\end{remark}

\section{Logarithmic Derivative of the Hankel Determinant}
In this section, we study the logarithmic derivative of the Hankel determinant. Usually this quantity satisfies a second-order differential equation, which is related to the Jimbo-Miwa-Okamoto $\s$-form of a particular Painlev\'{e} equation.

We introduce a quantity related to the logarithmic derivative of the Hankel determinant,
\be\label{def}
\s_n(t):=2t\frac{d}{dt}\ln D_n(t).
\ee
In view of (\ref{hankel}) and (\ref{eq1}), we find
\be\label{sn}
\s_n(t)=-\sum_{j=0}^{n-1}R_j(t).
\ee
Then equation (\ref{s2p3}) becomes
\be\label{re}
(2n+1+2\al)\bt_nR_{n-1}(t)+(2n-1+2\al)\bt_nR_{n}(t)=2\al t-2(-1)^n t(n+\al-r_n(t))+r_n^2(t)-\s_n(t),
\ee
where we have used (\ref{s2p1}).

The combination of (\ref{imp}) and (\ref{re}) shows that $\bt_n$ can be expressed in terms of $r_n(t)$ and $\s_n(t)$:
\be\label{be4}
\bt_n=\frac{n(n+2\al)+2(n+\al)r_n(t)+\s_n(t)+2t\left[(-1)^n(n+\al)-\al\right]}{(2n+1+2\al)(2n-1+2\al)}.
\ee
By taking a derivative in (\ref{be4}) and substituting it into (\ref{eq2}), we find
\be\label{re2}
\beta_{n}R_{n-1}(t)-\beta_{n}R_{n}(t)=\frac{2 t \left[2(n+\al)((-1)^n+r_n'(t))+\sigma_n'(t)-2 \alpha\right]}{(2n+1+2\al)(2n-1+2\al)}.
\ee

\begin{theorem}
The quantity $\s_n(t)$ satisfies the following second-order nonlinear ordinary differential equation:
\begin{small}
\bea\label{sode}
&&\Big\{4 t^3 (\sigma _n'')^2- 4 t^2\left[ 2t+2\alpha -2 (-1)^n(n+\al)-\sigma _n'\right]\sigma _n''+8 t^2 (\sigma _n')^3-t\big[4 t^2+40 \alpha\:  t-1+4 \sigma _n\nonumber\\
&+&24 (-1)^n t (n+\alpha)\big](\sigma _n')^2+4t\big[12 \alpha \: t^2 -(20 n^2+40 n\alpha+3) t-\alpha+(-1)^n(n+\al) (12 t^2+1)\nonumber\\
&+&4  \left(\alpha-t +3 (-1)^n(n+\al)\right)\sigma _n\big]\sigma _n'+8 [t-2 (-1)^n(n+\al)]\sigma _n^2+4t\big[2 t^2+1+14 n^2+28   n\alpha\nonumber\\
&+&8 \alpha ^2-4(-1)^n (n+\alpha) (3 t+2 \alpha)\big]\sigma _n-4t\big[4 \alpha\:  t^3- 2t^2(7 n^2+14   n\al+1)-4 \alpha t (3 n^2+6   n\alpha+1)\nonumber\\
&-&n^2-2   n\alpha-2 \alpha ^2+2(-1)^n\left(2t^3 ( n+\al)+2\left(3 n^3+9   n^2\alpha+ n(6 \alpha ^2+1)+\alpha\right)t +n\alpha  +\alpha ^2\right)\big]\Big\}^2\nonumber\\
&=&16 \Big\{t[t+2 \alpha-2 \sigma _n' -2 (-1)^n(n+\al)]+\sigma _n\Big\}\Big\{2 (-1)^n t^2 \sigma _n''+t\big[(-1)^n(4 n^2+8n\al-8\alpha t+1+4 \sigma _n)\nonumber\\
&-&8t(n+\al)\big]\sigma _n'-2 (-1)^n \sigma _n^2+2\left[4t(n+\al)-(-1)^n(n^2+2 n\alpha  +t^2)\right]\sigma _n+2t\big[(n+\alpha )(2 n^2+4  n\alpha\nonumber\\
&+&2 t^2+1)+(-1)^n\left(2 \alpha\: t^2-(5 n^2+10   n\alpha+1)t-\alpha  (2 n^2+4   n\alpha+1)\right)\big]\Big\}^2.
\eea
\end{small}
\end{theorem}
\begin{proof}
We define
$$
X:=\bt_nR_{n-1}(t),
$$
$$
Y:=\bt_nR_{n}(t).
$$
Then $X$ and $Y$ satisfy a linear system from (\ref{re}) and (\ref{re2}):
$$
(2n+1+2\al)X+(2n-1+2\al)Y=2\al t-2(-1)^n t(n+\al-r_n(t))+r_n^2(t)-\s_n(t),
$$
$$
X-Y=\frac{2 t \left[2(n+\al)((-1)^n+r_n'(t))+\sigma_n'(t)-2 \alpha\right]}{(2n+1+2\al)(2n-1+2\al)}.
$$
Solving this linear system, we get
\bea\label{x}
X&=&\frac{2 \alpha  t-2 (-1)^n t (n+\alpha-r_n(t))+r_n^2(t)-\sigma_n (t)}{4(n+\al)}\nonumber\\
&+&\frac{t\left[2(n+\al)\left((-1)^n+r_n'(t)\right)+\s_n'(t)-2\al\right]}{2(n+\al)(2n+1+2\al)},
\eea
\bea\label{y}
Y&=&\frac{2 (-1)^n t\: r_n(t)+r_n^2(t)-\sigma_n (t)}{4 (n+\alpha)}\nonumber\\
&-&\frac{ t \left[(2 n+1+2 \alpha) \left((-1)^n (n+\al)-\al\right)+2 (n+\alpha) r_n'(t)+\sigma_n'(t)\right]}{2 (n+\alpha) (2 n-1+2 \alpha)}.
\eea
Noting that from the definition of $X$ and $Y$, we have
\bea\label{xy}
X\cdot Y&=&\bt_n(\bt_nR_{n}(t)R_{n-1}(t))\nonumber\\
&=&-2(-1)^n t\bt_n r_{n}(t).
\eea
where we have used (\ref{s2p1}).

