DPSU-22-1

“Diophantine” and Factorisation Properties of Finite Orthogonal Polynomials in
the Askey Scheme

Satoru Odake${}^{\,a}$ and Ryu Sasaki${}^{\,b}$

a Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan

b Department of Physics, Tokyo University of Science, Noda 278-8510, Japan

###### Abstract

A new interpretation and applications of the “Diophantine” and factorisation
properties of finite orthogonal polynomials in the Askey scheme are explored.
The corresponding twelve polynomials are the ($q$-)Racah, (dual, $q$-)Hahn,
Krawtchouk and five types of $q$-Krawtchouk. These ($q$-)hypergeometric
polynomials, defined only for the degrees of $0,1,\ldots,N$, constitute the
main part of the eigenvectors of $N+1$-dimensional tri-diagonal real symmetric
matrices, which correspond to the difference equations governing the
polynomials. The monic versions of these polynomials all exhibit the
“Diophantine” and factorisation properties at higher degrees than $N$. This
simply means that these higher degree polynomials are zero-norm “eigenvectors”
of the $N+1$-dimensional tri-diagonal real symmetric matrices. A new type of
multi-indexed orthogonal polynomials belonging to these twelve polynomials
could be introduced by using the higher degree polynomials as the seed
solutions of the multiple Darboux transformations for the corresponding matrix
eigenvalue problems. The shape-invariance properties of the simplest type of
the multi-indexed polynomials are demonstrated. The explicit transformation
formulas are presented.

##  1 Introduction

It is noted for some time that some of the best known orthogonal polynomials,
the Laguerre, Jacobi, Wilson and Askey-Wilson, etc exhibit strange phenomena,
called “Diophantine” and factorisation properties [1, 2], when some parameters
are tuned at values which invalidate the orthogonality. In this paper we show
that these properties are universally shared, without tuning parameters, by a
not so well known group of orthogonal polynomials, the monic versions of
finite polynomials in the Askey scheme [3, 4, 5, 6].

A simple interpretation or explanation is that the monic versions of these
polynomials satisfy the same $(N+1)\times(N+1)$ matrix eigenvalue problem
(2.47) which govern the polynomials (2.4)–(2.12). Therefore at higher
$n=N+1+m$ ($m\in\mathbb{Z}_{\geq 0}$) degrees, the polynomials are zero-norm
solutions, which means the “Diophantine” and factorisation properties.

This paper is organised as follows. Section two start with some typical
examples of the “Diophantine” and factorisation properties together with the
background description. The finite orthogonal polynomials in the Askey scheme,
the main subject, are briefly introduced as the solutions of certain tri-
diagonal real symmetric $(N+1)\times(N+1)$ matrix eigenvalue problems [7] in §
2.1. The explicit data of the twelve polynomials are displayed according to
the five families of the sinusoidal coordinates in § 2.2. In § 2.3 we present
the main Theorem 2.1 stating that the “Diophantine” and factorisation is the
consequence of the zero norm nature of the higher degree $N+1+m$
($m\in\mathbb{Z}_{\geq 0}$) monic polynomials. Starting with the most generic
$q$-Racah polynomial, the explicit expressions of the “Diophantine” properties
and factorisation are displayed in § 2.3.1–§ 2.3.11. The general setting of
the multi-indexed orthogonal polynomials generated by multiple Darboux
transformations by using the zero-norm solutions is outlined in section three.
The very special cases of the new multi-indexed polynomials constructed by
using $M$ contiguous lowest degree zero-norm solutions are detailed in section
four. In these cases, the multi-indexed polynomials take the same form as the
original with $x$ and some parameters shifted, displaying the so-called shape
invariance. The general transformation rules for the five families are listed
in Theorem 4.1 together with the explicit forms of the transformation rules of
the polynomials in Theorem 4.2. Corresponding to the new type of shape
invariance, the forward $x$-shift operators are introduced. It is shown in
Theorem 4.3 that the multiple applications of the forward $x$-shift operators
reproduce the results of Theorem 4.2. The explicit forms of the corresponding
backward $x$-shift operators are listed. The final section is for a summary
and some comments.

##  2 “Diophantine” Properties and Factorisation

Many interesting examples of the “Diophantine” properties and factorisation of
various orthogonal polynomials belonging to the Askey scheme are reported and
explained by Calogero and collaborators [1] (to be cited as I) and Ismail and
a coauthor [2] (to be cited as II). Here we list some typical examples, which
are stated for the monic versions of the named polynomials: (typos in
(II.3.10) and (II.4.4) are corrected)

$\displaystyle\text{Racah with }\alpha=-n,$ $\displaystyle\qquad\quad
p_{n}(x;-n,\beta,\gamma,\delta)=\prod_{k=0}^{n-1}\bigl{(}x-k(k+\gamma+\delta+1)\bigr{)},$
(I.3.29a) $\displaystyle\text{Jacobi with }\alpha=-n,$
$\displaystyle\qquad\quad p_{n}(x;-n,\beta)=(x-1)^{n},$ (I.3.153)
$\displaystyle\text{Laguerre with }\alpha=-n,$ $\displaystyle\qquad\quad
p_{n}(x;-n)=x^{n},$ (I.3.162) $\displaystyle\text{Wilson with
}t_{3}=1-m-t_{4},\ \ 0\leq m\leq n,$ $\displaystyle\quad\ \
p_{n}(x;t_{1},t_{2},1-m-t_{4},t_{4})=\prod_{k=0}^{m-1}\bigl{(}x+(t_{4}+k)^{2}\bigr{)}\cdot
p_{n-m}(x;t_{1},t_{2},1-t_{4},t_{4}+m),$ (II.3.10) $\displaystyle\text{Askey-
Wilson with }t_{3}=q^{1-m}/t_{4},\ \
x\stackrel{{\scriptstyle\text{def}}}{{=}}\cos\theta,\ \ 0\leq m\leq n,$
$\displaystyle\qquad\quad
p_{n}(x;t_{1},t_{2},q^{1-m}/t_{4},t_{4})=(t_{4}e^{i\theta}\,;q)_{m}(t_{4}e^{-i\theta}\,;q)_{m}\,p_{n-m}(x;t_{1},t_{2},q/t_{4},q^{m}t_{4})$
$\displaystyle\phantom{\qquad\quad
p_{n}(x;t_{1},t_{2},q^{1-m}/t_{4},t_{4})=}\times(-1)^{m}t_{4}^{-m}q^{-\frac{1}{2}m(m-1)}2^{-m}.$
(II.4.4)

It should be stressed that in all these examples, some parameters are tuned to
a degree number $n$, in a rather ad-hoc manner. Therefore the “Diophantine”
and factorisation properties do not belong to the polynomials in general but
only to the particular degree polynomial to which the parameters are tuned.
Moreover, with those parameter assignments, the polynomials are no longer
orthogonal with each other. This situation makes it difficult to find
satisfactory interpretations of the “Diophantine” and factorisation properties
and tends to give a wrong impression that they are of haphazard or
unsystematic origin and having rather peripheral importance.

In this paper we present a different perspective and show that for a certain
group of orthogonal polynomials in the Askey scheme, the “Diophantine” and
factorisation properties are essential and inherent within the proper
parameter ranges in which the orthogonality holds. Let us first introduce the
general features of the polynomials belonging to this group.

###  2.1 Finite orthogonal polynomials of a discrete variable

The polynomials $\\{\check{P}_{n}(x;N,\boldsymbol{\lambda})\\}$ in the group
are also called finite orthogonal polynomials of a discrete variable [8]. They
are defined on a finite integer lattice $\mathcal{X}=\\{0,1,\ldots,N\\}$
satisfying

$\sum_{x\in\mathcal{X}}w(x;N,\boldsymbol{\lambda})\check{P}_{m}(x;N,\boldsymbol{\lambda})\check{P}_{n}(x;N,\boldsymbol{\lambda})=\frac{\delta_{m\,n}}{d_{n}(N,\boldsymbol{\lambda})^{2}}\quad(m,n\in\mathcal{X}),$
(2.1)

with a positive weight function $w(x;N,\boldsymbol{\lambda})>0$. Here
$\boldsymbol{\lambda}$ stands for the set of parameters other than the lattice
size $N$. The group consists of five families according to the type of the
sinusoidal coordinate $\eta(x;\boldsymbol{\lambda})$,

$\check{P}_{n}(x;N,\boldsymbol{\lambda})=P_{n}\bigl{(}\eta(x;\boldsymbol{\lambda});N,\boldsymbol{\lambda}\bigr{)},$
(2.2)

in which $P_{n}(\eta(x;\boldsymbol{\lambda});N,\boldsymbol{\lambda})$ is a
degree $n$ polynomial in $\eta(x;\boldsymbol{\lambda})$. There are five types
of the sinusoidal coordinates for the finite polynomials in the Askey scheme
[7],

$\displaystyle\text{(i)}:\ \eta(x)=x,\quad\text{(ii)}:\
\eta(x)=x(x+d),\quad\text{(iii)}:\ \eta(x)=1-q^{x},$
$\displaystyle\text{(iv)}:\ \eta(x)=q^{-x}-1,\quad\text{(v)}:\
\eta(x)=(q^{-x}-1)(1-dq^{x}),$ $\displaystyle\eta(0)=0,\quad\eta(x)>0\ \
(x\in\mathcal{X}\backslash\\{0\\}).$

In this paper we adopt the normalisation convention

$\check{P}_{n}(0;N,\boldsymbol{\lambda})=P_{n}(0;N,\boldsymbol{\lambda})=1,$
(2.3)

except for the monic polynomials to be introduced shortly.

