Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

Satoru ODAKE and Ryu SASAKI S. Odake and R. Sasaki Faculty of Science, Shinshu
University, Matsumoto 390-8621, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received December 30, 2016, in final form March 23, 2017; Published online
March 29, 2017

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.

multi-indexed orthogonal polynomials; Laguerre and Jacobi polynomials;
Wronskian formula; determinant formula

42C05; 33C45; 34A05

##  1 Introduction

The exceptional and multi-indexed orthogonal polynomials [2, 9, 10, 11, 15,
17, 18, 19, 21, 22] 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 [3, 23]. They are constructed from the original
quantum mechanical systems, the radial oscillator potential and the
Pöschl–Teller potential, by multiple application of Darboux transformations
[4, 5] in terms of seed solutions called virtual state wavefunctions which are
generated by two types of discrete symmetry transformations [17, 18, 19, 21].

The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions
[17] originating from multiple Darboux transformations [4]. 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 [24]. These simplified expressions show
explicitly the constituents of the multi-indexed orthogonal polynomials and
they are helpful for deeper understanding. In [6] and [7] Durán employed
similar simplified expressions as the starting point of his exposition of the
exceptional Laguerre and Jacobi polynomials. See also [8] by Durán and Pérez.
In their expressions, the matrix elements of the determinants are polynomials.
In the original expressions in [17], the matrix elements of the determinants
contain the non-polynomial factors, see (10)–(13). In the simplified
expressions presented in this paper in Sections 2.4 and 2.5, they are also all
polynomials.

This short note is organised as follows. In Section 2.1 the quantum mechanical
settings for the original Laguerre and Jacobi polynomials are recapitulated.
That is, the Hamiltonians with the radial oscillator potential and the
Pöschl–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
Section 2.2. 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 Section 2.3 the Wronskian forms of the multi-indexed
Laguerre and Jacobi polynomials derived in [17] and [14] are recapitulated as
the starting point. Sections 2.4 and 2.5 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 Section 2.4 by using
various identities of the original Laguerre and Jacobi polynomials. The second
simplified expressions, to be derived in Section 2.5, are the consequences of
the multi-linearity of determinants and the form of the Schrödinger equation
$\psi^{\prime\prime}(x)=(U(x)-\mathcal{E})\psi(x)$. Every even order
derivative $\psi^{(2m)}(x)$ in the Wronskian can be replaced by
$(-\mathcal{E})^{m}\psi(x)$. In Section 2.6 the parity transformation formula
of the multi-indexed Jacobi polynomials is presented. Section 3 is for a
summary and comments.

##  2 Multi-indexed Laguerre and Jacobi polynomials

The foundation of the theory of multi-indexed orthogonal polynomials is the
exactly solvable one-dimensional quantum mechanical system $\mathcal{H}$:

$\displaystyle\mathcal{H}\phi_{n}(x)=\mathcal{E}_{n}\phi_{n}(x),\quad
n=0,1,\ldots,\qquad\mathcal{H}=-\frac{d^{2}}{dx^{2}}+U(x),$ (1)

and its iso-spectrally deformed system

$\displaystyle\mathcal{H}_{\mathcal{D}}\phi_{\mathcal{D}\,n}(x)=\mathcal{E}_{n}\phi_{\mathcal{D}\,n}(x),\quad
n=0,1,\ldots,\qquad\mathcal{H}_{\mathcal{D}}=-\frac{d^{2}}{dx^{2}}+U_{\mathcal{D}}(x),$
(2)

in terms of multiple application of Darboux transformations [4, 17]. 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 [17] with
slight modifications for simplification sake.

###  2.1 Original Laguerre and Jacobi polynomials

####  2.1.1 Radial oscillator potential

The Hamiltonian with the radial oscillator potential

$\displaystyle
U(x)\stackrel{{\scriptstyle\text{def}}}{{=}}x^{2}+\frac{g(g-1)}{x^{2}}-2g-1,\qquad
0<x<\infty,\qquad g>\frac{1}{2},$ (3)

has the Laguerre polynomials $L^{(\alpha)}_{n}(\eta)$ as the main part of the
eigenfunctions:

$\displaystyle\phi_{n}(x;g)\stackrel{{\scriptstyle\text{def}}}{{=}}\phi_{0}(x;g)L^{(g-\frac{1}{2})}_{n}\bigl{(}\eta(x)\bigr{)},\qquad\mathcal{E}_{n}\stackrel{{\scriptstyle\text{def}}}{{=}}4n,\qquad\eta(x)\stackrel{{\scriptstyle\text{def}}}{{=}}x^{2},$
$\displaystyle\phi_{0}(x;g)\stackrel{{\scriptstyle\text{def}}}{{=}}e^{-\frac{1}{2}x^{2}}x^{g},\qquad
L_{n}^{(\alpha)}(\eta)=\frac{1}{n!}\sum_{k=0}^{n}\frac{(-n)_{k}}{k!}(\alpha+k+1)_{n-k}\eta^{k}.$

Here are some identities of the Laguerre polynomials [24] to be used in
Sections 2.4 and 2.5,

$\displaystyle\partial_{\eta}L_{n}^{(\alpha)}(\eta)=-L_{n-1}^{(\alpha+1)}(\eta),$
(4) $\displaystyle
L_{n}^{(\alpha)}(\eta)-L_{n}^{(\alpha-1)}(\eta)=L_{n-1}^{(\alpha)}(\eta),$ (5)
$\displaystyle\eta L_{n-1}^{(\alpha+1)}(\eta)-\alpha
L_{n-1}^{(\alpha)}(\eta)=-nL_{n}^{(\alpha-1)}(\eta).$ (6)

As is clear from (3), 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
$(a)_{n}\stackrel{{\scriptstyle\text{def}}}{{=}}\prod\limits_{j=1}^{n}(a+j-1)$
is the shifted factorial or the so-called Pochhammer’s symbol.