Substituting (\ref{x}), (\ref{y}) and (\ref{be4}) into (\ref{xy}), we obtain an equation satisfied by $r_n(t), r_n'(t), \s_n(t)$ and $\s_n'(t)$:
\begin{small}
\bea\label{com}
&&\big\{(2n+1+2\al)\left[2 \alpha  t-2 (-1)^n t (n+\alpha-r_n(t))+r_n^2(t)-\sigma_n (t)\right]\nonumber\\
&+&2t\left[2(n+\al)\left((-1)^n+r_n'(t)\right)+\s_n'(t)-2\al\right]\big\}\big\{(2n-1+2\al)\left[2 (-1)^n t\: r_n(t)+r_n^2(t)-\sigma_n (t)\right]\nonumber\\
&-&2t \left[(2 n+1+2 \alpha) \left((-1)^n (n+\al)-\al\right)+2 (n+\alpha) r_n'(t)+\sigma_n'(t)\right]\big\}\nonumber\\
&=&-32(-1)^n(n+\al)^2 t\:r_n(t)\big\{n(n+2\al)+2(n+\al)r_n(t)+\s_n(t)+2t\left[(-1)^n(n+\al)-\al\right]\big\}.
\eea
\end{small}\\[-40pt]

On the other hand, the combination of (\ref{s2}) and (\ref{be4}) shows that $\mathrm{p}(n,t)$ can be expressed in terms of $r_n(t)$ and $\s_n(t)$:
$$
\mathrm{p}(n,t)=\frac{n-n^2+\left[2 n-1+(-1)^n\right] t+r_n(t)+\sigma_n (t)}{2 (2 n-1+2 \alpha)}.
$$
Taking a derivative and substituting it into (\ref{pnt}), and replacing $\bt_n R_n(t)$ by (\ref{y}), we find a quadratic equation for $r_n(t)$ (the terms involving $r_n'(t)$ disappear!):
$$
r_n^2(t)+2 (-1)^n t\: r_n(t)-\sigma_n (t)+2 t \left[(-1)^n(n+\al)-\alpha +\sigma_n'(t)\right]=0.
$$
Then we have two solutions,
$$
r_n(t)=-(-1)^n t\pm\sqrt{\sigma_n(t)+t \left[t+2 \alpha -2 (-1)^n(n+\al)-2 \sigma_n '(t)\right]}.
$$
Substituting either solution into (\ref{com}), we obtain (\ref{sode}) after clearing the square root.
\end{proof}

Although the differential equation (\ref{sode}) for $\s_n(t)$ is very complicated, we find that it will be reduced to a very simple form under a suitable double scaling. Furthermore, we will show that $\s_n(t)$ can be expressed in terms of the Jimbo-Miwa-Okamoto $\s$-function of a particular Painlev\'{e} V in next section; see (\ref{rs1}) and (\ref{rs2}).
\begin{corollary}
If $n\rightarrow\infty$ and $t\rightarrow 0$ in the way such that $\lim\limits_{n\rightarrow\infty}n^4 t=s>0$ and we assume the limit
$$
\s(s):=\lim_{n\rightarrow\infty}\s_n\left(\frac{s}{n^4}\right)
$$
exists. Then $\sigma(s)$ satisfies the following second-order differential equation
\be\label{sf}
4s^2 (\sigma''(s))^2+4s\sigma'(s)\sigma''(s)+8s(\sigma'(s))^3-4\sigma(s)(\sigma'(s))^2+(\sigma'(s))^2=0.
\ee
\end{corollary}
\begin{proof}
With the change of variables $t\mapsto s/n^{4}, \s_n(t)\mapsto \s(s)$, equation (\ref{sode}) turns into
\bea
&&s\left[4s^2 (\sigma''(s))^2+4s\sigma'(s)\sigma''(s)+8s(\sigma'(s))^3-4\sigma(s)(\sigma'(s))^2+(\sigma'(s))^2\right]^2\nonumber\\
&+&\frac{8(-1)^n}{n^3}\Big\{8 s^4 (\sigma ''(s))^3-4 s^2 \left[6 s^2 (\sigma '(s))^2-12 s \sigma (s) \sigma '(s)-3 s \sigma '(s)+4 \sigma^2 (s)\right](\sigma ''(s))^2\nonumber\\
&-&2 s \sigma '(s) \left[4 s^2 (\sigma '(s))^2-20 s \sigma (s) \sigma '(s)-3 s \sigma '(s)+8 \sigma^2 (s)\right]\sigma ''(s)-48 s^3 (\sigma '(s))^5\nonumber\\
&+&2 s^2\left(60 \sigma (s)+1\right) (\sigma '(s))^4-s\left(80 \sigma^2 (s)-8 \sigma (s)-1\right) (\sigma '(s))^3\nonumber\\
&+&4 \sigma^2 (s)(4 \sigma (s)-1) (\sigma '(s))^2\Big\}+O\left(\frac{1}{n^4}\right)=0.\nonumber
\eea
Let $n\rightarrow\infty$, then we obtain (\ref{sf}) by retaining only the highest order terms.
\end{proof}