As the other orthogonal polynomials in the Askey scheme [3, 4, 5, 6], the
polynomials in this group satisfy second order difference equation

$\displaystyle\quad
B(x;N,\boldsymbol{\lambda})\bigl{(}\check{P}_{n}(x;N,\boldsymbol{\lambda})-\check{P}_{n}(x+1;N,\boldsymbol{\lambda})\bigr{)}+D(x;N,\boldsymbol{\lambda})\bigl{(}\check{P}_{n}(x;N,\boldsymbol{\lambda})-\check{P}_{n}(x-1;N,\boldsymbol{\lambda})\bigr{)}$
$\displaystyle=\mathcal{E}(n;\boldsymbol{\lambda})\check{P}_{n}(x;N,\boldsymbol{\lambda})\quad(x,n\in\mathcal{X}),$
(2.4)

on top of the three term recurrence relations. Here the coefficients
$B(x;N,\boldsymbol{\lambda})$ and $D(x;N,\boldsymbol{\lambda})$ are positive
in $\mathcal{X}$ [7] and vanishes only at the boundary of $\mathcal{X}$,

$\displaystyle B(x;N,\boldsymbol{\lambda})$ $\displaystyle>0\ \
(x\in\mathcal{X}\backslash\\{N\\}),$ $\displaystyle\quad
D(x;N,\boldsymbol{\lambda})$ $\displaystyle>0\ \
(x\in\mathcal{X}\backslash\\{0\\}),$ $\displaystyle
B(N;N,\boldsymbol{\lambda})$ $\displaystyle=0,$ $\displaystyle\quad
D(0;N,\boldsymbol{\lambda})$ $\displaystyle=0.$

As shown explicitly in [7], the difference equation (2.4) can be rewritten as
a matrix eigenvalue equation in terms of an $(N+1)\times(N+1)$ tri-diagonal
matrix
$\bigl{(}\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})_{x\,y}\bigr{)}_{x,y\in\mathcal{X}}$,

$\displaystyle\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})\check{P}_{n}(x;N,\boldsymbol{\lambda})=\mathcal{E}(n;\boldsymbol{\lambda})\check{P}_{n}(x;N,\boldsymbol{\lambda}),$
$\displaystyle\Bigl{(}\Longleftrightarrow\sum_{y\in\mathcal{X}}\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})_{x\,y}\check{P}_{n}(y;N,\boldsymbol{\lambda})=\mathcal{E}(n;\boldsymbol{\lambda})\check{P}_{n}(x;N,\boldsymbol{\lambda})\Bigr{)},$
(2.5)
$\displaystyle\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})_{x\,x+1}=-B(x;N,\boldsymbol{\lambda}),\quad\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})_{x\,x-1}=-D(x;N,\boldsymbol{\lambda}),$
$\displaystyle\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})_{x\,x}=B(x;N,\boldsymbol{\lambda})+D(x;N,\boldsymbol{\lambda}),\quad\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})_{x\,y}=0\
\ (|x-y|\geq 2),$ (2.6)

with the eigenvalue $\mathcal{E}(n;\boldsymbol{\lambda})$. By a similarity
transformation in terms of a positive diagonal matrix
$\Phi(x;N,\boldsymbol{\lambda})$, the matrix
$\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})$ is related to a real
symmetric tri-diagonal matrix $\mathcal{H}(N,\boldsymbol{\lambda})$,

$\displaystyle\mathcal{H}\stackrel{{\scriptstyle\text{def}}}{{=}}\Phi\widetilde{\mathcal{H}}\Phi^{-1}\Leftrightarrow\widetilde{\mathcal{H}}=\Phi^{-1}\mathcal{H}\Phi\
\Leftrightarrow\mathcal{H}_{x\,y}=\phi_{0}(x)\widetilde{\mathcal{H}}_{x\,y}\phi_{0}(y)^{-1},$
(2.7) $\displaystyle\Phi_{x\,x}=\phi_{0}(x),\quad\Phi_{x\,y}=0\ \ (x\neq y).$
(2.8)

In these formulas and hereafter the parameter dependence of
$\widetilde{\mathcal{H}}$, $\mathcal{H}$, $\Phi$, $\check{P}_{n}$, $B$, $D$,
$\eta$, $\mathcal{E}$ etc is omitted occasionally for simplicity of
presentation. Here a positive function $\phi_{0}(x)$ on $\mathcal{X}$ is
introduced by the ratios of $B(x)$ and $D(x+1)$,

$\phi_{0}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\sqrt{\prod_{y=0}^{x-1}\frac{B(y)}{D(y+1)}}\Leftrightarrow\frac{\phi_{0}(x+1)}{\phi_{0}(x)}=\frac{\sqrt{B(x)}}{\sqrt{D(x+1)}}\
\ (x\in\mathcal{X}\backslash\\{N\\}),$ (2.9)

where $\prod_{j=n}^{n-1}*\stackrel{{\scriptstyle\text{def}}}{{=}}1$
($\Rightarrow\phi_{0}(0)=1$). The tri-diagonal real symmetric matrix
$\mathcal{H}$ (2.7), expressed explicitly as

$\displaystyle\mathcal{H}_{x\,x+1}=-\sqrt{B(x)D(x+1)},\quad\mathcal{H}_{x\,x-1}=-\sqrt{B(x-1)D(x)},$
$\displaystyle\mathcal{H}_{x\,x}=B(x)+D(x),\quad\mathcal{H}_{x\,y}=0\ \
(|x-y|\geq 2),$ (2.10)

has the following eigenvectors

$\displaystyle\phi_{n}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\phi_{0}(x)\check{P}_{n}(x)\
\
(n\in\mathcal{X}),\quad\bigl{(}\phi_{n}(x)\bigr{)}_{x\in\mathcal{X}}\in\mathbb{R}^{N+1},$
(2.11)
$\displaystyle\mathcal{H}\phi_{n}(x)=\sum_{y\in\mathcal{X}}\mathcal{H}_{x\,y}\phi_{n}(y)=\sum_{y\in\mathcal{X}}\phi_{0}(x)\widetilde{\mathcal{H}}_{x\,y}\check{P}_{n}(y)=\mathcal{E}(n)\phi_{n}(x)\
\ (n\in\mathcal{X}).$ (2.12)

The norms are all finite as
$w(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{x\in\mathcal{X}}\phi_{0}(x)^{2}<\infty$
for all $B(x)$ and $D(x)$ in the group. We arrive at the complete set of
eigenvectors $\\{\phi_{n}(x)\\}$ $(n\in\mathcal{X})$, since the orthogonality
of the eigenvectors (2.1) is guaranteed by the simpleness of the eigenvalues
of the tri-diagonal matrices.

###  2.2 Polynomials data

Here we present the basic data of the polynomials in the group. For more
details of the polynomials, consult [3]–[6]. The parametrisations of some
polynomials are different from the conventional ones, see [7]. The polynomials
are divided into five families according to the forms of the sinusoidal
coordinate $\eta(x)$. The “Diophantine” properties are identical within the
same family and the factorisation is universal for all the polynomials in this
group.

##### (i) Family with $\boldsymbol{\eta(x)=x}$

Two polynomials belong to this family, the Krawtchouk and Hahn polynomials.
They are polynomials in $x$.

####  2.2.1 Krawtchouk (K)

The polynomial depends on one positive parameter $\boldsymbol{\lambda}=p$
($0<p<1$).

$\displaystyle B(x;N,p)=p(N-x),\quad
D(x;N,p)=(1-p)x,\quad\mathcal{E}(n)=n,\quad\eta(x)=x,$ (2.13)
$\displaystyle\check{P}_{n}(x;N,p)=P_{n}(x;N,p)={}_{2}F_{1}\Bigl{(}\genfrac{}{}{0.0pt}{}{-n,\,-x}{-N}\Bigm{|}p^{-1}\Bigr{)},\quad
P_{n}(x;N,p)=P_{x}(n;N,p),$ (2.14)
$\displaystyle\check{P}_{n}(N-x;N,p)=(-1)^{n}(p^{-1}-1)^{n}\check{P}_{n}(x;N,1-p)\
\ (\text{Mirror symmetry}).$ (2.15)

####  2.2.2 Hahn (H)

The Hahn polynomial depends on two positive parameters
$\boldsymbol{\lambda}=(a,b)$ ($a,b>0$).

