% os37 revised, 2 March 2017
% 25 March 2017 : equation numbering changed, reference order changed,
%                 keywords and MSC2010 added
%%%%%% LaTeX2e %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14 pages %%%%%
%                                                             %
%        Simplified Expressions of the Multi-indexed          %
%               Laguerre and Jacobi Polynomials               % 
%                                                             %
%                  S. Odake and R. Sasaki                     %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage{graphicx}
% Change page dimensions to match standard 8 1/2 X 11 inch size
\oddsidemargin=-0.1in
\evensidemargin=-0.1in
\topmargin=-0.2in
\textwidth=6.5in
\textheight=8.9in

\newcommand{\eqdef}{\stackrel{\text{def}}{=}}
%\newcommand{\eqdefrm}{\stackrel{\text{\rm def}}{=}}
\newcommand{\n}{\nonumber\\}
\newcommand{\bm}{\boldsymbol}
\newcommand{\ignore}[1]{}
%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
%\numberwithin{equation}{section}
\newcommand{\Romannumeral}[1]{\uppercase\expandafter{\romannumeral#1}}
\newcommand{\I}{\text{\Romannumeral{1}}}
\newcommand{\II}{\text{\Romannumeral{2}}}
\newcommand{\III}{\text{\Romannumeral{3}}}
\newcommand{\cF}{c_{\text{\tiny$\mathcal{F}$}}}
\newcommand{\norm}[1]{|\!|#1|\!|}

\allowdisplaybreaks[4]

\begin{document}

\baselineskip=20pt

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                          %
%  Title page                                              %
%                                                          %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\preprint}{
\vspace*{-20mm}
   \begin{flushright}\normalsize \sf
    DPSU-16-4\\
%     {\tt arXiv:1612.00927[math.CA]}\\
%     December 2016
  \end{flushright}}
\newcommand{\Title}[1]{{\baselineskip=26pt
  \begin{center} \Large \bf #1 \\ \ \\ \end{center}}}
\newcommand{\Author}{\begin{center}
  \large \bf Satoru Odake and Ryu Sasaki \end{center}}
\newcommand{\Address}{\begin{center}
%     Department of Physics, Shinshu University,\\
     Faculty of Science, Shinshu University,\\
     Matsumoto 390-8621, Japan
   \end{center}}
\newcommand{\Accepted}[1]{\begin{center}
  {\large \sf #1}\\ \vspace{1mm}{\small \sf Accepted for Publication}
  \end{center}}

\preprint
\thispagestyle{empty}

\Title{Simplified Expressions of the Multi-indexed Laguerre and Jacobi
Polynomials}

\Author

\Address
\vspace{1cm}

\begin{abstract}
The multi-indexed Laguerre and Jacobi polynomials form a complete set of
orthogonal polynomials. They satisfy second order differential equations but
not three term recurrence relations, because of the `holes' in their degrees.
The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions
originating from multiple Darboux transformations. For the ease of
applications, two different forms of simplified expressions of the
multi-indexed Laguerre and Jacobi polynomials are derived based on various
identities.
The parity transformation property of the multi-indexed Jacobi polynomials is
derived based on that of the Jacobi polynomial.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\paragraph{Keywords:}
\noindent{\bf Keywords:}
multi-indexed orthogonal polynomials,
Laguerre and Jacobi polynomials,
Wronskian formula,
determinant formula

%\paragraph{MSC2010:}
\noindent{\bf MSC2010:}
42C05, 33C45, 34A05
%
%Mathematics Subject Classifications (2010)
%
%42C05 Orthogonal functions and polynomials, general theory
%      [See also 33C45, 33C50, 33D45]
%33C45 Orthogonal polynomials and functions of hypergeometric type
%      (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
%      [See also 42C05 for general orthogonal polynomials and functions]
%34-XX ORDINARY DIFFERENTIAL EQUATIONS
%34A05 Explicit solutions and reductions
%34B24 Sturm-Liouville theory [See also 34Lxx]
%34Lxx Ordinary dierential operators [See also 47E05]
%34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions
%81Q05 Closed and approximate solutions to the Schodinger, Dirac,
%      Klein-Gordon and other equations of quantum mechanics
%81Q80 Special quantum systems, such as solvable systems
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                             %
%  1. Introduction                                            %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}

The exceptional and multi-indexed orthogonal polynomials
%\cite{gomez}--\cite{os35} seem to be a focal point of recent research on
\cite{gomez,gomez0,quesne,quesnebagchi,os16,gomez2,os25,os26,os27,os35}
seem to be a focal point of recent research on
exactly solvable quantum mechanics. They belong to a new type of orthogonal
polynomials which satisfy second order differential (difference) equations
and form complete orthogonal basis in an appropriate Hilbert space.
One of their characteristic features is that they do not satisfy three term
recurrence relations because of the `holes' in the degrees.
This is how they avoid the constraints due to Bochner \cite{routh,bochner}.
They are constructed from the original quantum mechanical systems, the radial
oscillator potential and the P\"oschl-Teller potential, by multiple
application of Darboux transformations \cite{darboux, crum} in terms of seed
solutions called virtual state wavefunctions which are generated by two types
%of discrete symmetry transformations \cite{os25}--\cite{os35}.
of discrete symmetry transformations \cite{os25,os26,os27,os35}.

The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions
\cite{os25} originating from multiple Darboux transformations \cite{crum}.
In this note we present two different forms of equivalent determinant
expressions without higher derivatives of the Wronskians by using various
identities of the Laguerre and Jacobi polynomials \cite{szego}. These
simplified expressions show explicitly the constituents of the multi-indexed
orthogonal polynomials and they are helpful for deeper understanding.
In \cite{d14} and \cite{d17} Dur\'an employed similar simplified expressions
as the starting point of his exposition of the exceptional Laguerre and Jacobi
polynomials.
See also \cite{dp15} by Dur\'an and P\'erez.
In their expressions, the matrix elements of the determinants are polynomials.
In the original expressions in \cite{os25}, the matrix elements of the
determinants contain the non-polynomial factors, see \eqref{XiD}--\eqref{Pn}.
In the simplified expressions presented in this paper in
\S\,\ref{sec:simpA}--\ref{sec:simpB}, they are also all polynomials.

This short note is organised as follows.
In \S\,\ref{sec:ori} the quantum mechanical settings for the original Laguerre
and Jacobi polynomials are recapitulated. That is, the Hamiltonians with the
radial oscillator potential and the P\"oschl-Teller potential are introduced
and their eigenvalues and eigenfunctions are presented. Type $\I$ and $\II$
discrete symmetry transformations for the Hamiltonians of the Laguerre and
Jacobi polynomials are explained in \S\,\ref{sec:sym}. The seed solutions for
Darboux transformations, to be called the virtual state wavefunctions of
type $\I$ and $\II$, for the Laguerre and Jacobi, are listed explicitly.
In \S\,\ref{sec:wron} the Wronskian forms of the multi-indexed Laguerre and
Jacobi polynomials derived in \cite{os25} and \cite{rrmiop3} are recapitulated
as the starting point.
\S\,\ref{sec:simpA} and \S\,\ref{sec:simpB} are the main content of this note.
Those who are familiar with the multi-indexed Laguerre and Jacobi polynomials
can directly go to this part. The first simplified expressions of the
multi-indexed Laguerre and Jacobi polynomials are derived in \S\,\ref{sec:simpA}
by using various identities of the original Laguerre and Jacobi polynomials.
The second simplified expressions, to be derived in \S\,\ref{sec:simpB}, are
the consequences of the multi-linearity of determinants and the form of the
Schr\"odinger equation $\psi''(x)=\bigl(U(x)-\mathcal{E}\bigr)\psi(x)$.
Every even order derivative $\psi^{(2m)}(x)$ in the Wronskian can be
replaced by $(-\mathcal{E})^m\psi(x)$.
In \S\,\ref{sec:pari} the parity transformation formula of the multi-indexed
Jacobi polynomials is presented.
%The final section is for a summary and comments.
Section \ref{sec:summary} is for a summary and comments.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                             %
%  2. Multi-indexed Laguerre and Jacobi Polynomials           %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Multi-indexed Laguerre and Jacobi Polynomials }
\label{sec:MILJ}

The foundation of the theory of multi-indexed orthogonal polynomials is the
exactly solvable one dimensional quantum mechanical system $\mathcal{H}$:
\begin{equation}
  \mathcal{H}\phi_n(x)=\mathcal{E}_n\phi_n(x)\ \ (n=0,1,\ldots),\quad
  \mathcal{H}=-\frac{d^2}{dx^2}+U(x),
  \label{scheq}
\end{equation}
and its iso-spectrally deformed system
\begin{equation}
  \mathcal{H}_{\mathcal{D}}\phi_{\mathcal{D}\,n}(x)
  =\mathcal{E}_n\phi_{\mathcal{D}\,n}(x)\ \ (n=0,1,\ldots),\quad
  \mathcal{H}_{\mathcal{D}}=-\frac{d^2}{dx^2}+U_{\mathcal{D}}(x),
  \label{Dscheq}
\end{equation}
in terms of multiple application of Darboux transformations \cite{os25,crum}.
In this note we discuss two systems which have the Laguerre and Jacobi
polynomials as the main parts of the eigenfunctions. We follow the notation
of \cite{os25} with slight modifications for simplification sake.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.1 Original Laguerre and Jacobi polynomials      %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Original Laguerre and Jacobi polynomials}
\label{sec:ori}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.1.1 radial oscillator potential                 %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{radial oscillator potential}