\begin{remark}
We also find the second-order differential equation (\ref{sf}) in the singular perturbed Gaussian weight problem under a suitable double scaling, and we call this equation the $\s$-form of Painlev\'{e} III\:$'$; see Theorem 3.3 in \cite{MLC}. This phenomenon is termed universality in random matrix theory \cite{Deift,Kui}. In addition, Forrester and Witte \cite{FW2002,FW2006} discovered that the generating function for the probability that an interval contains $k$ eigenvalues in the hard edge scaling limit of the Laguerre unitary ensemble can be evaluated in terms of another $\s$-form of Painlev\'{e} III\:$'$.
\end{remark}

\begin{theorem}
The quantity $\s_n(t)$ satisfies the following second-order nonlinear difference equation:
\bea\label{sd}
&&\left\{n-\frac{(\sigma _{n-1}-\sigma _n)(\sigma _n-\sigma _{n+1})\left[2 \alpha  (n-t)+2 (-1)^n t (n+\alpha)+n^2+\s_n\right]}{2 \left[(-1)^n t (2n+1+2\al)(2n-1+2\al)+(n+\alpha) (\sigma _{n-1}-\sigma _n) (\sigma _n-\sigma _{n+1})\right]}\right\}\nonumber\\[10pt]
&\times&\left\{n+2\al-\frac{(\sigma _{n-1}-\sigma _n)(\sigma _n-\sigma _{n+1})\left[2 \alpha  (n-t)+2 (-1)^n t (n+\alpha)+n^2+\s_n\right]}{2 \left[(-1)^n t (2n+1+2\al)(2n-1+2\al)+(n+\alpha) (\sigma _{n-1}-\sigma _n) (\sigma _n-\sigma _{n+1})\right]}\right\}\nonumber\\[10pt]
&=& t(2n-1+2\al+\sigma _{n-1}-\sigma _n)(2n+1+2\al+\sigma _{n}-\sigma _{n+1})\nonumber\\[10pt]
&\times&\frac{2 \alpha  (n-t)+2 (-1)^n t (n+\alpha)+n^2+\s_n}{ t (2n+1+2\al)(2n-1+2\al)+(-1)^n(n+\alpha) (\sigma _{n-1}-\sigma _n) (\sigma _n-\sigma _{n+1})},
\eea
with the initial conditions
$$
\s_1(t)=-R_0(t),\qquad \s_2(t)=-(R_0(t)+R_1(t))
$$
and $R_0(t)$ and $R_1(t)$ are evaluated in (\ref{ic1}) and (\ref{ic2}).
\end{theorem}
\begin{proof}
From (\ref{sn}) we have
$$
R_n(t)=\s_n(t)-\s_{n+1}(t).
$$
Then (\ref{s2p1}) and (\ref{be3}) becomes
\be\label{eq3}
-2(-1)^n t\: r_{n}(t)=\beta_{n}(\s_n(t)-\s_{n+1}(t))(\s_{n-1}(t)-\s_{n}(t))
\ee
and
\be\label{eq4}
\bt_n=\frac{(n+r_{n}(t))^2+2\al(n+r_{n}(t))}{(2n-1+2\al)(2n+1+2\al+\s_n(t)-\s_{n+1}(t))}+\frac{2(-1)^n t\: r_{n}(t)}{(2n-1+2\al)(\s_n(t)-\s_{n+1}(t))},
\ee
respectively.
Substituting (\ref{be4}) into (\ref{eq3}), we get a linear equation for $r_n(t)$:
\bea
&&-2(-1)^n(2n+1+2\al)(2n-1+2\al) t\: r_{n}(t)\nonumber\\
&=&\left[n(n+2\al)+2(n+\al)r_n(t)+\s_n(t)+2t\left((-1)^n(n+\al)-\al\right)\right]\nonumber\\
&\times&(\s_n(t)-\s_{n+1}(t))(\s_{n-1}(t)-\s_{n}(t)).\nonumber
\eea
So $r_n(t)$ can be expressed in terms of $\s_n(t), \s_{n+1}(t)$ and $\s_{n-1}(t)$:
\begin{small}
\be\label{rnsn}
r_n(t)=-\frac{(\sigma _{n-1}(t)-\sigma _n(t))(\sigma _n(t)-\sigma _{n+1}(t))\left[2 \alpha  (n-t)+2 (-1)^n t (n+\alpha)+n^2+\s_n(t)\right]}{2 \left[(-1)^n t (2n+1+2\al)(2n-1+2\al)+(n+\alpha) (\sigma _{n-1}(t)-\sigma _n(t)) (\sigma _n(t)-\sigma _{n+1}(t))\right]}.
\ee
\end{small}
Substituting (\ref{eq4}) into (\ref{eq3}), and using (\ref{rnsn}) to eliminate $r_n(t)$, we obtain (\ref{sd}). The initial conditions come from formula (\ref{sn}).
\end{proof}