$\displaystyle B(x;\boldsymbol{\lambda})=(x+a)(N-x),\quad
D(x;\boldsymbol{\lambda})=x(b+N-x),$ (2.16)
$\displaystyle\mathcal{E}(n;\boldsymbol{\lambda})=n(n+a+b-1),\quad\eta(x)=x,$
$\displaystyle\check{P}_{n}(x;N,\boldsymbol{\lambda})=P_{n}(x;N,\boldsymbol{\lambda})={}_{3}F_{2}\Bigl{(}\genfrac{}{}{0.0pt}{}{-n,\,n+a+b-1,\,-x}{a,\,-N}\Bigm{|}1\Bigr{)},$
(2.17)
$\displaystyle\check{P}_{n}(N-x;N,a,b)=(-1)^{n}\frac{(b)_{n}}{(a)_{n}}\check{P}_{n}(x;N,b,a)\
\ (\text{Mirror symmetry}).$ (2.18)

##### (ii) Family with $\boldsymbol{\eta(x)=x(x+d)}$

Two polynomials belong to this family, the Racah and dual Hahn polynomials.
They are polynomials in $x(x+d)$.

####  2.2.3 Racah (R)

The Racah polynomial depends on three parameters
$\boldsymbol{\lambda}=(b,c,d)$ ($0<d<b-N$, $0<c<1+d$) on top of the lattice
size $N$.

$\displaystyle
B(x;N,\boldsymbol{\lambda})=\frac{(x+b)(x+c)(x+d)(N-x)}{(2x+d)(2x+1+d)},$
(2.19) $\displaystyle
D(x;N,\boldsymbol{\lambda})=\frac{(b-d-x)(x+d-c)x(x+d+N)}{(2x-1+d)(2x+d)},$
(2.20)
$\displaystyle\mathcal{E}(n;\boldsymbol{\lambda})=n(n+\tilde{d}),\quad\eta(x;\boldsymbol{\lambda})=x(x+d),\quad\tilde{d}\stackrel{{\scriptstyle\text{def}}}{{=}}b+c-d-N-1,$
$\displaystyle\check{P}_{n}(x;N,\boldsymbol{\lambda})=P_{n}\bigl{(}\eta(x;\boldsymbol{\lambda});N,\boldsymbol{\lambda}\bigr{)}={}_{4}F_{3}\Bigl{(}\genfrac{}{}{0.0pt}{}{-n,\,n+\tilde{d},\,-x,\,x+d}{b,\,c,\,-N}\Bigm{|}1\Bigr{)}.$
(2.21)

####  2.2.4 dual Hahn (dH)

The dual Hahn polynomial depends on two positive parameters
$\boldsymbol{\lambda}=(a,b)$ ($a,b>0$), and the parameter $d$ is $d=a+b-1$.

$\displaystyle
B(x;N,\boldsymbol{\lambda})=\frac{(x+a)(x+a+b-1)(N-x)}{(2x-1+a+b)(2x+a+b)},$
(2.22) $\displaystyle
D(x;N,\boldsymbol{\lambda})=\frac{x(x+b-1)(x+a+b+N-1)}{(2x-2+a+b)(2x-1+a+b)},$
(2.23)
$\displaystyle\mathcal{E}(n)=n,\quad\eta(x;\boldsymbol{\lambda})=x(x+a+b-1),$
$\displaystyle\check{P}_{n}(x;N,\boldsymbol{\lambda})=P_{n}\bigl{(}\eta(x;\boldsymbol{\lambda});N,\boldsymbol{\lambda}\bigr{)}={}_{3}F_{2}\Bigl{(}\genfrac{}{}{0.0pt}{}{-n,\,x+a+b-1,\,-x}{a,\,-N}\Bigm{|}1\Bigr{)}.$
(2.24)

The remaining three families belong to the $q$-hypergeometric polynomial
category. The parameter $q$, $0<q<1$, dependence is not explicitly displayed.

##### (iii) Family with $\boldsymbol{\eta(x)=1-q^{x}}$

####  2.2.5 dual quantum $\boldsymbol{q}$-Krawtchouk (dq$\boldsymbol{q}$K)

The only polynomial belonging to this family is the dual quantum
$q$-Krawtchouk polynomial [7] depending on one positive parameter
$\boldsymbol{\lambda}=p>q^{-N}$.

$\displaystyle B(x;N,p)=p^{-1}q^{-x-N-1}(1-q^{N-x}),\quad
D(x;N,p)=(q^{-x}-1)(1-p^{-1}q^{-x}),$ (2.25)
$\displaystyle\mathcal{E}(n)=q^{-n}-1,\quad\eta(x)=1-q^{x},$
$\displaystyle\check{P}_{n}(x;N,p)=P_{n}\bigl{(}\eta(x);N,p\bigr{)}={}_{2}\phi_{1}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,q^{-x}}{q^{-N}}\Bigm{|}q\,;pq^{x+1}\Bigr{)}.$
(2.26)

##### (iv) Family with $\boldsymbol{\eta(x)=q^{-x}-1}$

On top of the $q$-Hahn polynomial, three sibling polynomials belong to this
family, the $q$-Krawtchouk, quantum $q$-Krawtchouk and affine $q$-Krawtchouk
polynomials, all depending on one positive parameter
$\boldsymbol{\lambda}=p>0$, but the parameter ranges are different, as shown
in each entry.

####  2.2.6 $\boldsymbol{q}$-Hahn ($\boldsymbol{q}$H)

The $q$-Hahn polynomial depends on two positive parameters
$\boldsymbol{\lambda}=(a,b)$ ($0<a,b<1$).

$\displaystyle B(x;N,\boldsymbol{\lambda})=(1-aq^{x})(q^{x-N}-1),\quad
D(x;N,\boldsymbol{\lambda})=aq^{-1}(1-q^{x})(q^{x-N}-b),$ (2.27)
$\displaystyle\mathcal{E}(n;\boldsymbol{\lambda})=(q^{-n}-1)(1-abq^{n-1}),\quad\eta(x)=q^{-x}-1,$
$\displaystyle\check{P}_{n}(x;N,\boldsymbol{\lambda})=P_{n}\bigl{(}\eta(x);N,\boldsymbol{\lambda}\bigr{)}={}_{3}\phi_{2}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,abq^{n-1},\,q^{-x}}{a,\,q^{-N}}\Bigm{|}q\,;q\Bigr{)}.$
(2.28)

####  2.2.7 $\boldsymbol{q}$-Krawtchouk ($\boldsymbol{q}$K)

$\displaystyle B(x;N,p)=q^{x-N}-1,\quad D(x;N,p)=p(1-q^{x}),$ (2.29)
$\displaystyle\mathcal{E}(n;p)=(q^{-n}-1)(1+pq^{n}),\quad\eta(x)=q^{-x}-1,\quad
p>0,$
$\displaystyle\check{P}_{n}(x;N,p)=P_{n}\bigl{(}\eta(x);N,p\bigr{)}={}_{3}\phi_{2}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,q^{-x},\,-pq^{n}}{q^{-N},\,0}\Bigm{|}q\,;q\Bigr{)}.$
(2.30)

####  2.2.8 quantum $\boldsymbol{q}$-Krawtchouk (q$\boldsymbol{q}$K)

$\displaystyle B(x;N,p)=p^{-1}q^{x}(q^{x-N}-1),\quad
D(x;N,p)=(1-q^{x})(1-p^{-1}q^{x-N-1}),$ (2.31)
$\displaystyle\mathcal{E}(n)=1-q^{n},\quad\eta(x)=q^{-x}-1,\quad p>q^{-N},$
$\displaystyle\check{P}_{n}(x;N,p)=P_{n}\bigl{(}\eta(x);N,p\bigr{)}={}_{2}\phi_{1}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,q^{-x}}{q^{-N}}\Bigm{|}q\,;pq^{n+1}\Bigr{)}.$
(2.32)

####  2.2.9 affine $\boldsymbol{q}$-Krawtchouk (a$\boldsymbol{q}$K)

$\displaystyle B(x;N,p)=(q^{x-N}-1)(1-pq^{x+1}),\quad
D(x;N,p)=pq^{x-N}(1-q^{x}),$ (2.33)
$\displaystyle\mathcal{E}(n)=q^{-n}-1,\quad\eta(x)=q^{-x}-1,\quad 0<p<q^{-1},$
$\displaystyle\check{P}_{n}(x;N,p)=P_{n}\bigl{(}\eta(x);N,p\bigr{)}={}_{3}\phi_{2}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,q^{-x},\,0}{pq,\,q^{-N}}\Bigm{|}q\,;q\Bigr{)}.$
(2.34)

##### (v) Family with $\boldsymbol{\eta(x)=(q^{-x}-1)(1-dq^{x})}$

Three polynomials belong to this family, the $q$-Racah, dual $q$-Hahn and dual
$q$-Krawtchouk polynomials.

####  2.2.10 $\boldsymbol{q}$-Racah ($\boldsymbol{q}$R)

The $q$-Racah polynomial depends on three parameters
$\boldsymbol{\lambda}=(b,c,d)$ ($0<bq^{-N}<d<1$, $qd<c<1$,
$\tilde{d}\stackrel{{\scriptstyle\text{def}}}{{=}}bcd^{-1}q^{-N-1}$).