The Hamiltonian with the radial oscillator potential 
\begin{equation}
  U(x)\eqdef x^2+\frac{g(g-1)}{x^2}-2g-1,\quad 0<x<\infty,\quad g>\frac12,
  \label{rad}
\end{equation}
has the Laguerre polynomials $L^{(\alpha)}_n(\eta)$ as the main part of the
eigenfunctions:
\begin{align*}
  \phi_n(x;g)&\eqdef\phi_0(x;g)L^{(g-\frac12)}_n\bigl(\eta(x)\bigr),\quad
  \mathcal{E}_n\eqdef 4n,\quad\eta(x)\eqdef x^2,\\
%  \label{lagsol}\\
%
  \phi_0(x;g)&\eqdef e^{-\frac12x^2}x^g,\quad 
  L_n^{(\alpha)}(\eta)=\frac{1}{n!}
  \sum_{k=0}^n\frac{(-n)_k}{k!}(\alpha+k+1)_{n-k}\eta^k.
%  \label{Lexp}
\end{align*}
Here are some identities of the Laguerre polynomials \cite{szego} to be used
in \S\,\ref{sec:simpA}--\ref{sec:simpB},
\begin{align}
  &\partial_{\eta}L_n^{(\alpha)}(\eta)=-L_{n-1}^{(\alpha+1)}(\eta),
  \label{Lforward}\\
%
  &L_n^{(\alpha)}(\eta)-L_n^{(\alpha-1)}(\eta)=L_{n-1}^{(\alpha)}(\eta),
  \label{Lid1}\\
%
  &\eta L_{n-1}^{(\alpha+1)}(\eta)-\alpha L_{n-1}^{(\alpha)}(\eta)
  =-nL_n^{(\alpha-1)}(\eta).
  \label{Lid2}
\end{align}
As is clear from \eqref{rad}, the lower boundary, $x=0$, is the regular
singular point with the characteristic exponents $g$ and $1-g$.
The upper boundary point, $x=\infty$, is an irregular singular point.
Here $(a)_n\eqdef\prod_{j=1}^n(a+j-1)$ is the shifted factorial or the
so-called Pochhammer's symbol.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.1.2 P\"oschl-Teller potential                   %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{P\"oschl-Teller potential}

The Hamiltonian with the P\"oschl-Teller potential
\begin{equation*}
  U(x)\eqdef\frac{g(g-1)}{\sin^2x}+\frac{h(h-1)}{\cos^2x}-(g+h)^2,\quad
  0<x<\frac{\pi}{2},\quad g>\frac12,\ h>\frac12,
%  \label{PTpot}
\end{equation*}
has the Jacobi polynomials $P_n^{(\alpha,\beta)}(\eta)$ as the main part of the
eigenfunctions:
\begin{align*}
  &\phi_n\bigl(x;(g,h)\bigr)\eqdef\phi_0\bigl(x;(g,h)\bigr)
  P_n^{(g-\frac12,h-\frac12)}\bigl(\eta(x)\bigr),\quad
  \eta(x)\eqdef\cos2x,\\[2pt]
%  \label{jacsol}\\[2pt]
%
  &\phi_0\bigl(x;(g,h)\bigr)\eqdef(\sin x)^g(\cos x)^h,\quad
  \mathcal{E}_n(g,h)\eqdef 4n(n+g+h),\\
%
  &P_n^{(\alpha,\beta)}(\eta)=\frac{(\alpha+1)_n}{n!}
  \sum_{k=0}^n\frac{1}{k!}\frac{(-n)_k(n+\alpha+\beta+1)_k}{(\alpha+1)_k}
  \Bigl(\frac{1-\eta}{2}\Bigr)^k.
%  \label{Jexp}
\end{align*}
Here are some identities of the Jacobi polynomials \cite{szego} to be used in
in \S\,\ref{sec:simpA}--\ref{sec:simpB},
\begin{align}
  &\partial_{\eta}P_n^{(\alpha,\beta)}(\eta)
  =\tfrac12(n+\alpha+\beta+1)P_{n-1}^{(\alpha+1,\beta+1)}(\eta),
  \label{Jforward}\\
%
  &(n+\beta)P_n^{(\alpha,\beta-1)}(\eta)
  -\beta P_n^{(\alpha-1,\beta)}(\eta)
  =(n+\alpha+\beta)\tfrac{1+\eta}{2}P_{n-1}^{(\alpha,\beta+1)}(\eta),
  \label{Jid1}\\
%
  &(n+\alpha)P_n^{(\alpha-1,\beta)}(\eta)
  -\alpha P_n^{(\alpha,\beta-1)}(\eta)
  =-(n+\alpha+\beta)\tfrac{1-\eta}{2}P_{n-1}^{(\alpha+1,\beta)}(\eta).
  \label{Jid1m}
\end{align}
The two boundary points, $x=0,\frac{\pi}2$ are the regular singular points
with the characteristic exponents $g$, $1-g$ and $h$, $1-h$, respectively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.2 Discrete symmetry transformations and virtual %
%     state wavefunctions                           %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discrete symmetry transformations and virtual state wavefunctions}
\label{sec:sym}

The seed solutions (virtual state wavefunctions) for Darboux transformations
can be constructed by applying discrete symmetry transformations of the
Hamiltonian to the eigenfunctions. They have negative energies and have no
node and they are not square integrable, see \cite{os25} for more detail.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.2.1 Laguerre (L) system                         %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Laguerre (L) system}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Type $\I$ transformation  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Type $\I$ transformation}
It is obvious that the transformation $x\to ix$ is the symmetry of the radial
oscillator system.
The seed solutions are
\begin{align*}
  \text{L1}:&\ \ \mathcal{H}\tilde{\phi}^{\I}_{\text{v}}(x)
  =\tilde{\mathcal{E}}^{\I}_{\text{v}}\tilde{\phi}^{\I}_{\text{v}}(x),\quad
  \tilde{\phi}^{\I}_{\text{v}}(x)\eqdef
  \tilde{\phi}^{\I}_0(x)\xi^{\I}_{\text{v}}\bigl(\eta(x)\bigr),\quad
  \tilde{\phi}^{\I}_0(x)\eqdef e^{\frac12x^2}x^g,\n
%
  &\ \ \xi^{\I}_{\text{v}}(\eta)\eqdef L^{(g-\frac12)}_{\text{v}}(-\eta),
  \quad\tilde{\mathcal{E}}^{\I}_{\text{v}}\eqdef-4(g+\text{v}+\tfrac12),\quad
  \text{v}\in\mathbb{Z}_{\ge0}.
\end{align*}
Hereafter we use $\text{v}$ for the degree of the seed polynomial
$\xi_{\text{v}}$ in order to distinguish it with the degree $n$ of
eigenpolynomial $P_n$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Type $\II$ transformation %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Type $\II$ transformation}
The exchange of the characteristic exponents of the regular singular point
$x=0$, $g\to1-g$, is another discrete symmetry transformation.
It generates the seed solutions,
\begin{align*}
  \text{L2}:&\ \ \mathcal{H}\tilde{\phi}^{\II}_{\text{v}}(x)
  =\tilde{\mathcal{E}}^{\II}_{\text{v}}\tilde{\phi}^{\II}_{\text{v}}(x),\quad
  \tilde{\phi}^{\II}_{\text{v}}(x)\eqdef
  \tilde{\phi}^{\II}_0(x)\xi^{\II}_{\text{v}}\bigl(\eta(x)\bigr),\quad
  \tilde{\phi}^{\II}_0(x)\eqdef e^{-\frac12x^2}x^{1-g},\n
%
  &\ \ \xi^{\II}_{\text{v}}(\eta)\eqdef L^{(\frac12-g)}_{\text{v}}(\eta),
  \quad\tilde{\mathcal{E}}^{\II}_{\text{v}}\eqdef-4(g-\text{v}-\tfrac12),\quad
  \text{v}=0,1,\ldots,[g-\tfrac12]',
\end{align*}
in which $[a]'$ denotes the greatest integer less than $a$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.2.2 Jacobi (J) system                           %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Jacobi (J) system}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Type $\I$ transformation  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Type $\I$ transformation}
The exchange of the characteristic exponents of the regular singular point
$x=\frac{\pi}{2}$, $h\to1-h$, is a discrete symmetry transformation:
\begin{align*}
  \text{J1}:&\ \ \mathcal{H}\tilde{\phi}^{\I}_{\text{v}}(x)
  =\tilde{\mathcal{E}}^{\I}_{\text{v}}\tilde{\phi}^{\I}_{\text{v}}(x),\quad
  \tilde{\phi}^{\I}_{\text{v}}(x)\eqdef
  \tilde{\phi}^{\I}_0(x)\xi^{\I}_{\text{v}}\bigl(\eta(x)\bigr),\quad
  \tilde{\phi}^{\I}_0(x)\eqdef(\sin x)^g(\cos x)^{1-h},\n
%
  &\ \xi^{\I}_{\text{v}}(\eta)\eqdef
  P^{(g-\frac12,\frac12-h)}_{\text{v}}(\eta),
  \ \tilde{\mathcal{E}}^{\I}_{\text{v}}\eqdef
  -4(g+\text{v}+\tfrac12)(h-\text{v}-\tfrac12),
  \ \ \text{v}=0,1,\ldots,[h-\tfrac12]'.
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Type $\II$ transformation %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Type $\II$ transformation}
Likewise the exchange of the characteristic exponents of the regular singular
point $x=0$, $g\to1-g$, is a discrete symmetry transformation:
\begin{align*}
  \text{J2}:&\ \ \mathcal{H}\tilde{\phi}^{\II}_{\text{v}}(x)
  =\tilde{\mathcal{E}}^{\II}_{\text{v}}\tilde{\phi}^{\II}_{\text{v}}(x),\quad
  \tilde{\phi}^{\II}_{\text{v}}(x)\eqdef
  \tilde{\phi}^{\II}_0(x)\xi^{\II}_{\text{v}}\bigl(\eta(x)\bigr),\quad
  \tilde{\phi}^{\II}_0(x)\eqdef(\sin x)^{1-g}(\cos x)^h,\n
%
  &\ \xi^{\II}_{\text{v}}(\eta)\eqdef
  P^{(\frac12-g,h-\frac12)}_{\text{v}}(\eta),
  \ \tilde{\mathcal{E}}^{\II}_{\text{v}}\eqdef
  -4(g-\text{v}-\tfrac12)(h+\text{v}+\tfrac12),
  \ \ \text{v}=0,1,\ldots,[g-\tfrac12]'.
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.3 Wronskian forms of the multi-indexed Laguerre %
%     and Jacobi polynomials                        %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Wronskian forms of the multi-indexed Laguerre and Jacobi
polynomials}
\label{sec:wron}