In the next theorem, we display the integral representation of $D_n(t)$ in terms of $R_n(t)$.
\begin{theorem}
\bea
\ln\frac{ D_n(t)}{D_{n}(0)}&=&\int_{0}^{t}\big[4 s^2 (2 n+1+2 \alpha) (R_n'(s))^2-4s R_n(s) (2 n +1+2 \alpha+R_n(s))R_n'(s)\nonumber\\
&-&( 2 n-1+2 \alpha) R_n^4(s)-2 \left(2 n^2+4 \alpha ( n-  s)-1\right)R_n^3(s)\nonumber\\
&-&(2 n+1+2 \alpha)  (4 s^2-8 \alpha  s-1)R_n^2(s)-8 s^2 (2 n+1+2 \alpha)^2 R_n(s)\nonumber\\
&-&4 s^2 (2 n+1+2 \alpha)^3\big]\frac{ds}{8s R_n^2(s) (2 n +1+2 \alpha+R_n(s))}.\nonumber
\eea
\end{theorem}
\begin{proof}
The combination of (\ref{be3}) and (\ref{be4}) shows that $\s_n(t)$ can be expressed in terms of $R_n(t)$ and $r_n(t)$:
\bea\label{sr}
\s_n(t)&=&-n^2-2\al(n-t)-2(n+\al)((-1)^n t+r_n(t))\nonumber\\
&+&(2n+1+2\al)\left[\frac{2 (-1)^n t\: r_n(t)}{R_n(t)}+\frac{(n+r_n(t)) (n+2 \alpha +r_n(t))}{2 n+1+2 \alpha +R_n(t)}\right].
\eea
Substituting (\ref{rexp}) into (\ref{sr}), we have
\bea
\s_n(t)&=&\frac{1}{4 R_n^2(t) (2 n +1+2 \alpha+R_n(t))}\big[4 t^2 (2 n+1+2 \alpha) (R_n'(t))^2\nonumber\\
&-&4 t R_n(t) (2 n +1+2 \alpha+R_n(t))R_n'(t)-( 2 n-1+2 \alpha) R_n^4(t)\nonumber\\
&-&2 \left(2 n^2+4 \alpha ( n-  t)-1\right)R_n^3(t)-(2 n+1+2 \alpha)  (4 t^2-8 \alpha  t-1)R_n^2(t)\nonumber\\
&-&8 t^2 (2 n+1+2 \alpha)^2 R_n(t)-4 t^2 (2 n+1+2 \alpha)^3\big].\nonumber
\eea
In view of the definition of $\s_n(t)$ in (\ref{def}), we have the desired result. Note that $D_n(0)$ is explicitly given by (\ref{bg}).
\end{proof}
\begin{remark}\label{rep}
In random matrix theory, the integral representation of $D_n(t)$ in terms of $R_n(t)$ is very important in the asymptotic analysis of the Hankel determinant. This is because $R_n(t)$ is usually related to a particular Painlev\'{e} transcendent. Then the asymptotics of $D_n(t)$ can be found via the Painlev\'{e} equations.
\end{remark}

\section{Double Scaling Analysis of the Hankel Determinant}
In this section, we consider the asymptotics of the Hankel determinant under a suitable double scaling. As is mentioned in Remark \ref{rep}, the asymptotics of $D_n(t)$ can be obtained from the integral representation involving a Painlev\'{e} transcendent. However, we do not intend to adopt this method, since we can establish the relation of our problem with a previous study by Chen and Dai.

In the paper of Chen and Dai \cite{ChenDai}, they studied the Hankel determinant generated by a Pollaczek-Jacobi type weight, i.e.,
$$
\tilde{D}_{n}(t,a,b):=\det\left(\int_{0}^{1}x^{i+j}x^{a}(1-x)^{b}\mathrm{e}^{-\frac{t}{x}}dx\right)_{i,j=0}^{n-1}.
$$
Note that we use the parameters $a$ and $b$ instead of $\al$ and $\bt$ in their paper to avoid confusion with our previous parameters.
The orthogonality with the Pollaczek-Jacobi type weight is
$$
\int_{0}^{1}\tilde{P}_{j}(x,a,b)\tilde{P}_{k}(x,a,b)x^{a}(1-x)^{b}\mathrm{e}^{-\frac{t}{x}}dx=\tilde{h}_{j}(t,a,b)\delta_{jk},
$$
where $\tilde{P}_{j}(x,a,b),\; j=0,1,2,\ldots$, are the monic polynomials of degree $j$ orthogonal with respect to the weight $x^{a}(1-x)^{b}\mathrm{e}^{-\frac{t}{x}}$. Obviously, $\tilde{P}_{j}(x,a,b)$ should also depend on $t$, while we do not write it down for simplicity.

Based on the ladder operator approach, Chen and Dai \cite{ChenDai} obtained the following result.
\begin{lemma}\label{cd}
The logarithmic derivative of the Hankel determinant with respect to $t$,
\be\label{def1}
H_{n}(t,a,b):=t \frac{d}{dt}\ln \tilde{D}_{n}(t,a,b),
\ee
satisfies the following nonlinear second-order differential equation,
\be\label{hn}
(t H_{n}'')^2=\left[n(n+a+b)-H_n+(a+t)H_n'\right]^2+4H_n'(tH_n'-H_n)(b-H_n').
\ee
Let
$$
\tilde{H}_{n}(t,a,b):=H_{n}(t,a,b)-n(n+a+b),
$$
then (\ref{hn}) becomes
\bea
(t\tilde{H}_{n}'')^2&=&-4t(\tilde{H}_{n}')^3+(\tilde{H}_{n}')^2\left[4\tilde{H}_{n}+(a+2b+t)^2+4n(n+a+b)-4b(a+b)\right]\nonumber\\
&-&2\tilde{H}_{n}'\left[(a+2b+t)\tilde{H}_{n}+2nb(n+a+b)\right]+\tilde{H}_{n}^2,\nonumber
\eea
which is the Jimbo-Miwa-Okamoto $\s$-form of Painlev\'{e} V, with parameters $\nu_{0}=0,\; \nu_{1}=-(n+a+b),\; \nu_{2}=n,\; \nu_{3}=-b$, following the convention in \cite[(C.45)]{Jimbo1981}.
\end{lemma}