$\displaystyle
B(x;N,\boldsymbol{\lambda})=\frac{(1-bq^{x})(1-cq^{x})(1-dq^{x})(q^{x-N}-1)}{(1-dq^{2x})(1-dq^{2x+1})}\,,$
(2.35) $\displaystyle
D(x;N,\boldsymbol{\lambda})=\tilde{d}\,\frac{(b^{-1}dq^{x}-1)(1-c^{-1}dq^{x})(1-q^{x})(1-dq^{x+N})}{(1-dq^{2x-1})(1-dq^{2x})},$
(2.36)
$\displaystyle\mathcal{E}(n;\boldsymbol{\lambda})=(q^{-n}-1)(1-\tilde{d}q^{n}),\quad\eta(x;\boldsymbol{\lambda})=(q^{-x}-1)(1-dq^{x}),$
$\displaystyle\check{P}_{n}(x;N,\boldsymbol{\lambda})=P_{n}\bigl{(}\eta(x;\boldsymbol{\lambda});N,\boldsymbol{\lambda}\bigr{)}={}_{4}\phi_{3}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,\tilde{d}q^{n},\,q^{-x},\,dq^{x}}{b,\,c,\,q^{-N}}\Bigm{|}q\,;q\Bigr{)}.$
(2.37)

####  2.2.11 dual $\boldsymbol{q}$-Hahn (d$\boldsymbol{q}$H)

The dual $q$-Hahn polynomial depends on two parameters
$\boldsymbol{\lambda}=(a,b)$ ($0<a,b<1$), and the parameter $d$ is
$d=abq^{-1}$.

$\displaystyle
B(x;N,\boldsymbol{\lambda})=\frac{(1-aq^{x})(1-abq^{x-1})(q^{x-N}-1)}{(1-abq^{2x-1})(1-abq^{2x})},$
(2.38) $\displaystyle
D(x;N,\boldsymbol{\lambda})=aq^{x-N-1}\frac{(1-q^{x})(1-bq^{x-1})(1-abq^{x+N-1})}{(1-abq^{2x-2})(1-abq^{2x-1})},$
(2.39)
$\displaystyle\mathcal{E}(n)=q^{-n}-1,\quad\eta(x;\boldsymbol{\lambda})=(q^{-x}-1)(1-abq^{x-1}),$
$\displaystyle\check{P}_{n}(x;N,\boldsymbol{\lambda})=P_{n}\bigl{(}\eta(x;\boldsymbol{\lambda});N,\boldsymbol{\lambda}\bigr{)}={}_{3}\phi_{2}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,abq^{x-1},\,q^{-x}}{a,\,q^{-N}}\Bigm{|}q\,;q\Bigr{)}.$
(2.40)

####  2.2.12 dual $\boldsymbol{q}$-Krawtchouk (d$\boldsymbol{q}$K)

Similar to the $q$-Krawtchouk, the dual $q$-Krawtchouk polynomial depends on
one parameter $\boldsymbol{\lambda}=p>0$, and the parameter $d$ is $d=-p$.

$\displaystyle
B(x;N,p)=\frac{(q^{x-N}-1)(1+pq^{x})}{(1+pq^{2x})(1+pq^{2x+1})},\ \
D(x;N,p)=pq^{2x-N-1}\frac{(1-q^{x})(1+pq^{x+N})}{(1+pq^{2x-1})(1+pq^{2x})},$
(2.41)
$\displaystyle\mathcal{E}(n)=q^{-n}-1,\quad\eta(x;p)=(q^{-x}-1)(1+pq^{x}),$
$\displaystyle\check{P}_{n}(x;N,p)=P_{n}\bigl{(}\eta(x;p);N,p\bigr{)}={}_{3}\phi_{2}\Bigl{(}\genfrac{}{}{0.0pt}{}{q^{-n},\,q^{-x},\,-pq^{x}}{q^{-N},\,0}\Bigm{|}q\,;q\Bigr{)}.$
(2.42)

###  2.3 Zero norm $=$ “Diophantine” and factorisation

The polynomials listed above § 2.2.1–§ 2.2.12 are finite polynomials, since
their degree $n$ is limited in $\mathcal{X}\ni n$ due to the presence of the
lower indices $-N$ or $q^{-N}$ in the ($q$)-hypergeometric expressions,
${}_{r+1}F_{r}$, ${}_{r+1}\phi_{r}$, of these polynomials, (2.14), (2.17),
(2.21), (2.24), (2.26), (2.28), (2.30), (2.32), (2.34), (2.37), (2.40) and
(2.42). For non-negative integer $m\geq 0$, the polynomial
$\check{P}_{N+1+m}(x;N,\boldsymbol{\lambda})$ is ill-defined due the presence
of the factors $(-N)_{N+1+k}=0$ and $(q^{-N}\,;q)_{N+1+k}=0$, $0\leq k\leq m$
in the denominator of the ($q$)-hypergeometric series expansion.

The situation is drastically changed by the introduction of the monic version
of each polynomial in § 2.2,

$\check{P}^{\text{monic}}_{n}(x;N,\boldsymbol{\lambda})=\frac{1}{c_{n}(N,\boldsymbol{\lambda})}\check{P}_{n}(x;N,\boldsymbol{\lambda}),$
(2.43)

where $c_{n}(N,\boldsymbol{\lambda})$ is the coefficient of the highest degree
term $\eta(x;\boldsymbol{\lambda})^{n}$. It is well defined for all degrees
$n=N+1+m$ ($m\in\mathbb{Z}_{\geq 0}$) and it satisfies the same difference
equation as (2.4), (2.5)

$\displaystyle\qquad\,B(x;N,\boldsymbol{\lambda})\bigl{(}\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})-\check{P}^{\text{monic}}_{N+1+m}(x+1;N,\boldsymbol{\lambda})\bigr{)}$
$\displaystyle\quad+D(x;N,\boldsymbol{\lambda})\bigl{(}\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})-\check{P}^{\text{monic}}_{N+1+m}(x-1;N,\boldsymbol{\lambda})\bigr{)}$
$\displaystyle=\mathcal{E}(N+1+m;\boldsymbol{\lambda})\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})\quad(x\in\mathcal{X}),$
(2.44)
$\displaystyle\Longrightarrow\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})=\mathcal{E}(N+1+m;\boldsymbol{\lambda})\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda}).$
(2.45)

This also means that the corresponding monic vector

$\phi^{\text{monic}}_{N+1+m}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\phi_{0}(x)\check{P}^{\text{monic}}_{N+1+m}(x)\quad(m\in\mathbb{Z}_{\geq
0}),$ (2.46)

satisfies the eigenvalue equation of the tri-diagonal real symmetric matrix
$\mathcal{H}$,

$\mathcal{H}\phi^{\text{monic}}_{N+1+m}(x)=\mathcal{E}(N+1+m;\boldsymbol{\lambda})\phi^{\text{monic}}_{N+1+m}(x)\quad(m\in\mathbb{Z}_{\geq
0}).$ (2.47)

If $\check{P}^{\text{monic}}_{N+1+m}(x)$ takes a non-vanishing value at some
point in $\mathcal{X}$, $\phi^{\text{monic}}_{N+1+m}(x)$ becomes an
eigenvector of $\mathcal{H}$, which contradicts the complete set of
eigenvectors $\\{\phi_{n}(x)\\}$ $(n\in\mathcal{X})$, (2.12). Thus we arrive
at

###### Theorem 2.1

For all the finite polynomials listed in § 2.2, the “Diophantine” property and
factorisation hold,

$\displaystyle\check{P}^{\text{\rm monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})$
$\displaystyle=\Lambda(x;N,\boldsymbol{\lambda})\check{Q}_{m}(x;N,\boldsymbol{\lambda})\quad(m\in\mathbb{Z}_{\geq
0}),$ (2.48) $\displaystyle\Lambda(x;N,\boldsymbol{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}\prod_{k=0}^{N}\bigl{(}\eta(x;\boldsymbol{\lambda})-\eta(k;\boldsymbol{\lambda})\bigr{)},$
(2.49)

in which $\check{Q}_{m}(x;N,\boldsymbol{\lambda})$ is a monic degree $m$
polynomial in $\eta(x;\boldsymbol{\lambda})$.

Of course, “Diophantine” property and factorisation can be demonstrated for
each polynomial by direct calculation of the ($q$)-hypergeometric series
expansion. Below we show the explicit derivation of the “Diophantine” and
factorisation property for the $q$-Racah polynomial, the most generic member
of the group.