We deform the original Hamiltonian system \eqref{scheq} by applying multiple
Darboux transformations \cite{os25,crum} in terms of $M_{\I}$ type $\I$ seed
solutions and $M_{\II}$ type $\II$ seed solutions specified by the degree set
$\mathcal{D}\eqdef\{d_1,\ldots,d_M\}$ (ordered set), $M\eqdef M_{\I}+M_{\II}$,
$M_{\I}\eqdef\#\{d_j|d_j\in\mathcal{D},\,\text{type $\I$}\}$,
$M_{\II}\eqdef\#\{d_j|d_j\in\mathcal{D},\,\text{type $\II$}\}$.
The multi-indexed orthogonal polynomials $\{P_{\mathcal{D},n}(\eta)\}$ are the
main parts of the eigenfunctions $\{\phi_{\mathcal{D}\,n}(x)\}$ of the
deformed Hamiltonian system $\mathcal{H}_{\mathcal{D}}$ \eqref{Dscheq}:
\begin{align*}
  U_{\mathcal{D}}(x)&\eqdef U(x)-2\partial_x^2
  \log\bigl|\text{W}[\tilde{\phi}_{d_1},\ldots,\tilde{\phi}_{d_M}](x)\bigr|,\\
%
  \phi_{\mathcal{D}\,n}(x)&\eqdef
  \frac{\text{W}[\tilde{\phi}_{d_1},\ldots,\tilde{\phi}_{d_M},\phi_n](x)}
  {\text{W}[\tilde{\phi}_{d_1},\ldots,\tilde{\phi}_{d_M}](x)}
  \ \ (n=0,1,\ldots)\\
%  \label{wronKA}\\
%
  &\eqdef\cF^{M}\psi_{\mathcal{D}}(x)
  P_{\mathcal{D},n}\bigl(\eta(x)\bigr),\quad
  \psi_{\mathcal{D}}(x)\eqdef\frac{\hat{\phi}_0(x)}
  {\Xi_{\mathcal{D}}\bigl(\eta(x)\bigr)},
\end{align*}
in which $\hat{\phi}_0(x)$ and $\cF$ are defined by
\begin{equation*}
  \hat{\phi}_0(x)\eqdef\left\{
  \begin{array}{ll}
  \phi_0(x;g+M_{\I}-M_{\II})&:\text{L}\\[2pt]
  \phi_0\bigl(x;(g+M_{\I}-M_{\II},h-M_{\I}+M_{\II})\bigr)&:\text{J}
  \end{array}\right.,\quad
  \cF\eqdef\left\{
  \begin{array}{cl}
  2&:\text{L}\\
  -4&:\text{J}
  \end{array}\right..
\end{equation*}
The superscript $\I$ and $\II$ of the seed solutions are suppressed for
simplicity of notation.
The Wronskian of $n$-functions $\{f_1,\ldots,f_n\}$ is defined by formula
\begin{equation*}
  \text{W}[f_1,\ldots,f_n](x)
  \eqdef\det\Bigl(\frac{d^{j-1}f_k(x)}{dx^{j-1}}\Bigr)_{1\leq j,k\leq n}.
%  \label{wron}
\end{equation*}
 
There are two different but equivalent Wronskian definitions of the
denominator polynomial $\Xi_{\mathcal{D}}(\eta)$ and the multi-indexed
orthogonal polynomial $P_{\mathcal{D},n}(\eta)$.
The first is based on the Wronskians of the `polynomials' \cite{os25}:
 \begin{align}
  \Xi_{\mathcal{D}}(\eta)
  &\eqdef\text{W}[\mu_{d_1},\ldots,\mu_{d_M}](\eta)\times\left\{
  \begin{array}{ll}
  \eta^{(M_{\I}+g-\frac12)M_{\II}}e^{-M_{\I}\eta}&:\text{L}\\
  \bigl(\frac{1-\eta}{2}\bigr)^{(M_{\I}+g-\frac12)M_{\II}}
  \bigl(\frac{1+\eta}{2}\bigr)^{(M_{\II}+h-\frac12)M_{\I}}&:\text{J}
  \end{array}\right.,
  \label{XiD}\\
%
  P_{\mathcal{D},n}(\eta)
  &\eqdef\text{W}[\mu_{d_1},\ldots,\mu_{d_M},P_n](\eta)\times\left\{
  \begin{array}{ll}
  \eta^{(M_{\I}+g+\frac12)M_{\II}}e^{-M_{\I}\eta}&:\text{L}\\
  \bigl(\frac{1-\eta}{2}\bigr)^{(M_{\I}+g+\frac12)M_{\II}}
  \bigl(\frac{1+\eta}{2}\bigr)^{(M_{\II}+h+\frac12)M_{\I}}&:\text{J}
  \end{array}\right.,
  \label{PDn}\\
%
  \mu_{\text{v}}(\eta)&\eqdef\left\{
  \begin{array}{ll}
  e^{\eta}\times L^{(g-\frac12)}_{\text{v}}(-\eta)
  &:\text{L, $\text{v}$ type $\I$}\\
  \eta^{\frac12-g}\times L^{(\frac12-g)}_{\text{v}}(\eta)
  &:\text{L, $\text{v}$ type $\II$}\\
  \bigl(\frac{1+\eta}{2}\bigr)^{\frac12-h}\times
  P^{(g-\frac12,\frac12-h)}_{\text{v}}(\eta)
  &:\text{J, $\text{v}$ type $\I$}\\[3pt]
  \bigl(\frac{1-\eta}{2}\bigr)^{\frac12-g}\times
  P^{(\frac12-g,h-\frac12)}_{\text{v}}(\eta)
  &:\text{J, $\text{v}$ type $\II$}
  \end{array}\right.,
  \label{muv}\\
%
  P_n(\eta)
  &\eqdef\left\{
  \begin{array}{ll}
  L^{(g-\frac12)}_n(\eta)&:\text{L}\\[2pt]
  P^{(g-\frac12,h-\frac12)}_n(\eta)&:\text{J}
  \end{array}\right..
  \label{Pn}
\end{align}
The second is based on the Wronskians of the virtual state wavefunctions
$\tilde{\phi}_{\text{v}}(x)$ and the eigenfunction $\phi_n(x)$ \cite{rrmiop3}:
\begin{align}
  &\Xi_{\mathcal{D}}(\eta)=\cF^{-\frac12M(M-1)}
  \text{W}[\tilde{\phi}_{d_1},\ldots,\tilde{\phi}_{d_M}](x)
  \times\left\{
  \begin{array}{ll}
  \eta^{-M'(M'+g-\frac12)}e^{-M'\eta}&:\text{L}\\
  \bigl(\frac{1-\eta}{2}\bigr)^{-M'(M'+g-\frac12)}
  \bigl(\frac{1+\eta}{2}\bigr)^{-M'(M'-h+\frac12)}
  &:\text{J}
  \end{array}\right.,
  \label{W=XiD}\\
%
  &P_{\mathcal{D},n}(\eta)=\cF^{-\frac12M(M+1)}
  \text{W}[\tilde{\phi}_{d_1},\ldots,\tilde{\phi}_{d_M},\phi_n](x)\n
%
  &\hspace{40mm}\times\left\{
  \begin{array}{ll}
  \eta^{-(M'+\frac12)(M'+g)}e^{-(M'-\frac12)\eta}&:\text{L}\\
  \bigl(\frac{1-\eta}{2}\bigr)^{-(M'+\frac12)(M'+g)}
  \bigl(\frac{1+\eta}{2}\bigr)^{-(M'-\frac12)(M'-h)}
  &:\text{J}
  \end{array}\right.,
  \label{W=PDn}
\end{align}
in which $M'\eqdef\frac12(M_{\I}-M_{\II})$ and $\eta=\eta(x)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.4 Simplified forms of the multi-indexed         %
%     Laguerre and Jacobi polynomials, A:           %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simplified forms of the multi-indexed Laguerre and Jacobi
polynomials, A:}
\label{sec:simpA}