####  2.1.2 Pöschl–Teller potential

The Hamiltonian with the Pöschl–Teller potential

$\displaystyle
U(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{g(g-1)}{\sin^{2}x}+\frac{h(h-1)}{\cos^{2}x}-(g+h)^{2},\qquad
0<x<\frac{\pi}{2},\qquad g>\frac{1}{2},\qquad h>\frac{1}{2},$

has the Jacobi polynomials $P_{n}^{(\alpha,\beta)}(\eta)$ as the main part of
the eigenfunctions:

$\displaystyle\phi_{n}\bigl{(}x;(g,h)\bigr{)}\stackrel{{\scriptstyle\text{def}}}{{=}}\phi_{0}\bigl{(}x;(g,h)\bigr{)}P_{n}^{(g-\frac{1}{2},h-\frac{1}{2})}\bigl{(}\eta(x)\bigr{)},\qquad\eta(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\cos
2x,$
$\displaystyle\phi_{0}\bigl{(}x;(g,h)\bigr{)}\stackrel{{\scriptstyle\text{def}}}{{=}}(\sin
x)^{g}(\cos
x)^{h},\qquad\mathcal{E}_{n}(g,h)\stackrel{{\scriptstyle\text{def}}}{{=}}4n(n+g+h),$
$\displaystyle
P_{n}^{(\alpha,\beta)}(\eta)=\frac{(\alpha+1)_{n}}{n!}\sum_{k=0}^{n}\frac{1}{k!}\frac{(-n)_{k}(n+\alpha+\beta+1)_{k}}{(\alpha+1)_{k}}\left(\frac{1-\eta}{2}\right)^{k}.$

Here are some identities of the Jacobi polynomials [24] to be used in in
Sections 2.4 and 2.5,

$\displaystyle\partial_{\eta}P_{n}^{(\alpha,\beta)}(\eta)=\tfrac{1}{2}(n+\alpha+\beta+1)P_{n-1}^{(\alpha+1,\beta+1)}(\eta),$
(7) $\displaystyle(n+\beta)P_{n}^{(\alpha,\beta-1)}(\eta)-\beta
P_{n}^{(\alpha-1,\beta)}(\eta)=(n+\alpha+\beta)\tfrac{1+\eta}{2}P_{n-1}^{(\alpha,\beta+1)}(\eta),$
(8) $\displaystyle(n+\alpha)P_{n}^{(\alpha-1,\beta)}(\eta)-\alpha
P_{n}^{(\alpha,\beta-1)}(\eta)=-(n+\alpha+\beta)\tfrac{1-\eta}{2}P_{n-1}^{(\alpha+1,\beta)}(\eta).$
(9)

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

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 [17] for more detail.

####  2.2.1 Laguerre (L) system

Type I transformation. It is obvious that the transformation $x\to ix$ is the
symmetry of the radial oscillator system. The seed solutions are

$\displaystyle\text{L1}\colon\quad$
$\displaystyle\mathcal{H}\tilde{\phi}^{\text{I}}_{\text{v}}(x)=\tilde{\mathcal{E}}^{\text{I}}_{\text{v}}\tilde{\phi}^{\text{I}}_{\text{v}}(x),\qquad\tilde{\phi}^{\text{I}}_{\text{v}}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\tilde{\phi}^{\text{I}}_{0}(x)\xi^{\text{I}}_{\text{v}}\bigl{(}\eta(x)\bigr{)},\qquad\tilde{\phi}^{\text{I}}_{0}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}e^{\frac{1}{2}x^{2}}x^{g},$
$\displaystyle\xi^{\text{I}}_{\text{v}}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}L^{(g-\frac{1}{2})}_{\text{v}}(-\eta),\qquad\tilde{\mathcal{E}}^{\text{I}}_{\text{v}}\stackrel{{\scriptstyle\text{def}}}{{=}}-4(g+\text{v}+\tfrac{1}{2}),\qquad\text{v}\in\mathbb{Z}_{\geq
0}.$

Hereafter we use 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. The exchange of the characteristic exponents of the
regular singular point $x=0$, $g\to 1-g$, is another discrete symmetry
transformation. It generates the seed solutions,

$\displaystyle\text{L2}\colon\quad$
$\displaystyle\mathcal{H}\tilde{\phi}^{\text{II}}_{\text{v}}(x)=\tilde{\mathcal{E}}^{\text{II}}_{\text{v}}\tilde{\phi}^{\text{II}}_{\text{v}}(x),\qquad\tilde{\phi}^{\text{II}}_{\text{v}}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\tilde{\phi}^{\text{II}}_{0}(x)\xi^{\text{II}}_{\text{v}}\bigl{(}\eta(x)\bigr{)},\qquad\tilde{\phi}^{\text{II}}_{0}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}e^{-\frac{1}{2}x^{2}}x^{1-g},$
$\displaystyle\xi^{\text{II}}_{\text{v}}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}L^{(\frac{1}{2}-g)}_{\text{v}}(\eta),\qquad\tilde{\mathcal{E}}^{\text{II}}_{\text{v}}\stackrel{{\scriptstyle\text{def}}}{{=}}-4\big{(}g-\text{v}-\tfrac{1}{2}\big{)},\qquad\text{v}=0,1,\ldots,\big{[}g-\tfrac{1}{2}\big{]}^{\prime},$

in which $[a]^{\prime}$ denotes the greatest integer less than $a$.

####  2.2.2 Jacobi (J) system

Type I transformation. The exchange of the characteristic exponents of the
regular singular point $x=\frac{\pi}{2}$, $h\to 1-h$, is a discrete symmetry
transformation:

$\displaystyle\text{J1}\colon\quad$
$\displaystyle\mathcal{H}\tilde{\phi}^{\text{I}}_{\text{v}}(x)=\tilde{\mathcal{E}}^{\text{I}}_{\text{v}}\tilde{\phi}^{\text{I}}_{\text{v}}(x),\qquad\tilde{\phi}^{\text{I}}_{\text{v}}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\tilde{\phi}^{\text{I}}_{0}(x)\xi^{\text{I}}_{\text{v}}\bigl{(}\eta(x)\bigr{)},\qquad\tilde{\phi}^{\text{I}}_{0}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}(\sin
x)^{g}(\cos x)^{1-h},$
$\displaystyle\xi^{\text{I}}_{\text{v}}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}P^{(g-\frac{1}{2},\frac{1}{2}-h)}_{\text{v}}(\eta),\qquad\tilde{\mathcal{E}}^{\text{I}}_{\text{v}}\stackrel{{\scriptstyle\text{def}}}{{=}}-4\big{(}g+\text{v}+\tfrac{1}{2}\big{)}\big{(}h-\text{v}-\tfrac{1}{2}\big{)},$
$\displaystyle\text{v}=0,1,\ldots,\big{[}h-\tfrac{1}{2}\big{]}^{\prime}.$