Since the coefficient of the highest degree of
$\check{P}_{n}(x\,;N,\boldsymbol{\lambda})$ (2.37) is

$c_{n}(N,\boldsymbol{\lambda})=\frac{(\tilde{d}q^{n}\,;q)_{n}}{(b,c,q^{-N}\,;q)_{n}},$
(2.50)

the monic $q$-Racah polynomial is

$\check{P}^{\text{monic}}_{n}(x;N,\boldsymbol{\lambda})=\frac{1}{c_{n}(N,\boldsymbol{\lambda})}\check{P}_{n}(x;N,\boldsymbol{\lambda})=\sum_{k=0}^{n}\frac{(bq^{k},cq^{k},q^{k-N}\,;q)_{n-k}}{(\tilde{d}q^{n+k}\,;q)_{n-k}}\frac{(q^{-n},q^{-x},dq^{x}\,;q)_{k}}{(q\,;q)_{k}}q^{k}.$
(2.51)

At $n=N+1+m$ ($m\in\mathbb{Z}_{\geq 0}$), $(q^{k-N}\,;q)_{n-k}$ vanishes for
$k=0,1,\ldots,N$, and the $k$ summation is reduced to $\sum_{k=N+1}^{N+1+m}$.
By changing $k=N+1+l$, we obtain

$\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})=\sum_{l=0}^{m}\frac{(bq^{N+1+l},cq^{N+1+l},q^{l+1}\,;q)_{m-l}}{(\tilde{d}q^{2(N+1)+m+l}\,;q)_{m-l}}\frac{(q^{-N-1-m},q^{-x},dq^{x}\,;q)_{N+1+l}}{(q\,;q)_{N+1+l}}q^{N+1+l}.$

From the explicit form of $\eta(x;\boldsymbol{\lambda})$

$\eta(x;\boldsymbol{\lambda})-\eta(k;\boldsymbol{\lambda})=-q^{-k}(1-q^{-x+k})(1-dq^{x+k}),$

we obtain

$\Lambda(x;N,\boldsymbol{\lambda})=\prod_{k=0}^{N}\bigl{(}\eta(x;\boldsymbol{\lambda})-\eta(k;\boldsymbol{\lambda})\bigr{)}=(-1)^{N+1}q^{-\binom{N+1}{2}}(q^{-x},dq^{x}\,;q)_{N+1}.$
(2.52)

By using the basic properties of the $q$-shifted factorial,

$(a\,;q)_{n+k}=(a\,;q)_{n}(aq^{n}\,;q)_{k},\quad(a\,;q)_{n}=(-a)^{n}q^{\binom{n}{2}}(a^{-1}q^{1-n}\,;q)_{n},$
(2.53)

we obtain

$\frac{(q^{-N-1-m},q^{-x},dq^{x}\,;q)_{N+1+l}}{(q\,;q)_{N+1+l}}=\frac{(q^{-N-1-m},q^{-x},dq^{x}\,;q)_{N+1}}{(q\,;q)_{l}}\frac{(q^{-m},q^{-x+N+1},dq^{x+N+1}\,;q)_{l}}{(q^{l+1}\,;q)_{N+1}}$

and

$\displaystyle\frac{(q^{-N-1-m}\,;q)_{N+1}}{(q^{l+1}\,;q)_{N+1}}$
$\displaystyle=\frac{(-q^{-N-1-m})^{N+1}q^{\binom{N+1}{2}}(q^{m+1}\,;q)_{N+1}}{(q^{l+1}\,;q)_{N+1}}$
$\displaystyle=(-q^{-N-1-m})^{N+1}q^{\binom{N+1}{2}}\frac{(q^{N+l+2}\,;q)_{m-l}}{(q^{l+1}\,;q)_{m-l}}.$

These lead to

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;\boldsymbol{\lambda})$
$\displaystyle=\Lambda(x;N,\boldsymbol{\lambda})\cdot q^{-(N+1)m}$
$\displaystyle\quad\times\sum_{l=0}^{m}\frac{(bq^{N+1+l},cq^{N+1+l},qq^{N+1+l}\,;q)_{m-l}}{(\tilde{d}q^{2(N+1)+m+l}\,;q)_{m-l}}\frac{(q^{-m},q^{-x+N+1},dq^{x+N+1}\,;q)_{l}}{(q\,;q)_{l}}q^{l}$
$\displaystyle=\Lambda(x;N,\boldsymbol{\lambda})\cdot
q^{-(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime}).$
(2.54)

in which the shifted parameters are

$\boldsymbol{\lambda}^{\prime}=(bq^{N+1},cq^{N+1},dq^{2(N+1)}).$ (2.55)

Below we show the “Diophantine” and factorisation property for the other
polynomials in the group.

##### (i) Family with $\boldsymbol{\eta(x)=x}$

This family possesses the true Diophantine property as all the zeros of
$\Lambda(x)$ are integers. Miki, Tsujimoto and Vinet showed the Diophantine
and factorisation property of the Krawtchouk polynomial by explicit
calculations in the seminal paper [9].

####  2.3.1 Krawtchouk (K)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,p)=\Lambda(x;N)\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p),\quad
c_{n}(N,p)=\frac{1}{(-N)_{n}\,p^{n}},$ (2.56)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p)=\sum_{k=0}^{m}(N+2+k)_{m-k}\frac{(-m,-x+N+1)_{k}}{k!}p^{m-k}.$
(2.57)

####  2.3.2 Hahn (H)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})=\Lambda(x;N)\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime}),$
(2.58) $\displaystyle\qquad\boldsymbol{\lambda}^{\prime}=(a+N+1,b+N+1),\quad
c_{n}(N,\boldsymbol{\lambda})=\frac{(n+a+b-1)_{n}}{(a,-N)_{n}},$ (2.59)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime})=\sum_{k=0}^{m}\frac{(a+N+1+k,N+2+k)_{m-k}}{(m+a+b+2N+1+k)_{m-k}}\frac{(-m,-x+N+1)_{k}}{k!}.$
(2.60)

##### (ii) Family with $\boldsymbol{\eta(x)=x(x+d)}$

####  2.3.3 Racah (R)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})=\Lambda(x;N,\boldsymbol{\lambda})\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime}),$
(2.61)
$\displaystyle\qquad\boldsymbol{\lambda}^{\prime}=\bigl{(}b+N+1,c+N+1,d+2(N+1)\bigr{)},\quad
c_{n}(N,\boldsymbol{\lambda})=\frac{(\tilde{d}+n)_{n}}{(b,c,-N)_{n}},$ (2.62)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime})=\sum_{k=0}^{m}\frac{(b+N+1+k,c+N+1+k,N+2+k)_{m-k}}{(\tilde{d}+2N+2+m+k)_{m-k}}$
$\displaystyle\phantom{\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime})=\sum_{k=0}^{m}}\times\frac{(-m,-x+N+1,x+N+1+d)_{k}}{k!}.$
(2.63)

####  2.3.4 dual Hahn (dH)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})=\Lambda(x;N,\boldsymbol{\lambda})\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime}),$
(2.64)
$\displaystyle\qquad\boldsymbol{\lambda}^{\prime}=\bigl{(}a+N+1,b+N+1\bigr{)},\quad
c_{n}(N,\boldsymbol{\lambda})=\frac{1}{(a,-N)_{n}},$ (2.65)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime})=\sum_{k=0}^{m}(a+N+1+k,N+2+k)_{m-k}$
$\displaystyle\phantom{\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime})=\sum_{k=0}^{m}}\times\frac{(-m,-x+N+1,x+a+b+N)_{k}}{k!}.$
(2.66)

##### (iii) Family with $\boldsymbol{\eta(x)=1-q^{x}}$

####  2.3.5 dual quantum $\boldsymbol{q}$-Krawtchouk (dq$\boldsymbol{q}$K)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,p)=\Lambda(x;N)\,q^{(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime}),$
(2.67) $\displaystyle\qquad p^{\prime}=pq^{N+1},\quad
c_{n}(N,p)=\frac{p^{n}q^{-\frac{1}{2}n(n-1)}}{(q^{-N}\,;q)_{n}},$ (2.68)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime})=\sum_{k=0}^{m}(q^{N+2+k}\,;q)_{m-k}\frac{(q^{-m},q^{-x+N+1}\,;q)_{k}}{(q\,;q)_{k}}$
$\displaystyle\phantom{\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime})=\sum_{k=0}^{m}}\times
p^{k-m}q^{kx+k+\frac{1}{2}m(m-1)-m(N+1)}.$ (2.69)

##### (iv) Family with $\boldsymbol{\eta(x)=q^{-x}-1}$

####  2.3.6 $\boldsymbol{q}$-Hahn ($\boldsymbol{q}$H)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})=\Lambda(x;N)\,q^{-(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime}),$
(2.70)
$\displaystyle\qquad\boldsymbol{\lambda}^{\prime}=(aq^{N+1},bq^{N+1}),\quad
c_{n}(N,\boldsymbol{\lambda})=\frac{(abq^{n-1}\,;q)_{n}}{(a,q^{-N}\,;q)_{n}},$
(2.71)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime})=\sum_{k=0}^{m}\frac{(aq^{N+1+k},q^{N+2+k}\,;q)_{m-k}}{(abq^{m+2N+1+k}\,;q)_{m-k}}\frac{(q^{-m},q^{-x+N+1}\,;q)_{k}}{(q\,;q)_{k}}q^{k}.\\!$
(2.72)