In this and the subsequent subsections we will derive simplified expressions
of the multi-indexed Laguerre and Jacobi polynomials $P_{\mathcal{D},n}(\eta)$
and the corresponding denominator polynomials $\Xi_\mathcal{D}(\eta)$ starting
from the Wronskian expressions \eqref{XiD}--\eqref{W=PDn} in the previous
subsection.

In this subsection we simplify the Wronskians of the `polynomials'
\eqref{XiD}--\eqref{Pn}.
Let us first transform the higher derivatives of the `seed polynomials'
$\mu_{\text{v}}(\eta)$ in \eqref{muv}.
For the L system, we obtain
\begin{alignat*}{2}
  \text{L1}&:\ \ &\partial_{\eta}
  \bigl(e^{\eta}L^{(g-\frac12)}_{\text{v}}(-\eta)\bigr)
  &=e^{\eta}L^{(g+1-\frac12)}_{\text{v}}(-\eta),\\
%  \label{Ldiff1}\\
%
  \text{L2}&:\ \ &\partial_{\eta}
  \bigl(\eta^{\frac12-g}L^{(\frac12-g)}_{\text{v}}(\eta)\bigr)
  &=(\text{v}-g+\tfrac12)\eta^{\frac12-(g+1)}
  L^{(\frac12-(g+1))}_{\text{v}}(\eta),
%  \label{Ldiff2}
\end{alignat*}
by using \eqref{Lforward}--\eqref{Lid1} and \eqref{Lforward}--\eqref{Lid2},
respectively.
By repeating these we arrive at for $j\in\mathbb{Z}_{\geq 1}$,
\begin{align*}
  &\partial_{\eta}^{j-1}\bigl(e^{\eta}L^{(g-\frac12)}_{\text{v}}(-\eta)\bigr)
  =e^{\eta}\times L^{(g+j-\frac32)}_{\text{v}}(-\eta),\\
%  \label{Ldiff1j}\\
%
  &\partial_{\eta}^{j-1}\bigl(\eta^{\frac12-g}
  L^{(\frac12-g)}_{\text{v}}(\eta)\bigr)
  =(-1)^{j-1}(g-\tfrac12-\text{v})_{j-1}\eta^{\frac32-g-j}
  L^{(\frac32-g-j)}_{\text{v}}(\eta)\n
%
  &\phantom{\partial_{\eta}^{j-1}\bigl(\eta^{\frac12-g}
  L^{(\frac12-g)}_{\text{v}}(\eta)\bigr)}
  =\eta^{\frac12-g-K}\times
  (-1)^{j-1}(g-\tfrac12-\text{v})_{j-1}\eta^{K+1-j}
  L^{(\frac32-g-j)}_{\text{v}}(\eta),
%  \label{Ldiff2j}
\end{align*}
in which $K$ is a positive integer. For the J system, we obtain
\begin{alignat*}{2}
  \text{J1}&:\ \ &\partial_{\eta}\bigl((\tfrac{1+\eta}{2})^{\frac12-h}
  P^{(g-\frac12,\frac12-h)}_{\text{v}}(\eta)\bigr)
  &=\tfrac12(\text{v}-h+\tfrac12)(\tfrac{1+\eta}{2})^{\frac12-(h+1)}
  P^{(g+1-\frac12,\frac12-(h+1))}_{\text{v}}(\eta),\\
%  \label{Jdiff1}\\
%
 \text{J2}&:\ \ &\partial_{\eta}\bigl((\tfrac{1-\eta}{2})^{\frac12-g}
  P^{(\frac12-g,h-\frac12)}_{\text{v}}(\eta)\bigr)
  &=-\tfrac12(\text{v}-g+\tfrac12)(\tfrac{1-\eta}{2})^{\frac12-(g+1)}
  P^{(\frac12-(g+1),h+1-\frac12)}_{\text{v}}(\eta),
%  \label{Jdiff2}
\end{alignat*}
by using \eqref{Jforward}\&\eqref{Jid1} and \eqref{Jforward}\&\eqref{Jid1m},
respectively.
Repeated applications of these formulas lead to 
\begin{align*}
  &\partial_{\eta}^{j-1}\bigl((\tfrac{1+\eta}{2})^{\frac12-h}
  P^{(g-\frac12,\frac12-h)}_{\text{v}}(\eta)\bigr)
  =\frac{(-1)^{j-1}}{2^{j-1}}(h-\tfrac12-\text{v})_{j-1}
  (\tfrac{1+\eta}{2})^{\frac32-h-j}
  P^{(g+j-\frac32,\frac32-h-j)}_{\text{v}}(\eta)\n
%
  &\qquad\qquad\quad
  =(\tfrac{1+\eta}{2})^{\frac12-h-K}\times
  \frac{(-1)^{j-1}}{2^{j-1}}(h-\tfrac12-\text{v})_{j-1}
  (\tfrac{1+\eta}{2})^{K+1-j}
  P^{(g+j-\frac32,\frac32-h-j)}_{\text{v}}(\eta),\\
%  \label{Jdiff1j}\\
%
  &\partial_{\eta}^{j-1}\bigl((\tfrac{1-\eta}{2})^{\frac12-g}
  P^{(\frac12-g,h-\frac12)}_{\text{v}}(\eta)\bigr)
  =\frac{1}{2^{j-1}}(g-\tfrac12-\text{v})_{j-1}
  (\tfrac{1-\eta}{2})^{\frac32-g-j}
  P^{(\frac32-g-j,h+j-\frac32)}_{\text{v}}(\eta)\n
%
  &\qquad\qquad\quad
  =(\tfrac{1-\eta}{2})^{\frac12-g-K}\times
  \frac{1}{2^{j-1}}(g-\tfrac12-\text{v})_{j-1}
  (\tfrac{1-\eta}{2})^{K+1-j}
  P^{(\frac32-g-j,h+j-\frac32)}_{\text{v}}(\eta).
%  \label{Jdiff2j}
\end{align*}
The higher derivatives of the eigenpolynomials $P_n(\eta)$ in \eqref{Pn} are
replaced simply through \eqref{Lforward} and \eqref{Jforward}, respectively,
\begin{align*}
  \partial_{\eta}^{j-1}L^{(g-\frac12)}_n(\eta)
  &=(-1)^{j-1}L^{(g+j-\frac32)}_{n+1-j}(\eta),\\
%  \label{Ldiff3j}\\
%
  \partial_{\eta}^{j-1}P^{(g-\frac12,h-\frac12)}_n(\eta)
  &=\frac{1}{2^{j-1}}(n+g+h)_{j-1}
  P^{(g+j-\frac32,h+j-\frac32)}_{n+1-j}(\eta),
%  \label{Jdiff3j}
\end{align*}
in which we adopt the convention
$L^{(\alpha)}_n(\eta)=P^{(\alpha,\beta)}_n(\eta)=0$ ($n\in\mathbb{Z}_{<0}$).