\begin{proposition}
We have
\be\label{de1}
\tilde{P}_j\left(x,-\frac{1}{2},\al\right)=P_{2j}(\sqrt{x}),\qquad j=0,1,2,\ldots,
\ee
\be\label{de2}
\tilde{P}_j\left(x,\frac{1}{2},\al\right)=\frac{P_{2j+1}(\sqrt{x})}{\sqrt{x}},\qquad j=0,1,2,\ldots,
\ee
where $\left\{\tilde{P}_j\left(x,-\frac{1}{2},\al\right)\right\}_{j=0}^{\infty}$ are the monic orthogonal polynomials with respect to the weight function $x^{-\frac{1}{2}}(1-x)^{\al}\mathrm{e}^{-\frac{t}{x}}$, and $\left\{\tilde{P}_j\left(x,\frac{1}{2},\al\right)\right\}_{j=0}^{\infty}$ are the monic orthogonal polynomials with respect to the weight $x^{\frac{1}{2}}(1-x)^{\al}\mathrm{e}^{-\frac{t}{x}}$. Here the subscript $j$ represents the degree of the corresponding polynomial. Furthermore, the following relations hold:
\be\label{rela1}
h_{2j}(t)=\tilde{h}_{j}\left(t,-\frac{1}{2},\alpha\right),
\ee
\be\label{rela2}
h_{2j+1}(t)=\tilde{h}_{j}\left(t,\frac{1}{2},\alpha\right).
\ee

\end{proposition}
\begin{proof}
At first, it is easy to see that the right hand side of (\ref{de1}) and (\ref{de2}) are indeed polynomials of degree of $j$.
From the orthogonality (\ref{or}), we have
\bea
h_{2j}(t)\delta_{2j,2k}&=&\int_{-1}^{1}P_{2j}(x)P_{2k}(x)(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}}dx\nonumber\\
&=&2\int_{0}^{1}P_{2j}(x)P_{2k}(x)(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}}dx\nonumber\\
&=&\int_{0}^{1}P_{2j}(\sqrt{y})P_{2k}(\sqrt{y})y^{-\frac{1}{2}}(1-y)^{\al}\mathrm{e}^{-\frac{t}{y}}dy,\qquad j,k=0,1,2,\ldots,\nonumber
\eea
which implies
$$
\tilde{P}_j\left(y,-\frac{1}{2},\al\right)=P_{2j}(\sqrt{y}),\qquad j=0,1,2,\ldots.
$$
Then
\bea
h_{2j}(t)
&=&\int_{0}^{1}\left[\tilde{P}_j\left(y,-\frac{1}{2},\al\right)\right]^{2}y^{-\frac{1}{2}}(1-y)^{\al}\mathrm{e}^{-\frac{t}{y}}dy\nonumber\\
&=&\tilde{h}_{j}\left(t,-\frac{1}{2},\alpha\right),\qquad j=0,1,2,\ldots.\nonumber
\eea
On the other hand,
\bea
h_{2j+1}(t)\delta_{2j+1,2k+1}&=&\int_{-1}^{1}P_{2j+1}(x)P_{2k+1}(x)(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}}dx\nonumber\\
&=&2\int_{0}^{1}P_{2j+1}(x)P_{2k+1}(x)(1-x^2)^\al\mathrm{e}^{-\frac{t}{x^{2}}}dx\nonumber\\
&=&\int_{0}^{1}\frac{P_{2j+1}(\sqrt{y})}{\sqrt{y}}\frac{P_{2k+1}(\sqrt{y})}{\sqrt{y}}y^{\frac{1}{2}}(1-y)^{\al}\mathrm{e}^{-\frac{t}{y}}dy.\nonumber
\eea
It follows that
$$
\tilde{P}_j\left(y,\frac{1}{2},\al\right)=\frac{P_{2j+1}(\sqrt{y})}{\sqrt{y}},\qquad j=0,1,2,\ldots.
$$
Hence,
\bea
h_{2j+1}(t)
&=&\int_{0}^{1}\left[\tilde{P}_j\left(y,\frac{1}{2},\al\right)\right]^{2}y^{\frac{1}{2}}(1-y)^{\al}\mathrm{e}^{-\frac{t}{y}}dy\nonumber\\
&=&\tilde{h}_{j}\left(t,\frac{1}{2},\alpha\right),\qquad j=0,1,2,\ldots.\nonumber
\eea
The proof is complete.
\end{proof}
From (\ref{rela1}) and (\ref{rela2}), and with the aid of (\ref{hankel}), we establish the relation between $D_{n}(t)$ and $\tilde{D}_{n}(t,a,b)$, with $n=0,1,2,\ldots$,
\be\label{hd1}
D_{2n}(t)=\tilde{D}_{n}\left(t,\frac{1}{2},\al\right)\tilde{D}_{n}\left(t,-\frac{1}{2},\al\right),
\ee
\be\label{hd2}
D_{2n+1}(t)=\tilde{D}_{n}\left(t,\frac{1}{2},\al\right)\tilde{D}_{n+1}\left(t,-\frac{1}{2},\al\right).
\ee
In addition, in view of (\ref{def}) and (\ref{def1}), we find for $n=0,1,2,\ldots,$
\be\label{re3}
\sigma_{2n}(t)=2\left[H_{n}\left(t,\frac{1}{2},\alpha\right)+H_{n}\left(t,-\frac{1}{2},\alpha\right)\right],
\ee
\be\label{re4}
\sigma_{2n+1}(t)=2\left[H_{n}\left(t,\frac{1}{2},\alpha\right)+H_{n+1}\left(t,-\frac{1}{2},\alpha\right)\right].
\ee
Hence, we can express $\s_n(t)$ in terms of the $\s$-function of a Painlev\'{e} V:
\be\label{rs1}
\sigma_{2n}(t)=2\left[\tilde{H}_{n}\left(t,\frac{1}{2},\alpha\right)+\tilde{H}_{n}\left(t,-\frac{1}{2},\alpha\right)+2n(n+\al)\right],
\ee
\be\label{rs2}
\sigma_{2n+1}(t)=2\left[\tilde{H}_{n}\left(t,\frac{1}{2},\alpha\right)+\tilde{H}_{n+1}\left(t,-\frac{1}{2},\alpha\right)+(2n+1)\left(n+\al+\frac{1}{2}\right)\right],
\ee
where the function $\tilde{H}_{n}(t,a,b)$, as shown in Lemma \ref{cd}, satisfies the Jimbo-Miwa-Okamoto $\s$-form of a particular Painlev\'{e} V.