####  2.3.7 $\boldsymbol{q}$-Krawtchouk ($\boldsymbol{q}$K)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,p)=\Lambda(x;N)\,q^{-(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime}),$
(2.73) $\displaystyle\qquad p^{\prime}=pq^{2(N+1)},\quad
c_{n}(N,p)=\frac{(-pq^{n}\,;q)_{n}}{(q^{-N}\,;q)_{n}},$ (2.74)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime})=\sum_{k=0}^{m}\frac{(q^{N+2+k}\,;q)_{m-k}}{(-pq^{2(N+1)+m+k}\,;q)_{m-k}}\frac{(q^{-m},q^{-x+N+1}\,;q)_{k}}{(q\,;q)_{k}}q^{k}.$
(2.75)

####  2.3.8 quantum $\boldsymbol{q}$-Krawtchouk (q$\boldsymbol{q}$K)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,p)=\Lambda(x;N)\,q^{-(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime}),$
(2.76) $\displaystyle\qquad p^{\prime}=pq^{N+1},\quad
c_{n}(N,p)=\frac{p^{n}q^{n^{2}}}{(q^{-N}\,;q)_{n}},$ (2.77)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime})=\sum_{k=0}^{m}(q^{N+2+k}\,;q)_{m-k}\frac{(q^{-m},q^{-x+N+1}\,;q)_{k}}{(q\,;q)_{k}}$
$\displaystyle\phantom{\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime})=\sum_{k=0}^{m}}\times
p^{k-m}q^{(N+m+2)k-m(m+N+1)}.$ (2.78)

####  2.3.9 affine $\boldsymbol{q}$-Krawtchouk (a$\boldsymbol{q}$K)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,p)=\Lambda(x;N)\,q^{-(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime}).$
(2.79) $\displaystyle\qquad p^{\prime}=pq^{N+1},\quad
c_{n}(N,p)=\frac{1}{(pq,q^{-N}\,;q)_{n}},$ (2.80)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime})=\sum_{k=0}^{m}(pq^{N+2+k},q^{N+2+k}\,;q)_{m-k}\frac{(q^{-m},q^{-x+N+1}\,;q)_{k}}{(q\,;q)_{k}}q^{k}.$
(2.81)

##### (v) Family with $\boldsymbol{\eta(x)=(q^{-x}-1)(1-dq^{x})}$

####  2.3.10 dual $\boldsymbol{q}$-Hahn (d$\boldsymbol{q}$H)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})=\Lambda(x;N,\boldsymbol{\lambda})\,q^{-(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime}),$
(2.82)
$\displaystyle\qquad\qquad\boldsymbol{\lambda}^{\prime}=(aq^{N+1},bq^{N+1}),\quad
c_{n}(N,\boldsymbol{\lambda})=\frac{1}{(a,q^{-N}\,;q)_{n}},$ (2.83)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,\boldsymbol{\lambda}^{\prime})=\sum_{k=0}^{m}(aq^{N+1+k},q^{N+2+k}\,;q)_{m-k}\frac{(q^{-m},abq^{x+N},q^{-x+N+1}\,;q)_{k}}{(q\,;q)_{k}}q^{k}.$
(2.84)

####  2.3.11 dual $\boldsymbol{q}$-Krawtchouk (d$\boldsymbol{q}$K)

$\displaystyle\check{P}^{\text{monic}}_{N+1+m}(x;N,p)=\Lambda(x;N,p)\,q^{-(N+1)m}\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime}),$
(2.85) $\displaystyle\qquad p^{\prime}=p^{2(N+1)},\quad
c_{n}(N,p)=\frac{1}{(q^{-N}\,;q)_{n}},$ (2.86)
$\displaystyle\check{P}^{\text{monic}}_{m}(x-N-1;-N-2,p^{\prime})=\sum_{k=0}^{m}(q^{N+2+k}\,;q)_{m-k}\frac{(q^{-m},q^{-x+N+1},-pq^{x+N+1}\,;q)_{k}}{(q\,;q)_{k}}q^{k}.$
(2.87)

###  2.4 Factorisation of multi-indexed and Krein-Adler systems

As shown in § 2.1, the “Diophantine” property and factorisation are a
consequence of a general fact that the finite orthogonal polynomials in the
Askey scheme are the eigenvectors of certain real symmetric matrices
$\mathcal{H}$ (2.10). Certain generalisation of the finite orthogonal
polynomials in the Askey scheme are known for some time. They are generated by
multiple Darboux transformations [10] of these polynomials by adopting certain
seed solutions. When the polynomial themselves are chosen as the seed
solutions [11], the obtained polynomials are called Krein-Adler [12, 13]
polynomials. When certain virtual state vectors are used as the seed
solutions, the new polynomials are called, for example, multi-indexed
($q$-)Racah polynomials, etc [14]. By construction, i.e. by appropriate
choices of the seed solutions and the parameter ranges, these new types of
orthogonal polynomials are the eigenvectors of certain real symmetric
matrices. Therefore, when $\check{P}_{n}(x)$ is changed to
$\check{P}^{\text{monic}}_{N+1+m}(x)$ in (3.17), the new types of orthogonal
polynomials have the “Diophantine” property and factorisation.

As will be discussed in the subsequent two sections, a new type of multi-
indexed polynomials can be generated by using
$\check{P}^{\text{monic}}_{N+1+m}(x)$ as seed solutions for the multiple
Darboux transformations. If the resulting orthogonality measures are positive
definite, which is the case for those introduced in § 4, such systems have
also the “Diophantine” property and factorisation.

##  3 New Multi-indexed Polynomials, General Setting

One possible application of the new polynomials satisfying the “Diophantine”
and factorisation properties (2.48) is the generation of new multi-indexed
orthogonal polynomials. It is well known that certain infinite norm solutions
of the Schrödinger equations with the radial oscillator and Pöschl-Teller
potentials are used as seed solutions for multiple Darboux transformations to
generate multi-indexed Laguerre and Jacobi polynomials [15, 16, 17, 18].
Likewise the non-eigenvector solutions of the tri-diagonal real symmetric
matrix $\mathcal{H}$ eigenvalue problem (2.12) are used as seed solutions to
generate the multi-indexed versions of the ($q$)-Racah polynomials [14] etc
through the difference equation analogues of the multiple Darboux
transformations. Now we have plenty of zero norm solutions

$\displaystyle\tilde{\phi}_{m}(x)$
$\displaystyle=\phi^{\text{monic}}_{N+1+m}(x)=\phi_{0}(x)\check{P}^{\text{monic}}_{N+1+m}(x;N,\boldsymbol{\lambda})\quad(m\in\mathbb{Z}_{\geq
0})$ (3.1)
$\displaystyle=\phi_{0}(x)\Lambda(x;N,\boldsymbol{\lambda})\check{Q}_{m}(x;N,\boldsymbol{\lambda}),\quad\bigl{(}\tilde{\phi}_{m}(x)\bigr{)}_{x\in\mathcal{X}}\in\mathbb{R}^{N+1},$
(3.2)

corresponding to (2.54), (2.56), (2.58), (2.61), (2.64), (2.67), (2.70),
(2.73), (2.76), (2.79), (2.82) and (2.85), which could be used as seed
solutions to generate the multi-indexed versions of the twelve polynomials
listed in § 2.2.1–§ 2.2.12.

Let us recapitulate the basic formulas of the multiple Darboux transformation
of the original eigenvectors $\\{\phi_{n}(x)\\}$ (2.11) by using the zero-norm
vectors $\\{\tilde{\phi}_{m}(x)\\}$ (3.1) as seed solutions. These formulas
apply for each of the finite polynomials listed in § 2.2. We employ the
formulas reported in § 2.3 of [19].