Let us define $M$ dimensional column vectors
$\vec{X}^{(M)}_{\text{v}}=\bigl(X^{(M)}_{\text{v},j}\bigr)_{j=1}^M$
and $\vec{Z}^{(M)}_n=\bigl(Z^{(M)}_{n,j}\bigr)_{j=1}^M$ by
\begin{align*}
  X^{(M)}_{\text{v},j}(\eta)&\eqdef\left\{
  \begin{array}{ll}
  L^{(g+j-\frac32)}_{\text{v}}(-\eta)
  &:\text{L, $\text{v}$ type $\I$}\\
  (-1)^{j-1}(g-\tfrac12-\text{v})_{j-1}\eta^{M-j}
  L^{(\frac32-g-j)}_{\text{v}}(\eta)
  &:\text{L, $\text{v}$ type $\II$}\\[4pt]
  \frac{(-1)^{j-1}}{2^{j-1}}(h-\tfrac12-\text{v})_{j-1}
  (\tfrac{1+\eta}{2})^{M-j}
  P^{(g+j-\frac32,\frac32-h-j)}_{\text{v}}(\eta)
  &:\text{J, $\text{v}$ type $\I$}\\[4pt]
  \frac{1}{2^{j-1}}(g-\tfrac12-\text{v})_{j-1}
  (\tfrac{1-\eta}{2})^{M-j}
  P^{(\frac32-g-j,h+j-\frac32)}_{\text{v}}(\eta)
  &:\text{J, $\text{v}$ type $\II$}
  \end{array}\right.,\\
%  \label{XM}\\
%
  Z^{(M)}_{n,j}(\eta)&\eqdef\left\{
  \begin{array}{ll}
  (-1)^{j-1}L^{(g+j-\frac32)}_{n+1-j}(\eta)
  &:\text{L}\\[4pt]
  \frac{1}{2^{j-1}}(n+g+h)_{j-1}
  P^{(g+j-\frac32,h+j-\frac32)}_{n+1-j}(\eta)
  &:\text{J}
  \end{array}\right..
%  \label{ZM}
\end{align*}
The Wronskians in \eqref{XiD}--\eqref{PDn} are replaced by ordinary
determinants consisting of these column vectors:
 \begin{align*}
  \text{W}[\mu_{d_1},\ldots,\mu_{d_M}](\eta)&=
  \Bigl|\vec{X}^{(M)}_{d_1}(\eta)\ \cdots\ \vec{X}^{(M)}_{d_M}(\eta)\Bigr|\n
%
  &\quad\times\left\{
  \begin{array}{ll}
  \bigl(e^{\eta}\bigr)^{M_{\I}}\bigl(\eta^{\frac32-g-M}\bigr)^{M_{\II}}
  &:\text{L}\\[6pt]
  \bigl((\frac{1+\eta}{2})^{\frac32-h-M}\bigr)^{M_{\I}}
  \bigl((\frac{1-\eta}{2})^{\frac32-g-M}\bigr)^{M_{\II}}
  &:\text{J}
  \end{array}\right.,\\[2pt]
%  \label{wronmu1}\\[2pt]
%
  \text{W}[\mu_{d_1},\ldots,\mu_{d_M},P_n](\eta)&=
  \Bigl|\vec{X}^{(M+1)}_{d_1}(\eta)\ \cdots\ \vec{X}^{(M+1)}_{d_M}(\eta)
  \ \vec{Z}^{(M+1)}_n(\eta)\Bigr|\n
%
  &\quad\times\left\{
  \begin{array}{ll}
  \bigl(e^{\eta}\bigr)^{M_{\I}}\bigl(\eta^{\frac12-g-M}\bigr)^{M_{\II}}
  &:\text{L}\\[6pt]
  \bigl((\frac{1+\eta}{2})^{\frac12-h-M}\bigr)^{M_{\I}}
  \bigl((\frac{1-\eta}{2})^{\frac12-g-M}\bigr)^{M_{\II}}
  &:\text{J}
  \end{array}\right..
%   \label{wronmu2}
\end{align*}
We arrive at the main result, the simple expressions of
$\Xi_{\mathcal{D}}(\eta)$ and $P_{\mathcal{D},n}(\eta)$
\begin{align}
  \Xi_{\mathcal{D}}(\eta)
  &=A\times
  \Bigl|\vec{X}^{(M)}_{d_1}(\eta)\ \cdots\ \vec{X}^{(M)}_{d_M}(\eta)\Bigr|,
  \label{XiDdet}\\
%
  P_{\mathcal{D},n}(\eta)
  &=A\times
  \Bigl|\vec{X}^{(M+1)}_{d_1}(\eta)\ \cdots\ \vec{X}^{(M+1)}_{d_M}(\eta)
  \ \vec{Z}^{(M+1)}_n(\eta)\Bigr|,
  \label{PDndet}\\[2pt]
%
  A&=\left\{
  \begin{array}{ll}
  \eta^{-M_{\II}(M_{\II}-1)}&:\text{L}\\[4pt]
  (\frac{1+\eta}{2})^{-M_{\I}(M_{\I}-1)}
  (\frac{1-\eta}{2})^{-M_{\II}(M_{\II}-1)}
  &:\text{J}
%  \end{array}\right..
  \end{array}\right..\nonumber
\end{align}
It should be stressed that the components of the matrices in \eqref{XiDdet}
and \eqref{PDndet} are all polynomials in $\eta$. This is a good contrast
with the starting Wronskians $\text{W}[\mu_{d_1},\ldots,\mu_{d_M}](\eta)$ and
$\text{W}[\mu_{d_1},\ldots,\mu_{d_M},P_n](\eta)$ in \eqref{XiD},
\eqref{PDn}, in which $\mu_{d_j}$'s have non-polynomial factors \eqref{muv}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.5 Simplified forms of the multi-indexed         %
%     Laguerre and Jacobi polynomials, B:           %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simplified forms of the multi-indexed Laguerre and Jacobi
polynomials, B:}
\label{sec:simpB}

Here we will simplify the Wronskians of the virtual state wavefunctions and
the eigenpolynomials \eqref{W=XiD}, \eqref{W=PDn}.
By replacing the even order derivatives of the virtual state wavefunctions
and eigenfunctions in the Wronskians \eqref{W=XiD}, \eqref{W=PDn}, by the
rule $\psi^{(2m)}(x)\to(-\mathcal{E})^m\psi(x)$, we obtain
\begin{align}
  &\text{W}[\tilde{\phi}_{d_1},\ldots,\tilde{\phi}_{d_M},\phi_n](x)
  =\det(a_{j,k})_{1\leq j,k\leq M+1},
  \label{W1}\\[2pt]
%
  &\ \ \left\{
  \begin{array}{rllrll}
  a_{2l-1,k}\!\!\!\!&=(-\tilde{\mathcal{E}}_{d_k})^{l-1}\tilde{\phi}_{d_k}(x)
  \ (1\leq k\leq M),
  &\!\!\!\!\!\!&a_{2l-1,M+1}\!\!\!\!&=(-\mathcal{E}_n)^{l-1}\phi_n(x)
  &\!(1\leq l\leq[\frac{M+2}{2}])\\[4pt]
  a_{2l,k}\!\!\!\!&=(-\tilde{\mathcal{E}}_{d_k})^{l-1}\tilde{\phi}'_{d_k}(x)
  \ (1\leq k\leq M),
  &\!\!\!\!\!\!&a_{2l,M+1}\!\!\!\!&=(-\mathcal{E}_n)^{l-1}\phi'_n(x)
  &\!(1\leq l\leq[\frac{M+1}{2}])
  \end{array}\right.\!\!,
  \nonumber
\end{align}
in which $[a]$ denotes the greatest integer not exceeding $a$.
The first derivatives in the $2l$-th row can be simplified by adding
$-\frac{\phi_0'(x)}{\phi_0(x)}\times(2l-1)$-th row,
\begin{align}
  \phi'_n(x)\to\Bigl(\frac{d}{dx}-\frac{\phi'_0(x)}{\phi_0(x)}\Bigr)\phi_n(x)
  &=\frac{\cF}{\eta'(x)}\,\phi_0(x)\zeta_n\bigl(\eta(x)\bigr)
  =\phi_0(x)\zeta_n\bigl(\eta(x)\bigr)\times A,\n
%  \label{Aphi}\\
%
  \tilde{\phi}'_{\text{v}}(x)\to
  \Bigl(\frac{d}{dx}-\frac{\phi'_0(x)}{\phi_0(x)}\Bigr)
  \tilde{\phi}_{\text{v}}(x)
  &=\frac{\cF}{\eta'(x)}\,\tilde{\phi}_0(x)
  \tilde{\zeta}_{\text{v}}\bigl(\eta(x)\bigr)
  =\tilde{\phi}_0(x)\tilde{\zeta}_{\text{v}}\bigl(\eta(x)\bigr)\times A,\n
%  \label{Aphi1}\\
%
  &A=\frac{\cF}{\eta'(x)}=\left\{
  \begin{array}{ll}
  x^{-1}&:\text{L}\\
  (\sin x\cos x)^{-1}&:\text{J}
  \end{array}\right.,
  \label{phi'}
\end{align}
in which $\zeta_n(\eta)$ and $\tilde{\zeta}_{\text v}(\eta)$ are polynomials
in $\eta$ defined by
\begin{align*}
  \zeta_n(\eta)&\eqdef\left\{
  \begin{array}{ll}
  -2\eta L^{(g+\frac12)}_{n-1}(\eta)
  &:\text{L}\\[4pt]
  -\tfrac12(n+g+h)(1-\eta^2)P^{(g+\frac12,h+\frac12)}_{n-1}(\eta)
  &:\text{J}
  \end{array}\right.,\\[2pt]
%
  \tilde{\zeta}_{\text{v}}(\eta)&\eqdef\left\{
  \begin{array}{ll}
  2\eta L^{(g+\frac12)}_{\text{v}}(-\eta)
  &:\text{L,\,$\I$}\\[4pt]
  -2(g-\tfrac12-\text{v})L^{(-g-\frac12)}_{\text{v}}(\eta)
  &:\text{L,\,$\II$}\\[4pt]
  (h-\tfrac12-\text{v})(1-\eta)
  P^{(g+\frac12,-h-\frac12)}_{\text{v}}(\eta)
  &:\text{J,\,$\I$}\\[4pt]
  -(g-\tfrac12-\text{v})(1+\eta)
  P^{(-g-\frac12,h+\frac12)}_{\text{v}}(\eta)
  &:\text{J,\,$\II$}
  \end{array}\right..
\end{align*}
Use is made of \eqref{Lforward}--\eqref{Lid2} for L and
\eqref{Jforward}--\eqref{Jid1m} for J.