Type II transformation. Likewise the exchange of the characteristic exponents
of the regular singular point $x=0$, $g\to 1-g$, is a discrete symmetry
transformation:

$\displaystyle\text{J2}\colon\quad$
$\displaystyle\mathcal{H}\tilde{\phi}^{\text{II}}_{\text{v}}(x)=\tilde{\mathcal{E}}^{\text{II}}_{\text{v}}\tilde{\phi}^{\text{II}}_{\text{v}}(x),\qquad\tilde{\phi}^{\text{II}}_{\text{v}}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\tilde{\phi}^{\text{II}}_{0}(x)\xi^{\text{II}}_{\text{v}}\bigl{(}\eta(x)\bigr{)},\qquad\tilde{\phi}^{\text{II}}_{0}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}(\sin
x)^{1-g}(\cos x)^{h},$
$\displaystyle\xi^{\text{II}}_{\text{v}}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}P^{(\frac{1}{2}-g,h-\frac{1}{2})}_{\text{v}}(\eta),\qquad\tilde{\mathcal{E}}^{\text{II}}_{\text{v}}\stackrel{{\scriptstyle\text{def}}}{{=}}-4\big{(}g-\text{v}-\tfrac{1}{2}\big{)}\big{(}h+\text{v}+\tfrac{1}{2}\big{)},$
$\displaystyle\text{v}=0,1,\ldots,\big{[}g-\tfrac{1}{2}\big{]}^{\prime}.$

###  2.3 Wronskian forms of the multi-indexed Laguerre and Jacobi polynomials

We deform the original Hamiltonian system (1) by applying multiple Darboux
transformations [4, 17] in terms of $M_{\text{I}}$ Type I seed solutions and
$M_{\text{II}}$ Type II seed solutions specified by the degree set
$\mathcal{D}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{d_{1},\ldots,d_{M}\\}$
(ordered set),
$M\stackrel{{\scriptstyle\text{def}}}{{=}}M_{\text{I}}+M_{\text{II}}$,
$M_{\text{I}}\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{d_{j}\,|\,d_{j}\in\mathcal{D},\,\text{Type
$\text{I}$}\\}$,
$M_{\text{II}}\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{d_{j}\,|\,d_{j}\in\mathcal{D},\,\text{Type
$\text{II}$}\\}$. 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
$\mathcal{H}_{\mathcal{D}}$ (2):

$\displaystyle
U_{\mathcal{D}}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}U(x)-2\partial_{x}^{2}\log\bigl{|}\text{W}\big{[}\tilde{\phi}_{d_{1}},\ldots,\tilde{\phi}_{d_{M}}\big{]}(x)\bigr{|},$
$\displaystyle\phi_{\mathcal{D}\,n}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{\text{W}\big{[}\tilde{\phi}_{d_{1}},\ldots,\tilde{\phi}_{d_{M}},\phi_{n}\big{]}(x)}{\text{W}\big{[}\tilde{\phi}_{d_{1}},\ldots,\tilde{\phi}_{d_{M}}\big{]}(x)}\stackrel{{\scriptstyle\text{def}}}{{=}}c_{\text{\tiny$\mathcal{F}$}}^{M}\psi_{\mathcal{D}}(x)P_{\mathcal{D},n}\bigl{(}\eta(x)\bigr{)},\qquad
n=0,1,\ldots,$
$\displaystyle\psi_{\mathcal{D}}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{\hat{\phi}_{0}(x)}{\Xi_{\mathcal{D}}\bigl{(}\eta(x)\bigr{)}},$

in which $\hat{\phi}_{0}(x)$ and $c_{\text{\tiny$\mathcal{F}$}}$ are defined
by

$\displaystyle\hat{\phi}_{0}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}\phi_{0}(x;g+M_{\text{I}}-M_{\text{II}}),&\text{L},\\\
\phi_{0}\bigl{(}x;(g+M_{\text{I}}-M_{\text{II}},h-M_{\text{I}}+M_{\text{II}})\bigr{)},&\text{J},\end{cases}\qquad
c_{\text{\tiny$\mathcal{F}$}}\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}\hphantom{-}2,&\text{L},\\\
-4,&\text{J}.\end{cases}$

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

$\displaystyle\text{W}[f_{1},\ldots,f_{n}](x)\stackrel{{\scriptstyle\text{def}}}{{=}}\det\left(\frac{d^{j-1}f_{k}(x)}{dx^{j-1}}\right)_{1\leq
j,k\leq n}.$

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’ [17]:

$\displaystyle\Xi_{\mathcal{D}}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\text{W}[\mu_{d_{1}},\ldots,\mu_{d_{M}}](\eta)\times\begin{cases}\eta^{(M_{\text{I}}+g-\frac{1}{2})M_{\text{II}}}e^{-M_{\text{I}}\eta},&\text{L},\\\
\bigl{(}\frac{1-\eta}{2}\bigr{)}^{(M_{\text{I}}+g-\frac{1}{2})M_{\text{II}}}\bigl{(}\frac{1+\eta}{2}\bigr{)}^{(M_{\text{II}}+h-\frac{1}{2})M_{\text{I}}},&\text{J},\end{cases}$
(10) $\displaystyle
P_{\mathcal{D},n}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\text{W}[\mu_{d_{1}},\ldots,\mu_{d_{M}},P_{n}](\eta)\times\begin{cases}\eta^{(M_{\text{I}}+g+\frac{1}{2})M_{\text{II}}}e^{-M_{\text{I}}\eta},&\text{L},\\\
\bigl{(}\frac{1-\eta}{2}\bigr{)}^{(M_{\text{I}}+g+\frac{1}{2})M_{\text{II}}}\bigl{(}\frac{1+\eta}{2}\bigr{)}^{(M_{\text{II}}+h+\frac{1}{2})M_{\text{I}}},&\text{J},\end{cases}$
(11)
$\displaystyle\mu_{\text{v}}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}e^{\eta}L^{(g-\frac{1}{2})}_{\text{v}}(-\eta),&\text{L,
$\text{v}$ Type $\text{I}$},\\\
\eta^{\frac{1}{2}-g}L^{(\frac{1}{2}-g)}_{\text{v}}(\eta),&\text{L, $\text{v}$
Type $\text{II}$},\\\
\bigl{(}\frac{1+\eta}{2}\bigr{)}^{\frac{1}{2}-h}P^{(g-\frac{1}{2},\frac{1}{2}-h)}_{\text{v}}(\eta),&\text{J,
$\text{v}$ Type $\text{I}$},\\\
\bigl{(}\frac{1-\eta}{2}\bigr{)}^{\frac{1}{2}-g}P^{(\frac{1}{2}-g,h-\frac{1}{2})}_{\text{v}}(\eta),&\text{J,
$\text{v}$ Type~{}$\text{II}$},\end{cases}$ (12) $\displaystyle
P_{n}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}L^{(g-\frac{1}{2})}_{n}(\eta),&\text{L},\\\
P^{(g-\frac{1}{2},h-\frac{1}{2})}_{n}(\eta),&\text{J}.\end{cases}$ (13)