Similarly, according to the definition of $R_{n}(t)$ in (\ref{Rnt}), we obtain
\be\label{dou1}
R_{2n}(t)=2R_{n}^*\left(t,-\frac{1}{2},\al\right)=2\left(\tilde{R}_n\left(t,-\frac{1}{2},\al\right)-2n-\frac{1}{2}-\al\right),
\ee
\be\label{dou2}
R_{2n+1}(t)=2R_{n}^*\left(t,\frac{1}{2},\al\right)=2\left(\tilde{R}_n\left(t,\frac{1}{2},\al\right)-2n-\frac{3}{2}-\al\right),
\ee
for $n=0,1,2,\ldots$, where $R_{n}^*(t,a,b)$ and $\tilde{R}_{n}(t,a,b)$ are defined by (2.16) and (2.17) in \cite{ChenDai},
$$
R_{n}^*(t,a,b):=\frac{t}{\tilde{h}_{n}(t,a,b)}\int_{0}^{1}\frac{\tilde{P}_{n}^2(y,a,b)}{y}y^{a}(1-y)^b\mathrm{e}^{-\frac{t}{y}}dy,
$$
$$
\tilde{R}_{n}(t,a,b):=\frac{b}{\tilde{h}_{n}(t,a,b)}\int_{0}^{1}\frac{\tilde{P}_{n}^2(y,a,b)}{1-y}y^{a}(1-y)^b\mathrm{e}^{-\frac{t}{y}}dy,
$$
and they have a simple relation given by formula (2.25) in \cite{ChenDai},
$$
R_{n}^*(t,a,b)=\tilde{R}_{n}(t,a,b)-2n-1-a-b.
$$

By employing the general linear statistics results from \cite{Chen1998}, based on Dyson's Coulomb fluid interpretation \cite{Dyson}, Chen et al. \cite{CCF} studied the asymptotic behavior of $\tilde{R}_n(t,a,b)$ and $\tilde{D}_{n}(t,a,b)$ under a suitable double scaling. According to their results, we obtain the asymptotics of our scaled $R_n(t)$ and $D_n(t)$.