By extracting the functions $\phi_0(x)$, $\tilde{\phi}_0(x)$ from each column
of the matrix $a_{j,k}$ \eqref{W1} and the factor $A$ of \eqref{phi'} from
the even rows, we arrive at another set of simplified determinant expressions
for the multi-indexed polynomials:
\begin{align}
  &P_{\mathcal{D},n}(\eta)=\cF^{-\frac12M(M+1)}
  \det(a_{j,k})_{1\leq j,k\leq M+1}\times A,
  \label{PDn1}\\[2pt]
%
  &\ \ \left\{
  \begin{array}{rllrll}
  a_{2l-1,k}\!\!\!\!&=(-\tilde{\mathcal{E}}_{d_k})^{l-1}\xi_{d_k}(\eta)
  \ (1\leq k\leq M),
  &\!\!\!\!\!\!&a_{2l-1,M+1}\!\!\!\!&=(-\mathcal{E}_n)^{l-1}P_n(\eta)
  &\!(1\leq l\leq[\frac{M+2}{2}])\\[4pt]
  a_{2l,k}\!\!\!\!&=(-\tilde{\mathcal{E}}_{d_k})^{l-1}\tilde{\zeta}_{d_k}(\eta)
  \ (1\leq k\leq M),
  &\!\!\!\!\!\!&a_{2l,M+1}\!\!\!\!&=(-\mathcal{E}_n)^{l-1}\zeta_n(\eta)
  &\!(1\leq l\leq[\frac{M+1}{2}])
  \end{array}\right.\!\!,\n[2pt]
%
  &\quad A=\left\{
  \begin{array}{ll}
  \eta^{-([M']+1)([M']+M-2[\frac{M}{2}])}
  &:\text{L}\\[2pt]
  \bigl(\tfrac{1-\eta}{2}\bigr)^{-([M']+1)([M']+M-2[\frac{M}{2}])}
  \bigl(\tfrac{1+\eta}{2}\bigr)^{-([-M']+1)([-M']+M-2[\frac{M}{2}])}
  &:\text{J}
  \end{array}\right..
  \nonumber
\end{align}
In a similar way, we obtain
\begin{align}
  &\Xi_{\mathcal{D}}(\eta)=\cF^{-\frac12M(M-1)}
  \det(a_{j,k})_{1\leq j,k\leq M}\times A,
  \label{XiD1}\\[2pt]
%
  &\ \ \left\{
  \begin{array}{rll}
  a_{2l-1,k}\!\!\!\!&=(-\tilde{\mathcal{E}}_{d_k})^{l-1}\xi_{d_k}(\eta)
  &\!(1\leq l\leq[\frac{M+1}{2}])\\[4pt]
  a_{2l,k}\!\!\!\!&=(-\tilde{\mathcal{E}}_{d_k})^{l-1}\tilde{\zeta}_{d_k}(\eta)
  &\!(1\leq l\leq[\frac{M}{2}])
  \end{array}\right.\!\!,
  \ \ A=\left\{
  \begin{array}{ll}
  \eta^{-[M']([M']+M-2[\frac{M}{2}])}
  &:\text{L}\\[2pt]
  \bigl(\tfrac{1-\eta}{2}\tfrac{1+\eta}{2}\bigr)^{-[M']([M']+M-2[\frac{M}{2}])}
  &:\text{J}
  \end{array}\right.\!\!.
  \nonumber
\end{align}
Again all the components of the matrices $a_{j,k}$ in \eqref{PDn1} and
\eqref{XiD1} are polynomials in $\eta$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                   %
% 2.6 Parity transformation of the multi-indexed    %
%     Jacobi polynomials                            %
%                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Parity transformation of the multi-indexed Jacobi polynomials}
\label{sec:pari}

The Jacobi polynomial has the parity transformation property \cite{szego}
\begin{equation}
  P^{(\alpha,\beta)}_n(-x)=(-1)^nP^{(\beta,\alpha)}_n(x).
  \label{P:J}
\end{equation}
We will show that this property is inherited by the multi-indexed Jacobi
polynomials.
It is based on the property of the Wronskian
\begin{equation*}
  \text{W}[f_1,\ldots,f_n](-\eta)
  =(-1)^{\frac12n(n-1)}\text{W}[g_1,\ldots,g_n](\eta),\quad
  g_k(\eta)\eqdef f_k(-\eta).
%  \label{Wid}
\end{equation*}

In this subsection, we indicate the types of the virtual states explicitly
by $(\text{v},\text{t})$, in which $\text{t}$ stands for type $\I$ or $\II$.
Based on \eqref{P:J}, we obtain
\begin{align*}
  \mu_{(\text{v},\text{t})}\bigl(-\eta;(g,h)\bigr)
  &=(-1)^{\text{v}}\mu_{(\text{v},\bar{\text{t}})}\bigl(\eta;(h,g)\bigr),\quad
  \bar{\text{t}}\eqdef\left\{
  \begin{array}{ll}
  \II&:\text{t}=\I\\
  \I&:\text{t}=\II
  \end{array}\right.,\\
%  \label{P:mu}\\
%
  P_n\bigl(-\eta;(g,h)\bigr)&=(-1)^nP_n\bigl(\eta;(h,g)\bigr).
%  \label{P:Pn}
\end{align*}
For the multi-index set
$\mathcal{D}=\{(d_1,\text{t}_1),\ldots,(d_M,\text{t}_M)\}$ of the virtual
state wavefunctions, let us define the `mirror reflected' multi-index set 
 $\mathcal{D}'\eqdef\{(d_1,\bar{\text{t}}_1),\ldots,(d_M,\bar{\text{t}}_M)\}$.
Corresponding to $M_{\I}\eqdef\#\{d_j|(d_j,\I)\in\mathcal{D}\}$,
$M_{\II}\eqdef\#\{d_j|(d_j,\II)\in\mathcal{D}\}$, we have
$M'_{\I}\eqdef\#\{d_j|(d_j,\I)\in\mathcal{D}'\}=M_{\II}$,
$M'_{\II}\eqdef\#\{d_j|(d_j,\II)\in\mathcal{D}'\}=M_{\I}$.
By parity transformation $\eta\to-\eta$, the multi-indexed Jacobi polynomial 
$P_{\mathcal{D},n}\bigl(\eta;(g,h)\bigr)$ is mapped to
$P_{\mathcal{D}',n}\bigl(\eta;(h,g)\bigr)$ with the `mirror reflected'
multi-index set $\mathcal{D}'$:
\begin{align*}
  &\quad P_{\mathcal{D},n}\bigl(-\eta;(g,h)\bigr)\n
%
  &=(\tfrac{1-\eta}{2})^{(M_{\II}+h+\frac12)M_{\I}}
  (\tfrac{1+\eta}{2})^{(M_{\I}+g+\frac12)M_{\II}}
  \text{W}[\mu_{(d_1,\text{t}_1)},\ldots,\mu_{(d_M,\text{t}_M)},P_n]
  \bigl(-\eta;(g,h)\bigr)\n
%
  &=(\tfrac{1-\eta}{2})^{(M_{\II}+h+\frac12)M_{\I}}
  (\tfrac{1+\eta}{2})^{(M_{\I}+g+\frac12)M_{\II}}
  (-1)^{\frac12M(M+1)}\n
%
  &\quad\times
  \text{W}[(-1)^{d_1}\mu_{(d_1,\bar{\text{t}}_1)},\ldots,
  (-1)^{d_M}\mu_{(d_M,\bar{\text{t}}_M)},(-1)^nP_n]\bigl(\eta;(h,g)\bigr)\n
%
  &=(\tfrac{1-\eta}{2})^{(M_{\II}+h+\frac12)M_{\I}}
  (\tfrac{1+\eta}{2})^{(M_{\I}+g+\frac12)M_{\II}}
  (-1)^{\frac12M(M+1)}(-1)^{d_1+\cdots+d_M+n}\n
%
  &\quad\times
  \text{W}[\mu_{(d_1,\bar{\text{t}}_1)},\ldots,\mu_{(d_M,\bar{\text{t}}_M)},
  P_n]\bigl(\eta;(h,g)\bigr)\n
%
  &=(-1)^{n+\frac12M(M+1)+\sum_{k=1}^Md_k}
  P_{\mathcal{D}',n}\bigl(\eta;(h,g)\bigr).
%  \label{P:PDn}
\end{align*}
Similarly, the denominator polynomial
$\Xi_{\mathcal{D}}\bigl(\eta;(g,h)\bigr)$ is mapped to
$\Xi_{\mathcal{D}'}\bigl(\eta;(h,g)\bigr)$ with a sign factor:
\begin{equation*}
  \Xi_{\mathcal{D}}\bigl(-\eta;(g,h)\bigr)
  =(-1)^{\frac12M(M-1)+\sum_{k=1}^Md_k}
  \Xi_{\mathcal{D}'}\bigl(\eta;(h,g)\bigr).
%  \label{P:XiD}
\end{equation*}
For the special case of `mirror symmetric' multi-index set
$\mathcal{D}'=\mathcal{D}$ (as a set), {\em i.e.\/}
$\{d_j|(d_j,\I)\in\mathcal{D}\}=\{d_j|(d_j,\II)\in\mathcal{D}\}$ (as a set),
we have $P_{\mathcal{D}',n}(\eta)=\pm P_{\mathcal{D},n}(\eta)$.
In fact, this formula turns out to be
\begin{equation*}
  P_{\mathcal{D}',n}(\eta)=(-1)^{(\frac{M}{2})^2}P_{\mathcal{D},n}(\eta).
%  \label{PD'n=PDn}
\end{equation*}
For this special case the parity transformation gives
\begin{equation*}
  P_{\mathcal{D},n}\bigl(-\eta;(g,h)\bigr)
  =(-1)^nP_{\mathcal{D},n}\bigl(\eta;(h,g)\bigr),\quad
   \Xi_{\mathcal{D}}\bigl(-\eta;(g,h)\bigr)
  =\Xi_{\mathcal{D}}\bigl(\eta;(h,g)\bigr).
%  \label{P:PDnspecial}
\end{equation*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                             %
%  3. Summary and Comments                                    %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Comments}
\label{sec:summary}