We choose $M$ distinct zero-norm seed solutions

$\\{\tilde{\phi}_{m_{1}}(x),\tilde{\phi}_{m_{2}}(x),\ldots,\tilde{\phi}_{m_{M}}(x)\\},\quad
0\leq m_{1}<m_{2}<\cdots<m_{M},$ (3.3)

which correspond to the index set

$\mathcal{D}=\\{m_{1},\ldots,m_{M}\\}.$ (3.4)

The deformed set of eigenvectors $\\{\bar{\phi}_{\mathcal{D},n}\\}$
($n\in\mathcal{X}$) satisfy the difference equation

$\displaystyle\mathcal{H}_{\mathcal{D}}\bar{\phi}_{\mathcal{D},n}(x)=\mathcal{E}(n)\bar{\phi}_{\mathcal{D},n}(x),$
(3.5)
$\displaystyle\mathcal{H}_{\mathcal{D}}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{A}^{\dagger}_{\mathcal{D}}\mathcal{A}_{\mathcal{D}},\quad\mathcal{A}_{\mathcal{D}}\stackrel{{\scriptstyle\text{def}}}{{=}}\sqrt{\bar{B}(x)}-e^{\partial}\sqrt{\bar{D}(x)},\quad\mathcal{A}_{\mathcal{D}}^{\dagger}\stackrel{{\scriptstyle\text{def}}}{{=}}\sqrt{\bar{B}(x)}-\sqrt{\bar{D}(x)}e^{-\partial},$
(3.6)
$\displaystyle(\bar{\phi}_{\mathcal{D},n},\bar{\phi}_{\mathcal{D},\ell})=\prod_{j=1}^{M}\bigl{(}\mathcal{E}(n)-\mathcal{E}(N+1+m_{j})\bigr{)}\cdot\frac{\delta_{n\,\ell}}{d_{n}^{2}}.$
(3.7)

In these formulas $e^{\partial}$
($\partial\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{d}{dx}$) is a finite
shift operator acting on smooth functions,

$e^{\partial}f(x)=f(x+1),\quad(e^{\partial})^{j}=e^{j\partial},\quad
e^{j\partial}f(x)=f(x+j)\quad(j\in\mathbb{Z}).$ (3.8)

As an operator it passes a smooth function $g(x)$ as

$e^{j\partial}g(x)=g(x+j)e^{j\partial}.$

Here $\bar{B}(x)$, $\bar{D}(x)$ and $\bar{\phi}_{\mathcal{D},n}(x)$ are
expressed in terms of the Casoratians involving the seed solutions
$\\{\tilde{\phi}_{m}\\}$, eigenvector $\phi_{n}$ and the groundstate
eigenvector $\phi_{0}$,

$\displaystyle\text{W}_{\text{C}}[f_{1},\ldots,f_{n}](x)\stackrel{{\scriptstyle\text{def}}}{{=}}\det\Bigl{(}f_{k}(x+j-1)\Bigr{)}_{1\leq
j,k\leq n},$ (3.9)
$\displaystyle\bar{B}(x)=\sqrt{B(x+M)D(x+M+1)}\,\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x+1)}$
$\displaystyle\phantom{\bar{B}(x)=}\times\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},{\phi}_{0}](x+1)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},\phi_{0}](x)},$
(3.10)
$\displaystyle\bar{D}(x)=\sqrt{B(x-1)D(x)}\,\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x+1)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x)}$
$\displaystyle\phantom{\bar{D}(x)=}\times\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},\phi_{0}](x-1)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},\phi_{0}](x)},$
(3.11) $\displaystyle
F(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{\sqrt{\prod_{k=1}^{M}B(x+k-1)D(x+k)}}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x)\,\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x+1)},$
$\displaystyle\bar{\phi}_{\mathcal{D},n}(x)=(-1)^{M}\sqrt{F(x)}\,\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},\phi_{n}](x).$
(3.12)

By using the Casoratian identity

$\displaystyle\text{W}_{\text{C}}[gf_{1},gf_{2},\ldots,gf_{n}](x)=\prod_{k=0}^{n-1}g(x+k)\cdot\text{W}_{\text{C}}[f_{1},f_{2},\ldots,f_{n}](x),$
(3.13)

some formulas are simplified,

$\displaystyle\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x+1)}$
$\displaystyle=\frac{\phi_{0}(x)\Lambda(x)}{\phi_{0}(x+M)\Lambda(x+M)}\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)},$
$\displaystyle\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x+1)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x)}$
$\displaystyle=\frac{\phi_{0}(x+M)\Lambda(x+M)}{\phi_{0}(x)\Lambda(x)}\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)},$
$\displaystyle\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},{\phi}_{0}](x+1)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},\phi_{0}](x)}$
$\displaystyle=\frac{\phi_{0}(x+M+1)\Lambda(x+M+1)}{\phi_{0}(x)\Lambda(x)}$
$\displaystyle\quad\times\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x+1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x)},$
$\displaystyle\frac{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},{\phi}_{0}](x-1)}{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},\phi_{0}](x)}$
$\displaystyle=\frac{\phi_{0}(x-1)\Lambda(x-1)}{\phi_{0}(x+M)\Lambda(x+M)}\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x-1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x)},$
$\displaystyle{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}},\phi_{n}](x)}$
$\displaystyle=\prod_{k=0}^{M}\phi_{0}(x+k)\Lambda(x+k)\cdot{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}\check{P}_{n}](x)}.$

These lead to

$\displaystyle\bar{B}(x)$
$\displaystyle=\sqrt{B(x+M)D(x+M+1)}\,\frac{\phi_{0}(x+M+1)\Lambda(x+M+1)}{\phi_{0}(x+M)\Lambda(x+M)}$
$\displaystyle\quad\times\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)}\,\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x+1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x)}$
$\displaystyle=B(x+M)\frac{\Lambda(x+M+1)}{\Lambda(x+M)}$
$\displaystyle\quad\times\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)}\,\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x+1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x)},$
(3.14) $\displaystyle\bar{D}(x)$
$\displaystyle=\sqrt{B(x-1)D(x)}\,\frac{\phi_{0}(x-1)\Lambda(x-1)}{\phi_{0}(x)\Lambda(x)}$
$\displaystyle\quad\times\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)}\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x-1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x)}$
$\displaystyle=D(x)\frac{\Lambda(x-1)}{\Lambda(x)}$
$\displaystyle\quad\times\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)}\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x-1)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}](x)}.$
(3.15)

Furthermore, we have

$\displaystyle\quad{\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x)\,\text{W}_{\text{C}}[\tilde{\phi}_{m_{1}},\ldots,\tilde{\phi}_{m_{M}}](x+1)}$
$\displaystyle=\bigl{(}\phi_{0}(x)\phi_{0}(x+M)\Lambda(x)\Lambda(x+M)\bigr{)}^{-1}\prod_{k=0}^{M}\phi_{0}(x+k)^{2}\Lambda(x+k)^{2}$
$\displaystyle\quad\times{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)},$
$\displaystyle\sqrt{\prod_{k=1}^{M}B(x+k-1)D(x+k)}\
\phi_{0}(x)\phi_{0}(x+M)=\phi_{0}(x)^{2}\prod_{k=1}^{M}B(x+k-1).$

In terms of these we arrive at

$\displaystyle F(x)$
$\displaystyle=\prod_{k=1}^{M}B(x+k-1)\cdot\frac{\Lambda(x+M)}{\Lambda(x)}\frac{1}{\prod_{k=1}^{M}\phi_{0}(x+k)^{2}\Lambda(x+k)^{2}}$
$\displaystyle\quad\times\frac{1}{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)},$
(3.16) $\displaystyle\bar{\phi}_{\mathcal{D},n}(x)$
$\displaystyle=(-1)^{M}\sqrt{\prod_{k=1}^{M}B(x+k-1)}\
\phi_{0}(x)\sqrt{\Lambda(x)\Lambda(x+M)}$
$\displaystyle\quad\times\frac{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}},\Lambda^{-1}\check{P}_{n}](x)}{\sqrt{\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x)\text{W}_{\text{C}}[\check{Q}_{m_{1}},\ldots,\check{Q}_{m_{M}}](x+1)}}.$
(3.17)

For each type of the polynomials, the multi-indexed polynomials are identified
by removing various kinematical factors from $\bar{\phi}_{\mathcal{D},n}$. As
shown by (3.7), the orthogonality is built in. After the identification and
the orthogonality, securing the positivity of the orthogonality measures is
essential. It must be verified for each polynomial, for each choice of the set
$\mathcal{D}$ of the zero-norm seed solutions and for the appropriate ranges
of the involved parameters. The situation is much more complicated than the
multi-indexed Laguerre and Jacobi polynomials [18] and ($q$)-Racah polynomials
[14], etc. In these established cases, the seed solutions can be chosen to
have a definite sign, and that is closely related to the positive definite
orthogonality measures. We hope these detailed tasks for each type of
polynomials will be carried out in future. A few examples of single-indexed
exceptional Krawtchouk polynomials are reported in [9] and their “Diophantine”
property and factorisation are discussed.

##  4 Shape-invariant Cases

Here we report on multi-indexed polynomials corresponding to a very special
choice of the seed solutions, $M$ contiguous lowest degree zero-norm
solutions,

$\mathcal{D}=\\{0,1,\ldots,M-1\\}.$ (4.1)

For the Schrödinger equations, for which the Laguerre and Jacobi polynomials
are the eigen polynomials, the multiple Darboux transformations by using $M$
contiguous lowest degree eigenfunctions produce remarkable effects [10, 12,
13],

$\displaystyle\text{Laguerre}:$ $\displaystyle
L_{n}^{(\alpha)}\bigl{(}\eta(x)\bigr{)}$
$\displaystyle\,\to\,(\text{const})\times
L_{n-M}^{(\alpha+M)}\bigl{(}\eta(x)\bigr{)}$ $\displaystyle\ \ (n\geq M),$
$\displaystyle\text{Jacobi}:$ $\displaystyle
P_{n}^{(\alpha,\beta)}\bigl{(}\eta(x)\bigr{)}$
$\displaystyle\,\to\,(\text{const})\times
P_{n-M}^{(\alpha+M,\beta+M)}\bigl{(}\eta(x)\bigr{)}$ $\displaystyle\ \ (n\geq
M).$

Under the transformations, the polynomials keep their identity with shifted
parameters and the positivity of the orthogonality measure unchanged. This
phenomenon is called shape-invariance [20]. Shape-invariance also holds for
all the polynomials listed in § 2.2 and the others in the Askey scheme. It is
called the forward shift relation [5]. For example

$\displaystyle\text{Racah}:$ $\displaystyle\check{P}_{n}(x;N,b,c,d)$
$\displaystyle\,\to\,(\text{const})\times\check{P}_{n-M}(x;N-M,b+M,c+M,d+M)$
$\displaystyle\ \ (n\geq M),$ $\displaystyle\text{$q$-Racah}:$
$\displaystyle\check{P}_{n}(x;N,b,c,d)$
$\displaystyle\,\to\,(\text{const})\times\check{P}_{n-M}(x;N-M,bq^{M},cq^{M},dq^{M})$
$\displaystyle\ \ (n\geq M).$

For more details, see [7].