The second is based on the Wronskians of the virtual state wavefunctions
$\tilde{\phi}_{\text{v}}(x)$ and the eigenfunction $\phi_{n}(x)$ [14]:

$\displaystyle\Xi_{\mathcal{D}}(\eta)=c_{\text{\tiny$\mathcal{F}$}}^{-\frac{1}{2}M(M-1)}\text{W}\big{[}\tilde{\phi}_{d_{1}},\ldots,\tilde{\phi}_{d_{M}}\big{]}(x)$
$\displaystyle\hphantom{\Xi_{\mathcal{D}}(\eta)=}{}\times\begin{cases}\eta^{-M^{\prime}(M^{\prime}+g-\frac{1}{2})}e^{-M^{\prime}\eta},&\text{L},\\\
\bigl{(}\frac{1-\eta}{2}\bigr{)}^{-M^{\prime}(M^{\prime}+g-\frac{1}{2})}\bigl{(}\frac{1+\eta}{2}\bigr{)}^{-M^{\prime}(M^{\prime}-h+\frac{1}{2})},&\text{J},\end{cases}$
(14) $\displaystyle
P_{\mathcal{D},n}(\eta)=c_{\text{\tiny$\mathcal{F}$}}^{-\frac{1}{2}M(M+1)}\text{W}\big{[}\tilde{\phi}_{d_{1}},\ldots,\tilde{\phi}_{d_{M}},\phi_{n}\big{]}(x)$
$\displaystyle\hphantom{P_{\mathcal{D},n}(\eta)=}{}\times\begin{cases}\eta^{-(M^{\prime}+\frac{1}{2})(M^{\prime}+g)}e^{-(M^{\prime}-\frac{1}{2})\eta},&\text{L},\\\
\bigl{(}\frac{1-\eta}{2}\bigr{)}^{-(M^{\prime}+\frac{1}{2})(M^{\prime}+g)}\bigl{(}\frac{1+\eta}{2}\bigr{)}^{-(M^{\prime}-\frac{1}{2})(M^{\prime}-h)},&\text{J},\end{cases}$
(15)

in which
$M^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{2}(M_{\text{I}}-M_{\text{II}})$
and $\eta=\eta(x)$.

###  2.4 Simplified forms of the multi-indexed Laguerre
and Jacobi polynomials, A

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 (10)–(15) in the previous subsection.

In this subsection we simplify the Wronskians of the ‘polynomials’ (10)–(13).
Let us first transform the higher derivatives of the ‘seed polynomials’
$\mu_{\text{v}}(\eta)$ in (12). For the L system, we obtain

$\displaystyle\text{L1}\colon\quad$
$\displaystyle\partial_{\eta}\bigl{(}e^{\eta}L^{(g-\frac{1}{2})}_{\text{v}}(-\eta)\bigr{)}=e^{\eta}L^{(g+1-\frac{1}{2})}_{\text{v}}(-\eta),$
$\displaystyle\text{L2}\colon\quad$
$\displaystyle\partial_{\eta}\bigl{(}\eta^{\frac{1}{2}-g}L^{(\frac{1}{2}-g)}_{\text{v}}(\eta)\bigr{)}=\big{(}\text{v}-g+\tfrac{1}{2}\big{)}\eta^{\frac{1}{2}-(g+1)}L^{(\frac{1}{2}-(g+1))}_{\text{v}}(\eta),$

by using (4)–(5) and (4)–(6), respectively. By repeating these we arrive at
for $j\in\mathbb{Z}_{\geq 1}$,

$\displaystyle\partial_{\eta}^{j-1}\bigl{(}e^{\eta}L^{(g-\frac{1}{2})}_{\text{v}}(-\eta)\bigr{)}=e^{\eta}L^{(g+j-\frac{3}{2})}_{\text{v}}(-\eta),$
$\displaystyle\partial_{\eta}^{j-1}\bigl{(}\eta^{\frac{1}{2}-g}L^{(\frac{1}{2}-g)}_{\text{v}}(\eta)\bigr{)}=(-1)^{j-1}\big{(}g-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\eta^{\frac{3}{2}-g-j}L^{(\frac{3}{2}-g-j)}_{\text{v}}(\eta)$
$\displaystyle\phantom{\partial_{\eta}^{j-1}\bigl{(}\eta^{\frac{1}{2}-g}L^{(\frac{1}{2}-g)}_{\text{v}}(\eta)\bigr{)}}=\eta^{\frac{1}{2}-g-K}(-1)^{j-1}\big{(}g-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\eta^{K+1-j}L^{(\frac{3}{2}-g-j)}_{\text{v}}(\eta),$

in which $K$ is a positive integer. For the J system, we obtain

$\displaystyle\text{J1}\colon\quad$
$\displaystyle\partial_{\eta}\bigl{(}\big{(}\tfrac{1+\eta}{2}\big{)}^{\frac{1}{2}-h}P^{(g-\frac{1}{2},\frac{1}{2}-h)}_{\text{v}}(\eta)\bigr{)}=\tfrac{1}{2}\big{(}\text{v}-h+\tfrac{1}{2}\big{)}\big{(}\tfrac{1+\eta}{2}\big{)}^{\frac{1}{2}-(h+1)}P^{(g+1-\frac{1}{2},\frac{1}{2}-(h+1))}_{\text{v}}(\eta),$
$\displaystyle\text{J2}\colon\quad$
$\displaystyle\partial_{\eta}\bigl{(}\big{(}\tfrac{1-\eta}{2}\big{)}^{\frac{1}{2}-g}P^{(\frac{1}{2}-g,h-\frac{1}{2})}_{\text{v}}(\eta)\bigr{)}=-\tfrac{1}{2}\big{(}\text{v}-g+\tfrac{1}{2}\big{)}\big{(}\tfrac{1-\eta}{2}\big{)}^{\frac{1}{2}-(g+1)}P^{(\frac{1}{2}-(g+1),h+1-\frac{1}{2})}_{\text{v}}(\eta),$

by using (7), (8) and (7), (9), respectively. Repeated applications of these
formulas lead to