The multi-indexed Laguerre and Jacobi polynomials are defined by the
Wronskian expressions originating from multiple Darboux transformations.
Two simplified determinant expressions of them, \eqref{XiDdet}--\eqref{PDndet}
and \eqref{PDn1}--\eqref{XiD1}, which do not contain derivatives,
are derived based on the properties of the Wronskian and identities of the
Laguerre and Jacobi polynomials.
For \eqref{PDn1}--\eqref{XiD1}, the Schr\"odinger equation is used.
For \eqref{XiDdet}--\eqref{PDndet}, various identities of the
Laguerre and Jacobi polynomials are used, which are essentially forward shift
relations. 
Although the calculation in \S\,\ref{sec:simpA} is performed for polynomials,
it can be done for wavefunctions just like \cite{os29}, in which simplified
determinant expressions are presented for the multi-indexed polynomials
obtained by multiple Darboux transformations with pseudo virtual states
wavefunctions as seed solutions.
The parity transformation property of the multi-indexed Jacobi polynomials is
also derived.

Multi-indexed orthogonal polynomials have been constructed for the classical
orthogonal polynomials in the Askey scheme \cite{askey,kls}, {\em i.e.\/}, 
the Wilson, Askey-Wilson, Meixner, little $q$-Jacobi and ($q$-)\,Racah
polynomials, etc \cite{os26,os27,os35}.
These polynomials belong to `discrete' quantum mechanics \cite{os24}, in
which the Schr\"odinger equations are second order difference equations.
The Casoratian expressions of these multi-indexed polynomials can also be
simplified by using various identities as demonstrated here.
These simplifications will be published elsewhere \cite{odake16}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                             %
%  Acknowledgments                                            %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}

S.\,O. is supported in part by Grant-in-Aid for Scientific Research
from the Ministry of Education, Culture, Sports, Science and Technology
(MEXT), No.25400395.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                             %
%  References                                                 %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

%%%%% Askey-scheme
\bibitem{askey}
G.\,E.\,Andrews, R.\,Askey and R.\,Roy,
{\it Special Functions\/},
vol. 71 of Encyclopedia of mathematics and its applications,
Cambridge Univ. Press, Cambridge, (1999).

\bibitem{quesnebagchi}
B.\,Bagchi, C.\,Quesne and R.\,Roychoudhury,
``Isospectrality of conventional and new extended potentials,
second-order supersymmetry and role of PT symmetry,''
Pramana J. Phys. {\bf 73} (2009) 337-347,
{\tt arXiv:0812.1488[quant-ph]}.

\bibitem{bochner}
S.\,Bochner,
``\"Uber Sturm-Liouvillesche Polynomsysteme,"
Math. Zeit. {\bf 29} (1929) 730-736.
%%%%%  -------------------- 

\bibitem{crum}
M.\,M.\,Crum,
``Associated Sturm-Liouville systems,"
Quart. J. Math. Oxford Ser. (2) {\bf 6} (1955) 121-127,
{\tt arXiv:physics/9908019}.
%%%%%  -------------------- 

%%%%%%%% Darboux-Crum
\bibitem{darboux}
G.\,Darboux,
``Sur une proposition relative aux \'equations lin\'eaires,"
{\it C. R. Acad. Paris} {\bf 94} (1882) 1456-1459.

%%%%% Duran
\bibitem{d14}
A.\,J.\,Dur\'{a}n,
``Exceptional Meixner and Laguerre orthogonal polynomials,"
J. Approx. Theory {\bf 184} (2014) 176-208,
{\tt arXiv:1310.4658[math.CA]}.
%%%%%  -------------------- 

%%%% added Duran and Perez
\bibitem{d17}
A.\,J.\,Dur\'{a}n,
``Exceptional Hahn and Jacobi orthogonal polynomials,"
J. Approx. Theory {\bf 214} (2017) 9-48,
{\tt arXiv:1510.02579[math.CA]}.

\bibitem{dp15}
A.\,J.\,Dur\'{a}n and M.\,P\'{e}rez,
``Admissibility condition for exceptional Laguerre polynomials,''
J. Math. Anal. Appl. {\bf 424} (2015) 1042-1053,
{\tt arXiv:1409.4901[math.CA]}.
%%%%%  --------------------

\bibitem{gomez0}
D.\,G\'{o}mez-Ullate, N.\,Kamran and R.\,Milson,
``An extended class of orthogonal polynomials defined by a
Sturm-Liouville problem,''
J. Math. Anal. Appl. {\bf 359} (2009) 352-367,
{\tt arXiv:0807.3939[math-\hspace{0pt}ph]}.

\bibitem{gomez}
D.\,G\'{o}mez-Ullate, N.\,Kamran and R.\,Milson,
``An extension of Bochner's problem: exceptional invariant subspaces,''
J. Approx Theory {\bf 162} (2010) 987-1006,
{\tt arXiv:0805.\hspace{0pt}3376[math-ph]};

\bibitem{gomez2}
D.\,G\'{o}mez-Ullate, N.\,Kamran and R.\,Milson,
``Two-step Darboux transformations and exceptional Laguerre polynomials,"
J. Math. Anal. Appl. {\bf 387} (2012) 410-418,
{\tt arXiv:\hspace{0pt}1103.5724[math-ph]}.

\bibitem{kls}
R.\,Koekoek, P.\,A.\,Lesky and R.\,F.\,Swarttouw,
{\it Hypergeometric orthogonal polynomials and their $q$-analogues,\/}
Springer-Verlag Berlin-Heidelberg (2010).
%%%%%  --------------------

\bibitem{odake16}
S.\,Odake, in preparation. ($\to$ appeared as {\tt arXiv:1702.03078[math-ph]})

\bibitem{rrmiop3}
S.\,Odake,
``Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : $\III$,"
J. Math. Phys. {\bf 57} (2016) 023514 (24pp),
{\tt arXiv:1509.08213[math-ph]}.

\bibitem{os16}
S.\,Odake and R.\,Sasaki,
``Infinitely many shape invariant potentials and new orthogonal polynomials,''
Phys. Lett. {\bf B679} (2009) 414-417,
{\tt arXiv:0906.0142[math-ph]}.