\begin{theorem}
Assume that $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=2n^2 t$ is fixed and the following limits
$$
g_1(s):=\lim_{n\rightarrow\infty}n\: R_{2n}\left(\frac{s}{2n^2}\right)
$$
$$
g_2(s):=\lim_{n\rightarrow\infty}n\:R_{2n+1}\left(\frac{s}{2n^2}\right)
$$
exist. Then for $s\rightarrow 0$,
$$
g_1(s)=-4 s+\frac{32 s^2}{3}-\frac{256 s^3}{15}+\frac{8192 s^4}{315}-\frac{311296 s^5}{8505}+\frac{7733248 s^6}{155925}+O(s^7),
$$
$$
g_2(s)=4 s+\frac{32 s^2}{3}+\frac{256 s^3}{15}+\frac{8192 s^4}{315}+\frac{311296 s^5}{8505}+\frac{7733248 s^6}{155925}+O(s^7),
$$
and for $s\rightarrow \infty$,
$$
g_1(s)=2 s^{\frac{2}{3}}+\frac{1}{3}s^{\frac{1}{3}}+\frac{1}{108s^{\frac{1}{3}}}-\frac{1}{648s^{\frac{2}{3}}}+\frac{1}{324 s}-\frac{7}{5832 s^{\frac{4}{3}}}+O(s^{-\frac{5}{3}}),
$$
$$
g_2(s)=2 s^{\frac{2}{3}}-\frac{1}{3}s^{\frac{1}{3}}-\frac{1}{108s^{\frac{1}{3}}}-\frac{1}{648s^{\frac{2}{3}}}-\frac{1}{324 s}-\frac{7}{5832 s^{\frac{4}{3}}}+O(s^{-\frac{5}{3}}).
$$
\end{theorem}
\begin{proof}
According to (\ref{dou1}) and (\ref{dou2}), we have
\be\label{re11}
g_1(s)=4g\left(s,-\frac{1}{2},\al\right),
\ee
\be\label{re12}
g_2(s)=4g\left(s,\frac{1}{2},\al\right),
\ee
where $g(s,a,b)$ is defined by \cite[p.6]{CCF}
$$
g(s,a,b)=\lim_{n\rightarrow\infty}n^2\left(\frac{\tilde{R}_{n}\left(\frac{s}{2n^2},a,b\right)}{2n+1+a+b}-1\right).
$$
and $g(s,a,b)$ satisfies a particular Painlev\'{e} III$'$ \cite[(A.45.4$'$)]{Mehta}:
$$
g''=\frac{(g')^2}{g}-\frac{g'}{s}+\frac{2g^2}{s^2}+\frac{a}{2s}-\frac{1}{4g},
$$
i.e., $P_{III'}(8,2a,0,-1)$, with the initial conditions $g(0,a,b)=0, g'(0,a,b)=\frac{1}{2a}$. Then Chen et al. obtained the small $s$ and large $s$ asymptotics of $g(s,a,b)$, which are given by formulas (2.10) and (2.11) in \cite{CCF}:
\bea
g(s,a,b)&=&\frac{s}{2a}-\frac{s^2}{2a^2(a^2-1)}+\frac{3s^3}{2a^3(a^2-1)(a^2-4)}+\frac{3(3-2 a^2) s^4}{a^4 (a^2-1)^2(a^2-4)(a^2-9) }\nonumber\\[10pt]
&+&\frac{5 (11 a^2-36) s^5}{2 a^5(a^2-1)^2(a^2-4)  (a^2-9)  (a^2-16)}\nonumber\\[10pt]
&-&\frac{3 (91 a^6-1115 a^4+4219 a^2-3600) s^6}{2 a^6 (a^2-1)^3(a^2-4)^2 (a^2-9)(a^2-16)(a^2-25)}
+O(s^7),\quad s\rightarrow 0,\quad a\notin \mathbb{Z},\nonumber
\eea
\bea
g(s,a,b)&=&\frac{1}{2}s^{\frac{2}{3}}-\frac{a}{6}s^{\frac{1}{3}}+\frac{a(a^2-1)}{162s^{\frac{1}{3}}}+\frac{a^2(a^2-1)}{486s^{\frac{2}{3}}}+\frac{a(a^2-1)}{486 s}-\frac{a^2(a^2-1)(2a^2-11)}{6561s^{\frac{4}{3}}}\nonumber\\
&+&O(s^{-\frac{5}{3}}),\qquad s\rightarrow \infty.\nonumber
\eea
Here we produce more terms in the asymptotic expansions by using their method. In view of the relations (\ref{re11}) and (\ref{re12}), we obtain the desired results of the theorem.
\end{proof}

\begin{theorem}\label{thm}
Suppose that $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=2n^2 t$ is fixed and the following limits
$$
\Delta_{1}(s):=\lim_{n\rightarrow\infty}\frac{D_{2n}\left(\frac{s}{2n^2}\right)}{D_{2n}(0)},
$$
$$
\Delta_{2}(s):=\lim_{n\rightarrow\infty}\frac{D_{2n+1}\left(\frac{s}{2n^2}\right)}{D_{2n+1}(0)}
$$
exist. We have for $s\rightarrow 0$,
$$
\Delta_{1}(s)=\Delta_{2}(s)=\exp\left(-\frac{4 s^2}{3}-\frac{256 s^4}{315}-\frac{966656 s^6}{1403325}+O(s^8)\right),
$$
while for $s\rightarrow\infty$,
$$
\Delta_{1}(s)=\Delta_{2}(s)=\exp\left(\frac{\ln 2}{12}+3\zeta'(-1)-\frac{\ln s}{36}-\frac{9}{4} s^{\frac{2}{3}}+\frac{1}{576}s^{-\frac{2}{3}}+\frac{7}{20736}s^{-\frac{4}{3}}+O(s^{-2})\right),
$$
where  $\zeta(\cdot)$ is the Riemann zeta function. Here the constant term $\frac{\ln 2}{12}+3\zeta'(-1)$ is called Dyson's constant \cite{Dyson1976}.
\end{theorem}
\begin{proof}
From (\ref{hd1}) and (\ref{hd2}) we have
\be\label{re13}
\Delta_{1}(s)=\Delta_{2}(s)=\Delta\left(s,\frac{1}{2},\al\right)\Delta\left(s,-\frac{1}{2},\al\right),
\ee
where
$$
\Delta(s,a,b):=\lim_{n\rightarrow\infty}\frac{\tilde{D}_{n}\left(\frac{s}{2n^2},a,b\right)}{\tilde{D}_{n}(0,a,b)}.
$$
By using the integral representation of $\Delta(s,a,b)$ in terms of $g(s,a,b)$, Chen et al. obtained the small $s$ and large $s$ asymptotics of $\Delta(s,a,b)$ in \cite[Thm. 4]{CCF}: for $s\rightarrow 0$,
\bea
\Delta(s,a,b)&=&\exp\Big[-\frac{s}{2 a}+\frac{s^2}{8 a^2 (a^2-1)}-\frac{s^3}{6 a^3(a^2-1)(a^2-4)}\nonumber\\[10pt]
&+&\frac{3 (2 a^2-3) s^4}{16 a^4 (a^2-1)^2 (a^2-4)(a^2-9) }-\frac{(11 a^2-36)s^5}{10 a^5 (a^2-1)^2  (a^2-4)(a^2-9) (a^2-16)}\nonumber\\[10pt]
&+&\frac{(91 a^6-1115 a^4+4219 a^2-3600)s^6}{24 a^6 (a^2-1)^3 (a^2-4)^2(a^2-9)(a^2-16)(a^2-25) }+O(s^7)\Big],\nonumber
\eea
and for $s\rightarrow\infty$,
\bea
\Delta(s,a,b)&=&\exp\Big[c(a)-\frac{9}{8}s^{\frac{2}{3}}+\frac{3a}{2}s^{\frac{1}{3}}+\frac{1-6a^2}{36}\ln s-\frac{a(a^2-1)}{18}s^{-\frac{1}{3}}-\frac{a^2(a^2-1)}{216}s^{-\frac{2}{3}}\nonumber\\[10pt]
&-&\frac{a(a^2-1)}{486}s^{-1}+\frac{a^2(a^2-1)(2a^2-11)}{11664}s^{-\frac{4}{3}}+\frac{a(a^2-1)(a^4-a^2-15)}{21870}s^{-\frac{5}{3}}\nonumber\\[10pt]
&+&O(s^{-2})\Big],\nonumber
\eea
where $c(a)$ is an integration constant, independent of $s$, given by
\be\label{cons}
c(a)=\ln\frac{G(a+1)}{(2\pi)^{\frac{a}{2}}},
\ee
and $G(\cdot)$ is the Barnes $G$-function. Here we show more terms in the asymptotic expansions of $\Delta(s,a,b)$.