$\displaystyle\partial_{\eta}^{j-1}\bigl{(}\big{(}\tfrac{1+\eta}{2}\big{)}^{\frac{1}{2}-h}P^{(g-\frac{1}{2},\frac{1}{2}-h)}_{\text{v}}(\eta)\bigr{)}=\frac{(-1)^{j-1}}{2^{j-1}}\big{(}h-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\big{(}\tfrac{1+\eta}{2}\big{)}^{\frac{3}{2}-h-j}P^{(g+j-\frac{3}{2},\frac{3}{2}-h-j)}_{\text{v}}(\eta)$
$\displaystyle\qquad{}=\big{(}\tfrac{1+\eta}{2}\big{)}^{\frac{1}{2}-h-K}\frac{(-1)^{j-1}}{2^{j-1}}\big{(}h-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\big{(}\tfrac{1+\eta}{2}\big{)}^{K+1-j}P^{(g+j-\frac{3}{2},\frac{3}{2}-h-j)}_{\text{v}}(\eta),$
$\displaystyle\partial_{\eta}^{j-1}\bigl{(}\big{(}\tfrac{1-\eta}{2}\big{)}^{\frac{1}{2}-g}P^{(\frac{1}{2}-g,h-\frac{1}{2})}_{\text{v}}(\eta)\bigr{)}=\frac{1}{2^{j-1}}\big{(}g-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\big{(}\tfrac{1-\eta}{2}\big{)}^{\frac{3}{2}-g-j}P^{(\frac{3}{2}-g-j,h+j-\frac{3}{2})}_{\text{v}}(\eta)$
$\displaystyle\qquad{}=\big{(}\tfrac{1-\eta}{2}\big{)}^{\frac{1}{2}-g-K}\frac{1}{2^{j-1}}\big{(}g-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\big{(}\tfrac{1-\eta}{2}\big{)}^{K+1-j}P^{(\frac{3}{2}-g-j,h+j-\frac{3}{2})}_{\text{v}}(\eta).$

The higher derivatives of the eigenpolynomials $P_{n}(\eta)$ in (13) are
replaced simply through (4) and (7), respectively,

$\displaystyle\partial_{\eta}^{j-1}L^{(g-\frac{1}{2})}_{n}(\eta)=(-1)^{j-1}L^{(g+j-\frac{3}{2})}_{n+1-j}(\eta),$
$\displaystyle\partial_{\eta}^{j-1}P^{(g-\frac{1}{2},h-\frac{1}{2})}_{n}(\eta)=\frac{1}{2^{j-1}}(n+g+h)_{j-1}P^{(g+j-\frac{3}{2},h+j-\frac{3}{2})}_{n+1-j}(\eta),$

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

$\displaystyle
X^{(M)}_{\text{v},j}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}L^{(g+j-\frac{3}{2})}_{\text{v}}(-\eta),&\text{L,
$\text{v}$ Type $\text{I}$},\\\
(-1)^{j-1}\big{(}g-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\eta^{M-j}L^{(\frac{3}{2}-g-j)}_{\text{v}}(\eta),&\text{L,
$\text{v}$ Type $\text{II}$},\\\
\frac{(-1)^{j-1}}{2^{j-1}}\big{(}h-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\big{(}\tfrac{1+\eta}{2}\big{)}^{M-j}P^{(g+j-\frac{3}{2},\frac{3}{2}-h-j)}_{\text{v}}(\eta),&\text{J,
$\text{v}$ Type $\text{I}$},\\\
\frac{1}{2^{j-1}}\big{(}g-\tfrac{1}{2}-\text{v}\big{)}_{j-1}\big{(}\tfrac{1-\eta}{2}\big{)}^{M-j}P^{(\frac{3}{2}-g-j,h+j-\frac{3}{2})}_{\text{v}}(\eta),&\text{J,
$\text{v}$ Type $\text{II}$},\end{cases}$ $\displaystyle
Z^{(M)}_{n,j}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}(-1)^{j-1}L^{(g+j-\frac{3}{2})}_{n+1-j}(\eta),&\text{L},\\\
\frac{1}{2^{j-1}}(n+g+h)_{j-1}P^{(g+j-\frac{3}{2},h+j-\frac{3}{2})}_{n+1-j}(\eta),&\text{J}.\end{cases}$

The Wronskians in (10)–(11) are replaced by ordinary determinants consisting
of these column vectors:

$\displaystyle\text{W}[\mu_{d_{1}},\ldots,\mu_{d_{M}}](\eta)=\bigl{|}\vec{X}^{(M)}_{d_{1}}(\eta)\cdots\vec{X}^{(M)}_{d_{M}}(\eta)\bigr{|}$
$\displaystyle\hphantom{\text{W}[\mu_{d_{1}},\ldots,\mu_{d_{M}}](\eta)=}{}\times\begin{cases}\bigl{(}e^{\eta}\bigr{)}^{M_{\text{I}}}\bigl{(}\eta^{\frac{3}{2}-g-M}\bigr{)}^{M_{\text{II}}},&\text{L},\\\
\bigl{(}\big{(}\frac{1+\eta}{2}\big{)}^{\frac{3}{2}-h-M}\bigr{)}^{M_{\text{I}}}\bigl{(}\big{(}\frac{1-\eta}{2}\big{)}^{\frac{3}{2}-g-M}\bigr{)}^{M_{\text{II}}},&\text{J},\end{cases}$
$\displaystyle\text{W}[\mu_{d_{1}},\ldots,\mu_{d_{M}},P_{n}](\eta)=\bigl{|}\vec{X}^{(M+1)}_{d_{1}}(\eta)\cdots\vec{X}^{(M+1)}_{d_{M}}(\eta)\vec{Z}^{(M+1)}_{n}(\eta)\bigr{|}$
$\displaystyle\hphantom{\text{W}[\mu_{d_{1}},\ldots,\mu_{d_{M}},P_{n}](\eta)=}{}\times\begin{cases}\bigl{(}e^{\eta}\bigr{)}^{M_{\text{I}}}\bigl{(}\eta^{\frac{1}{2}-g-M}\bigr{)}^{M_{\text{II}}},&\text{L},\\\
\bigl{(}\big{(}\frac{1+\eta}{2}\big{)}^{\frac{1}{2}-h-M}\bigr{)}^{M_{\text{I}}}\bigl{(}\big{(}\frac{1-\eta}{2}\big{)}^{\frac{1}{2}-g-M}\bigr{)}^{M_{\text{II}}},&\text{J}.\end{cases}$

We arrive at the main result, the simple expressions of
$\Xi_{\mathcal{D}}(\eta)$ and $P_{\mathcal{D},n}(\eta)$

$\displaystyle\Xi_{\mathcal{D}}(\eta)=A\bigl{|}\vec{X}^{(M)}_{d_{1}}(\eta)\cdots\vec{X}^{(M)}_{d_{M}}(\eta)\bigr{|},$
(16) $\displaystyle
P_{\mathcal{D},n}(\eta)=A\bigl{|}\vec{X}^{(M+1)}_{d_{1}}(\eta)\cdots\vec{X}^{(M+1)}_{d_{M}}(\eta)\vec{Z}^{(M+1)}_{n}(\eta)\bigr{|},$
(17) $\displaystyle
A=\begin{cases}\eta^{-M_{\text{II}}(M_{\text{II}}-1)},&\text{L},\\\
\big{(}\frac{1+\eta}{2}\big{)}^{-M_{\text{I}}(M_{\text{I}}-1)}\big{(}\frac{1-\eta}{2}\big{)}^{-M_{\text{II}}(M_{\text{II}}-1)},&\text{J}.\end{cases}$

It should be stressed that the components of the matrices in (16) and (17) 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 (10), (11), in which
$\mu_{d_{j}}$’s have non-polynomial factors (12).

###  2.5 Simplified forms of the multi-indexed Laguerre
and Jacobi polynomials, B

Here we will simplify the Wronskians of the virtual state wavefunctions and
the eigenpolynomials (14), (15). By replacing the even order derivatives of
the virtual state wavefunctions and eigenfunctions in the Wronskians (14),
(15), by the rule $\psi^{(2m)}(x)\to(-\mathcal{E})^{m}\psi(x)$, we obtain

$\displaystyle\text{W}\big{[}\tilde{\phi}_{d_{1}},\ldots,\tilde{\phi}_{d_{M}},\phi_{n}\big{]}(x)=\det(a_{j,k})_{1\leq
j,k\leq M+1},$ (18) $\displaystyle
a_{2l-1,k}=(-\tilde{\mathcal{E}}_{d_{k}})^{l-1}\tilde{\phi}_{d_{k}}(x),\qquad
1\leq k\leq M,\qquad 1\leq l\leq\big{[}\tfrac{M+2}{2}\big{]},$ $\displaystyle
a_{2l-1,M+1}=(-\mathcal{E}_{n})^{l-1}\phi_{n}(x),\qquad 1\leq
l\leq\big{[}\tfrac{M+2}{2}\big{]},$ $\displaystyle
a_{2l,k}=(-\tilde{\mathcal{E}}_{d_{k}})^{l-1}\tilde{\phi}^{\prime}_{d_{k}}(x),\qquad
1\leq k\leq M,\qquad 1\leq l\leq\big{[}\tfrac{M+1}{2}\big{]},$ $\displaystyle
a_{2l,M+1}=(-\mathcal{E}_{n})^{l-1}\phi^{\prime}_{n}(x),\qquad 1\leq
l\leq\big{[}\tfrac{M+1}{2}\big{]},$

in which $[a]$ denotes the greatest integer not exceeding $a$. The first
derivatives in the $2l$-th row can be simplified by adding
$-\frac{\phi_{0}^{\prime}(x)}{\phi_{0}(x)}\times(2l-1)$-th row,

$\displaystyle\phi^{\prime}_{n}(x)\to\left(\frac{d}{dx}-\frac{\phi^{\prime}_{0}(x)}{\phi_{0}(x)}\right)\phi_{n}(x)=\frac{c_{\text{\tiny$\mathcal{F}$}}}{\eta^{\prime}(x)}\phi_{0}(x)\zeta_{n}\bigl{(}\eta(x)\bigr{)}=\phi_{0}(x)\zeta_{n}\bigl{(}\eta(x)\bigr{)}A,$
$\displaystyle\tilde{\phi}^{\prime}_{\text{v}}(x)\to\left(\frac{d}{dx}-\frac{\phi^{\prime}_{0}(x)}{\phi_{0}(x)}\right)\tilde{\phi}_{\text{v}}(x)=\frac{c_{\text{\tiny$\mathcal{F}$}}}{\eta^{\prime}(x)}\,\tilde{\phi}_{0}(x)\tilde{\zeta}_{\text{v}}\bigl{(}\eta(x)\bigr{)}=\tilde{\phi}_{0}(x)\tilde{\zeta}_{\text{v}}\bigl{(}\eta(x)\bigr{)}A,$
$\displaystyle
A=\frac{c_{\text{\tiny$\mathcal{F}$}}}{\eta^{\prime}(x)}=\begin{cases}x^{-1},&\text{L},\\\
(\sin x\cos x)^{-1},&\text{J},\end{cases}$ (19)

in which $\zeta_{n}(\eta)$ and $\tilde{\zeta}_{\text{v}}(\eta)$ are
polynomials in $\eta$ defined by