%%%%% dQM
\bibitem{os24}
S.\,Odake and R.\,Sasaki,
``Discrete quantum mechanics,'' (Topical Review)
J. Phys. {\bf A44} (2011) 353001 (47pp),
{\tt arXiv:1104.0473[math-ph]}.
%%%%%  --------------------

\bibitem{os25}
S.\,Odake and R.\,Sasaki,
``Exactly solvable quantum mechanics and infinite families of
multi-indexed orthogonal polynomials,"
Phys. Lett. {\bf B702} (2011) 164-170,
{\tt arXiv:1105.\hspace{0pt}0508[math-ph]}.

\bibitem{os26}
S.\,Odake and R.\,Sasaki,
``Multi-indexed ($q$-)Racah polynomials,"
J. Phys. {\bf A 45} (2012) 385201 (21pp),
{\tt arXiv:1203.5868[math-ph]}.

\bibitem{os27}
S.\,Odake and R.\,Sasaki,
``Multi-indexed Wilson and Askey-Wilson polynomials,"
J. Phys. {\bf A46} (2013) 045204 (22pp),
{\tt arXiv:1207.5584[math-ph]}.

%%%%%% pseudovirtual
\bibitem{os29}
S.\,Odake and R.\,Sasaki,
``Krein-Adler transformations for shape-invariant potentials and pseudo
virtual states,"
J. Phys. {\bf A46} (2013) 245201 (24pp),
{\tt arXiv:1212.6595[math-\hspace{0pt}ph]}.
%%%%%  -------------------- 

\bibitem{os35}
S.\,Odake and R.\,Sasaki,
``Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials,"
J. Phys. {\bf A50} (2017) 165204 (23pp),
{\tt arXiv:1610.09854[math.CA]}.

\bibitem{quesne}
C.\,Quesne,
``Exceptional orthogonal polynomials, exactly solvable potentials
and supersymmetry,''
J. Phys. {\bf A41} (2008) 392001 (6pp),
{\tt arXiv:0807.4087[quant-ph]};

%%%% Routh-Bochner
\bibitem{routh}
E.\,Routh,
``On some properties of certain solutions of a differential equation of
the second order,"
Proc. London Math. Soc. {\bf 16} (1884) 245-261.

%%%%% szego
\bibitem{szego}
G.\,Szeg\"o,
{\it Orthogonal polynomials},
Amer. Math. Soc. New York (1939).
%%%%%  -------------------- 

\end{thebibliography}

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  End                     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  references : original order                            %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{gomez}
D.\,G\'{o}mez-Ullate, N.\,Kamran and R.\,Milson,
``An extension of Bochner's problem: exceptional invariant subspaces,''
J. Approx Theory {\bf 162} (2010) 987-1006,
{\tt arXiv:0805.\hspace{0pt}3376[math-ph]};
%
``An extended class of orthogonal polynomials defined by a
Sturm-Liouville problem,''
J. Math. Anal. Appl. {\bf 359} (2009) 352-367,
{\tt arXiv:0807.3939[math-\hspace{0pt}ph]}.

\bibitem{quesne}
C.\,Quesne,
``Exceptional orthogonal polynomials, exactly solvable potentials
and supersymmetry,''
J. Phys. {\bf A41} (2008) 392001,
{\tt arXiv:0807.4087[quant-ph]};
%
B.\,Bagchi, C.\,Quesne and R.\,Roychoudhury,
``Isospectrality of conventional and new extended potentials,
second-order supersymmetry and role of PT symmetry,''
Pramana J. Phys. {\bf 73} (2009) 337-347,
{\tt arXiv:0812.1488[quant-ph]}.

\bibitem{os16}
S.\,Odake and R.\,Sasaki,
``Infinitely many shape invariant potentials and new orthogonal polynomials,''
Phys. Lett. {\bf B679} (2009) 414-417,
{\tt arXiv:0906.0142[math-ph]}.

\bibitem{gomez2}
D.\,G\'{o}mez-Ullate, N.\,Kamran and R.\,Milson,
``Two-step Darboux transformations and exceptional Laguerre polynomials,"
J. Math. Anal. Appl. {\bf 387} (2012) 410-418,
{\tt arXiv:\hspace{0pt}1103.5724[math-ph]}.

\bibitem{os25}
S.\,Odake and R.\,Sasaki,
``Exactly solvable quantum mechanics and infinite families of
multi-indexed orthogonal polynomials,"
Phys. Lett. {\bf B702} (2011) 164-170,
{\tt arXiv:1105.\hspace{0pt}0508[math-ph]}.

\bibitem{os26}
S.\,Odake and R.\,Sasaki,
``Multi-indexed ($q$-)Racah polynomials,"
J. Phys. {\bf A 45} (2012) 385201 (21pp),
{\tt arXiv:1203.5868[math-ph]}.

\bibitem{os27}
S.\,Odake and R.\,Sasaki,
``Multi-indexed Wilson and Askey-Wilson polynomials,"
J. Phys. {\bf A46} (2013) 045204 (22pp),
{\tt arXiv:1207.5584[math-ph]}.

\bibitem{os35}
S.\,Odake and R.\,Sasaki,
``Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials,"
{\tt arXiv:1610.09854[math.CA]}.

%%%% Routh-Bochner
\bibitem{routh}
E.\,Routh,
``On some properties of certain solutions of a differential equation of
the second order,"
Proc. London Math. Soc. {\bf 16} (1884) 245-261.

\bibitem{bochner}
S.\,Bochner,
``\"Uber Sturm-Liouvillesche Polynomsysteme,"
Math. Zeit. {\bf 29} (1929) 730-736.
%%%%%  -------------------- 

%%%%%%%% Darboux-Crum
\bibitem{darboux}
G.\,Darboux,
``Sur une proposition relative aux \'equations lin\'eaires,"
{\it C. R. Acad. Paris} {\bf 94} (1882) 1456-1459.

\bibitem{crum}
M.\,M.\,Crum,
``Associated Sturm-Liouville systems,"
Quart. J. Math. Oxford Ser. (2) {\bf 6} (1955) 121-127,
{\tt arXiv:physics/9908019}.
%%%%%  -------------------- 

%%%%% szego
\bibitem{szego}
G.\,Szeg\"o,
{\it Orthogonal polynomials},
Amer. Math. Soc. New York (1939).
%%%%%  -------------------- 

%%%%% Duran
\bibitem{d14}
A.\,J.\,Dur\'{a}n,
``Exceptional Meixner and Laguerre orthogonal polynomials,"
J. Approx. Theory {\bf 184} (2014) 176-208,
{\tt arXiv:1310.4658[math.CA]}.
%%%%%  -------------------- 

%%%% added Duran and Perez
\bibitem{d17}
A.\,J.\,Dur\'{a}n,
``Exceptional Hahn and Jacobi orthogonal polynomials,"
J. Approx. Theory {\bf 214} (2017) 9-48,
{\tt arXiv:1510.02579[math.CA]}.

\bibitem{dp15}
A.\,J.\,Dur\'{a}n and M.\,P\'{e}rez,
``Admissibility condition for exceptional Laguerre polynomials,''
J. Math. Anal. Appl. {\bf 424} (2015) 1042-1053,
{\tt arXiv:1409.4901[math.CA]}.
%%%%%  --------------------

\bibitem{rrmiop3}
S.\,Odake,
``Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : $\III$,"
J. Math. Phys. {\bf 57} (2016) 023514 (24pp),
{\tt arXiv:1509.08213[math-ph]}.

%%%%%% pseudovirtual
\bibitem{os29}
S.\,Odake and R.\,Sasaki,
``Krein-Adler transformations for shape-invariant potentials and pseudo
virtual states,"
J. Phys. {\bf A46} (2013) 245201 (24pp),
{\tt arXiv:1212.6595[math-\hspace{0pt}ph]}.
%%%%%  -------------------- 

%%%%% Askey-scheme
\bibitem{askey}
G.\,E.\,Andrews, R.\,Askey and R.\,Roy,
{\it Special Functions\/},
vol. 71 of Encyclopedia of mathematics and its applications,
Cambridge Univ. Press, Cambridge, (1999).

\bibitem{kls}
R.\,Koekoek, P.\,A.\,Lesky and R.\,F.\,Swarttouw,
{\it Hypergeometric orthogonal polynomials and their $q$-analogues,\/}
Springer-Verlag Berlin-Heidelberg (2010).
%%%%%  --------------------

%%%%% dQM
\bibitem{os24}
S.\,Odake and R.\,Sasaki,
``Discrete quantum mechanics,'' (Topical Review)
J. Phys. {\bf A44} (2011) 353001 (47pp),
{\tt arXiv:1104.0473[math-ph]}.
%%%%%  --------------------

\bibitem{odake16}
S.\,Odake, in preparation. ($\to$ appeared as {\tt arXiv:1702.03078[math-ph]})

\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  references : original order                            %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