In view of (\ref{re13}), we find for $s\rightarrow 0$,
$$
\Delta_{1}(s)=\Delta_{2}(s)=\exp\left(-\frac{4 s^2}{3}-\frac{256 s^4}{315}-\frac{966656 s^6}{1403325}+O(s^8)\right),
$$
and for $s\rightarrow\infty$,
$$
\Delta_{1}(s)=\Delta_{2}(s)=\exp\left(c\left(\frac{1}{2}\right)+c\left(-\frac{1}{2}\right)-\frac{\ln s}{36}-\frac{9}{4} s^{\frac{2}{3}}+\frac{1}{576}s^{-\frac{2}{3}}+\frac{7}{20736}s^{-\frac{4}{3}}+O(s^{-2})\right).
$$
It follows from (\ref{cons}) that
$$
c\left(\frac{1}{2}\right)+c\left(-\frac{1}{2}\right)=\ln \left(G\left(\frac{1}{2}\right)G\left(\frac{3}{2}\right)\right)=\ln \left(G^2\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)\right).
$$
With the aid of the expression of $G\left(\frac{1}{2}\right)$ in \cite[(6.39)]{Voros}, i.e.,
$$
G\left(\frac{1}{2}\right)=2^{\frac{1}{24}}\pi^{-\frac{1}{4}}\mathrm{e}^{\frac{3\zeta'(-1)}{2}},
$$
and $\zeta(\cdot)$ is the Riemann zeta function, we have
$$
c\left(\frac{1}{2}\right)+c\left(-\frac{1}{2}\right)=\frac{\ln 2}{12}+3\zeta'(-1).
$$
Hence, the theorem is established.
\end{proof}
\begin{remark}
We find that the small $s$ and large $s$ asymptotic behaviors of the scaled Hankel determinant in Theorem \ref{thm} are identical with the asymptotics of the Hankel determinant for the singularly perturbed Gaussian weight under a different double scaling \cite{MLC}. Our results demonstrate the universality in random matrix theory. That is, although the Hankel determinants for different weights are quite distinct in the finite dimensional case, they may have the same characteristics when considering the limiting behavior.
\end{remark}

\section*{Acknowledgments}
Chao Min was supported by the Scientific Research Funds of Huaqiao University under grant number 17BS402.
Yang Chen was supported by the Macau Science and Technology Development Fund under grant number FDCT 023/2017/A1 and by the University of Macau under grant number MYRG 2018-00125-FST.

config: arxiv


TITLE: Rectifiable curves in $\mathbb{R}^n$ have measure $0$, but line integral is nonzero.
QUESTION [1 upvotes]: I covered a bit of Lebesgue measure and came across the property that states that the integral of a function $f$ over some nullset $E$ is always $0$. If rectifiable curves have measure $0$ in $\mathbb{R}^n$  , how come every line integral is not vanishing?
I know this is very elementary and I must be making a big mistake somewhere. I am trying to get the grasp of Lebesgue measure/some intuition for this problem. Any help is appreciated!

REPLY [1 votes]: You are confusing one and two dimensional measures. The line segment joining $(0,0)$ to $(1,0)$ in the plane has area (planar Lebesgue measure) $0$ but still has length $1$. Line integrals along curves use the arclength along the curve as the underlying measure.


TITLE: Is this a Projection operator on Hilbert space?
QUESTION [3 upvotes]: Let $T$ be a bounded operator on a Hilbert space with the property that $T^*(T-I)= 0$. I'd like to show that $T$ is an orthogonal projection.
I'm not really sure how to show that an operator is an orthogonal projection.
If $A:X\rightarrow U$ is a projection onto a closed subspace of X, then $\langle x- Ax, u_i\rangle = 0 \;\;\forall u_i \in U$ ?
Expanding we get $T'T = T' \Longleftrightarrow TT' = T\ (T'' = T$ in Hilbert spaces?)
$$Tx = T'(Tx) \Longleftrightarrow y = T'y$$
Hence $T'$ has $\lambda = 1$ after $Tx$, do $T'$ and $T$ have the same eigenvalues?
There are a lot of question marks here.

REPLY [4 votes]: Your equation is $T^*T=T^*$. The key observation here is that $(T^*T)^*=T^*T$. Then
$$
T=(T^*)^*=(T^*T)^*=T^*T=T^*.
$$
so $T$ is selfadjoint, and moreover now your original equation reads $T^2=T$, so $T$ is an orthogonal projection.