$\displaystyle\zeta_{n}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}-2\eta
L^{(g+\frac{1}{2})}_{n-1}(\eta),&\text{L},\\\
-\tfrac{1}{2}(n+g+h)\big{(}1-\eta^{2}\big{)}P^{(g+\frac{1}{2},h+\frac{1}{2})}_{n-1}(\eta),&\text{J},\end{cases}$
$\displaystyle\tilde{\zeta}_{\text{v}}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}2\eta
L^{(g+\frac{1}{2})}_{\text{v}}(-\eta),&\text{L, $\text{I}$},\\\
-2\big{(}g-\tfrac{1}{2}-\text{v}\big{)}L^{(-g-\frac{1}{2})}_{\text{v}}(\eta),&\text{L,
$\text{II}$},\\\
\big{(}h-\tfrac{1}{2}-\text{v}\big{)}(1-\eta)P^{(g+\frac{1}{2},-h-\frac{1}{2})}_{\text{v}}(\eta),&\text{J,
$\text{I}$},\\\
-\big{(}g-\tfrac{1}{2}-\text{v}\big{)}(1+\eta)P^{(-g-\frac{1}{2},h+\frac{1}{2})}_{\text{v}}(\eta),&\text{J,
$\text{II}$}.\end{cases}$

Use is made of (4)–(6) for L and (7)–(9) for J.

By extracting the functions $\phi_{0}(x)$, $\tilde{\phi}_{0}(x)$ from each
column of the matrix $a_{j,k}$ (18) and the factor $A$ of (19) from the even
rows, we arrive at another set of simplified determinant expressions for the
multi-indexed polynomials:

$\displaystyle
P_{\mathcal{D},n}(\eta)=c_{\text{\tiny$\mathcal{F}$}}^{-\frac{1}{2}M(M+1)}\det(a_{j,k})_{1\leq
j,k\leq M+1}A,$ (20) $\displaystyle
a_{2l-1,k}=(-\tilde{\mathcal{E}}_{d_{k}})^{l-1}\xi_{d_{k}}(\eta),\qquad 1\leq
k\leq M,\qquad 1\leq l\leq\big{[}\tfrac{M+2}{2}\big{]},$ $\displaystyle
a_{2l-1,M+1}=(-\mathcal{E}_{n})^{l-1}P_{n}(\eta),\qquad 1\leq
l\leq\big{[}\tfrac{M+2}{2}\big{]},$ $\displaystyle
a_{2l,k}=(-\tilde{\mathcal{E}}_{d_{k}})^{l-1}\tilde{\zeta}_{d_{k}}(\eta),\qquad
1\leq k\leq M,\qquad 1\leq l\leq\big{[}\tfrac{M+1}{2}\big{]},$ $\displaystyle
a_{2l,M+1}=(-\mathcal{E}_{n})^{l-1}\zeta_{n}(\eta),\qquad 1\leq
l\leq\big{[}\tfrac{M+1}{2}\big{]},$ $\displaystyle
A=\begin{cases}\eta^{-([M^{\prime}]+1)([M^{\prime}]+M-2[\frac{M}{2}])},&\text{L},\\\
\bigl{(}\tfrac{1-\eta}{2}\bigr{)}^{-([M^{\prime}]+1)([M^{\prime}]+M-2[\frac{M}{2}])}\bigl{(}\tfrac{1+\eta}{2}\bigr{)}^{-([-M^{\prime}]+1)([-M^{\prime}]+M-2[\frac{M}{2}])},&\text{J}.\end{cases}$

In a similar way, we obtain

$\displaystyle\Xi_{\mathcal{D}}(\eta)=c_{\text{\tiny$\mathcal{F}$}}^{-\frac{1}{2}M(M-1)}\det(a_{j,k})_{1\leq
j,k\leq M}A,$ (21) $\displaystyle
a_{2l-1,k}=(-\tilde{\mathcal{E}}_{d_{k}})^{l-1}\xi_{d_{k}}(\eta),\qquad 1\leq
l\leq\big{[}\tfrac{M+1}{2}\big{]},$ $\displaystyle
a_{2l,k}=(-\tilde{\mathcal{E}}_{d_{k}})^{l-1}\tilde{\zeta}_{d_{k}}(\eta),\qquad
1\leq l\leq\big{[}\tfrac{M}{2}\big{]},$ $\displaystyle
A=\begin{cases}\eta^{-[M^{\prime}]([M^{\prime}]+M-2[\frac{M}{2}])},&\text{L},\\\
\bigl{(}\tfrac{1-\eta}{2}\tfrac{1+\eta}{2}\bigr{)}^{-[M^{\prime}]([M^{\prime}]+M-2[\frac{M}{2}])},&\text{J}.\end{cases}$

Again all the components of the matrices $a_{j,k}$ in (20) and (21) are
polynomials in $\eta$.

###  2.6 Parity transformation of the multi-indexed Jacobi polynomials

The Jacobi polynomial has the parity transformation property [24]

$\displaystyle P^{(\alpha,\beta)}_{n}(-x)=(-1)^{n}P^{(\beta,\alpha)}_{n}(x).$
(22)

We will show that this property is inherited by the multi-indexed Jacobi
polynomials. It is based on the property of the Wronskian

$\displaystyle\text{W}[f_{1},\ldots,f_{n}](-\eta)=(-1)^{\frac{1}{2}n(n-1)}\text{W}[g_{1},\ldots,g_{n}](\eta),\qquad
g_{k}(\eta)\stackrel{{\scriptstyle\text{def}}}{{=}}f_{k}(-\eta).$

In this subsection, we indicate the types of the virtual states explicitly by
$(\text{v},\text{t})$, in which t stands for Type I or II. Based on (22), we
obtain

$\displaystyle\mu_{(\text{v},\text{t})}\bigl{(}{-}\eta;(g,h)\bigr{)}=(-1)^{\text{v}}\mu_{(\text{v},\bar{\text{t}})}\bigl{(}\eta;(h,g)\bigr{)},\qquad\bar{\text{t}}\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{cases}\text{II},&\text{t}=\text{I},\\\
\text{I},&\text{t}=\text{II},\end{cases}$ $\displaystyle
P_{n}\bigl{(}{-}\eta;(g,h)\bigr{)}=(-1)^{n}P_{n}\bigl{(}\eta;(h,g)\bigr{)}.$

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
$\mathcal{D}^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(d_{1},\bar{\text{t}}_{1}),\ldots,(d_{M},\bar{\text{t}}_{M})\\}$.
Corresponding to
$M_{\text{I}}\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{d_{j}\,|\,(d_{j},\text{I})\in\mathcal{D}\\}$,
$M_{\text{II}}\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{d_{j}\,|\,(d_{j},\text{II})\in\mathcal{D}\\}$,
we have
$M^{\prime}_{\text{I}}\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{d_{j}\,|\,(d_{j},\text{I})\in\mathcal{D}^{\prime}\\}=M_{\text{II}}$,
$M^{\prime}_{\text{II}}\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{d_{j}\,|\,(d_{j},\text{II})\in\mathcal{D}^{\prime}\\}=M_{\text{I}}$.
By parity transformation $\eta\to-\eta$, the multi-indexed Jacobi polynomial
$P_{\mathcal{D},n}\bigl{(}\eta;(g,h)\bigr{)}$ is mapped to
$P_{\mathcal{D}^{\prime},n}\bigl{(}\eta;(h,g)\bigr{)}$ with the ‘mirror
reflected’ multi-index set $\mathcal{D}^{\prime}$:

$\displaystyle P_{\mathcal{D},n}\bigl{(}{-}\eta;(g,h)\bigr{)}$
$\displaystyle\qquad{}=\big{(}\tfrac{1-\eta}{2}\big{)}^{(M_{\text{II}}+h+\frac{1}{2})M_{\text{I}}}\big{(}\tfrac{1+\eta}{2}\big{)}^{(M_{\text{I}}+g+\frac{1}{2})M_{\text{II}}}\text{W}\big{[}\mu_{(d_{1},\text{t}_{1})},\ldots,\mu_{(d_{M},\text{t}_{M})},P_{n}\big{]}\bigl{(}{-}\eta;(g,h)\bigr{)}$
$\displaystyle\qquad{}=\big{(}\tfrac{1-\eta}{2}\big{)}^{(M_{\text{II}}+h+\frac{1}{2})M_{\text{I}}}\big{(}\tfrac{1+\eta}{2}\big{)}^{(M_{\text{I}}+g+\frac{1}{2})M_{\text{II}}}(-1)^{\frac{1}{2}M(M+1)}$
$\displaystyle\qquad\quad{}\times\text{W}\big{[}(-1)^{d_{1}}\mu_{(d_{1},\bar{\text{t}}_{1})},\ldots,(-1)^{d_{M}}\mu_{(d_{M},\bar{\text{t}}_{M})},(-1)^{n}P_{n}\big{]}\bigl{(}\eta;(h,g)\bigr{)}$
$\displaystyle\qquad{}=\big{(}\tfrac{1-\eta}{2}\big{)}^{(M_{\text{II}}+h+\frac{1}{2})M_{\text{I}}}\big{(}\tfrac{1+\eta}{2}\big{)}^{(M_{\text{I}}+g+\frac{1}{2})M_{\text{II}}}(-1)^{\frac{1}{2}M(M+1)}(-1)^{d_{1}+\cdots+d_{M}+n}$
$\displaystyle\qquad\quad{}\times\text{W}\big{[}\mu_{(d_{1},\bar{\text{t}}_{1})},\ldots,\mu_{(d_{M},\bar{\text{t}}_{M})},P_{n}\big{]}\bigl{(}\eta;(h,g)\bigr{)}$
$\displaystyle\qquad{}=(-1)^{n+\frac{1}{2}M(M+1)+\sum\limits_{k=1}^{M}d_{k}}P_{\mathcal{D}^{\prime},n}\bigl{(}\eta;(h,g)\bigr{)}.$

Similarly, the denominator polynomial
$\Xi_{\mathcal{D}}\bigl{(}\eta;(g,h)\bigr{)}$ is mapped to
$\Xi_{\mathcal{D}^{\prime}}\bigl{(}\eta;(h,g)\bigr{)}$ with a sign factor:

$\displaystyle\Xi_{\mathcal{D}}\bigl{(}{-}\eta;(g,h)\bigr{)}=(-1)^{\frac{1}{2}M(M-1)+\sum\limits_{k=1}^{M}d_{k}}\Xi_{\mathcal{D}^{\prime}}\bigl{(}\eta;(h,g)\bigr{)}.$

For the special case of ‘mirror symmetric’ multi-index set
$\mathcal{D}^{\prime}=\mathcal{D}$ (as a set), i.e.,
$\\{d_{j}\,|\,(d_{j},\text{I})\in\mathcal{D}\\}=\\{d_{j}\,|\,(d_{j},\text{II})\in\mathcal{D}\\}$
(as a set), we have $P_{\mathcal{D}^{\prime},n}(\eta)=\pm
P_{\mathcal{D},n}(\eta)$. In fact, this formula turns out to be

$\displaystyle
P_{\mathcal{D}^{\prime},n}(\eta)=(-1)^{(\frac{M}{2})^{2}}P_{\mathcal{D},n}(\eta).$

For this special case the parity transformation gives

$\displaystyle
P_{\mathcal{D},n}\bigl{(}{-}\eta;(g,h)\bigr{)}=(-1)^{n}P_{\mathcal{D},n}\bigl{(}\eta;(h,g)\bigr{)},\qquad\Xi_{\mathcal{D}}\bigl{(}{-}\eta;(g,h)\bigr{)}=\Xi_{\mathcal{D}}\bigl{(}\eta;(h,g)\bigr{)}.$

##  3 Summary and comments

The multi-indexed Laguerre and Jacobi polynomials are defined by the Wronskian
expressions originating from multiple Darboux transformations. Two simplified
determinant expressions of them, (16), (17) and (20), (21), which do not
contain derivatives, are derived based on the properties of the Wronskian and
identities of the Laguerre and Jacobi polynomials. For (20), (21), the
Schrödinger equation is used. For (16), (17), various identities of the
Laguerre and Jacobi polynomials are used, which are essentially forward shift
relations. Although the calculation in Section 2.4 is performed for
polynomials, it can be done for wavefunctions just like [20], 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 [1, 12], i.e., the Wilson,
Askey–Wilson, Meixner, little $q$-Jacobi and ($q$-)Racah polynomials, etc.
[18, 19, 21]. These polynomials belong to ‘discrete’ quantum mechanics [16],
in which the Schrödinger 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 [13].

### 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.

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