We will demonstrate similar effects for the present case of using $M$
contiguous lowest zero-norm states (4.1) for the finite orthogonal
polynomials. The explicit formulas of the transformation of $\bar{B}$ (3.14),
$\bar{D}$ (3.15) and $\bar{\phi}_{\mathcal{D},n}$ (3.17), depend on the type
of the sinusoidal coordinates $\eta(x)$. For three families (i), (iii) and
(iv), in which the sinusoidal coordinate contains no parameter other than $q$,
(i) $\eta(x)=x$, (iii) $\eta(x)=1-q^{x}$ and (iv) $\eta(x)=q^{-x}-1$. For the
other two families (ii) and (v), in which $\eta(x)$ contains $d$, (ii)
$\eta(x)=x(x+d)$, (v) $\eta(x)=(q^{-x}-1)(1-dq^{x})$, the parameter $d$ shifts
in the formulas. They are summarised in the following

###### Theorem 4.1

The results of the multiple Darboux transformation are

$\displaystyle\text{Family {\rm(i),\,(iv)}}:$ $\displaystyle\quad
B(x;N,\boldsymbol{\lambda})$
$\displaystyle\,\to\,B(x+M;N+M,\boldsymbol{\lambda}),$ (4.2) $\displaystyle
D(x;N,\boldsymbol{\lambda})$
$\displaystyle\,\to\,D(x+M;N+M,\boldsymbol{\lambda}),$ (4.3)
$\displaystyle\check{P}_{n}(x;N,\boldsymbol{\lambda})$
$\displaystyle\,\to\,(\text{\rm
const})\times\check{P}_{n}(x+M;N+M,\boldsymbol{\lambda}),$ (4.4)
$\displaystyle\text{Family {\rm(ii)}}:$ $\displaystyle\quad
B(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle\,\to\,B(x+M;N+M,d-M,\bar{\boldsymbol{\lambda}}),$ (4.5)
$\displaystyle D(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle\,\to\,D(x+M;N+M,d-M,\bar{\boldsymbol{\lambda}}),$ (4.6)
$\displaystyle\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle\,\to\,(\text{\rm
const})\times\check{P}_{n}(x+M;N+M,d-M,\bar{\boldsymbol{\lambda}}),$ (4.7)
$\displaystyle\text{Family {\rm(iii)}}:$ $\displaystyle\quad B(x;N,p)$
$\displaystyle\,\to\,B(x+M;N+M,pq^{-M}),$ (4.8) $\displaystyle D(x;N,p)$
$\displaystyle\,\to\,D(x+M;N+M,pq^{-M}),$ (4.9)
$\displaystyle\check{P}_{n}(x;N,p)$ $\displaystyle\,\to\,(\text{\rm
const})\times\check{P}_{n}(x+M;N+M,pq^{-M}),$ (4.10)
$\displaystyle\text{Family {\rm(v)}}:$ $\displaystyle\quad
B(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle\,\to\,B(x+M;N+M,dq^{-M},\bar{\boldsymbol{\lambda}}),$ (4.11)
$\displaystyle D(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle\,\to\,D(x+M;N+M,dq^{-M},\bar{\boldsymbol{\lambda}}),$ (4.12)
$\displaystyle\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle\,\to\,(\text{\rm
const})\times\check{P}_{n}(x+M;N+M,dq^{-M},\bar{\boldsymbol{\lambda}}),$
(4.13)

in which $n\in\mathcal{X}$ and $\bar{\boldsymbol{\lambda}}$ stands for the
parameters other than $N$ and $d$. The positivity of the orthogonality
measures is unchanged for the formulas (4.2)–(4.4). The positivity also holds
for the formulas (4.5)–(4.13) for an appropriate range of the parameter $p$ or
$d$.

In the rest of this section we present the outline of the derivation of the
formulas (4.2)–(4.13).

In order to obtain the explicit forms for $\bar{B}(x)$ (3.14), $\bar{D}(x)$
(3.15) and $\bar{\phi}_{\mathcal{D},n}$ (3.17) we need to evaluate only four
formulas

$\displaystyle\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1}](x)$
,
$\displaystyle\Lambda(x){\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1},\Lambda^{-1}](x)},$
$\displaystyle\Lambda(x+M){\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1},\Lambda^{-1}](x)}$
,
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1},\Lambda^{-1}\check{P}_{n}](x).$

The other components are obtained by shifting $x\to x\pm 1$. As shown in
Theorem 2.1, $\check{Q}_{m}(x;N,\boldsymbol{\lambda})$ is a monic degree $m$
polynomial in $\eta(x;\boldsymbol{\lambda})$. Therefore these formulas are
simplified by sweeping the Casoratian determinants successively,

$\displaystyle\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1}](x)$
$\displaystyle=\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1}](x),$
(4.14)
$\displaystyle\Lambda(x){\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1},\Lambda^{-1}](x)}$
$\displaystyle=\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x),$
(4.15)
$\displaystyle\Lambda(x+M){\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1},\Lambda^{-1}](x)}$
$\displaystyle=\Lambda(x+M)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x),$
(4.16)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\check{Q}_{1},\ldots,\check{Q}_{M-1},\Lambda^{-1}\check{P}_{n}](x)$
$\displaystyle=\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}\check{P}_{n}](x).$
(4.17)

It should be stressed that the explicit expressions of $\check{Q}_{m}(x)$
listed in § 2.2.1–§ 2.2.12 are not necessary. Now that the r.h.s. of (4.14) is
explicitly known as it is a Vandermonde determinant. Likewise (4.15) and
(4.16) are easily evaluated as $\Lambda(x)/\Lambda(x+j-1)$ and
$\Lambda(x+M)/\Lambda(x+j-1)$ have simple expressions. It is important to
stress that these quantities depend on $N$ and $\eta(x;\boldsymbol{\lambda})$
only, except for the one (4.17) containing $\check{P}_{n}$.

Now we list the explicit forms of the expressions of (4.14)–(4.17) for each
family (i)–(v). By using the following constants,

$\displaystyle c(M)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\prod_{k=1}^{M-1}k!,\quad
c(N,M)\stackrel{{\scriptstyle\text{def}}}{{=}}(-1)^{M}\,(N+1)_{M}\,c(M),$
(4.18) $\displaystyle c_{q}(M)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}q^{-\frac{1}{6}M(M-1)(2M-1)}\prod_{k=1}^{M-1}(q\,;q)_{k},$
(4.19) $\displaystyle c_{q}(N,M)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(-1)^{M}q^{-\frac{1}{2}M(M-1)}(q^{N+1}\,;q)_{M}\,c_{q}(M),$
(4.20) $\displaystyle\binom{M}{j}$
$\displaystyle=\frac{M!}{j!\,(M-j)!},\quad\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{(q\,;q)_{M}}{(q\,;q)_{j}(q\,;q)_{M-j}},$
(4.21)

they are given as follows.
Family (i) :

$\displaystyle\text{W}_{\text{C}}[1,x,x^{2},\ldots,x^{M-1}](x)=c(M),$ (4.22)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,x,x^{2},\ldots,x^{M-1},\Lambda^{-1}](x)=\frac{c(N,M)}{(x+1)_{M}},$
(4.23)
$\displaystyle\Lambda(x+M)\text{W}_{\text{C}}[1,x,x^{2},\ldots,x^{M-1},\Lambda^{-1}](x)=\frac{c(N,M)}{(x-N)_{M}},$
(4.24)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,x,x^{2},\ldots,x^{M-1},\Lambda^{-1}\check{P}_{n}](x)$
$\displaystyle=\frac{(-1)^{M}c(M)}{(x+1)_{M}}\sum_{j=0}^{M}(-1)^{j}\binom{M}{j}(x+1+j)_{M-j}(x-N)_{j}\check{P}_{n}(x+j),$
(4.25)

Family (ii) :

$\displaystyle\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1}](x)=c(M)\prod_{k=1}^{M-1}(2x+k+d)_{k},$
(4.26)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c(N,M)\prod_{k=1}^{M}(2x+k+d)_{k}}{(x+1,x+N+1+d)_{M}},$
(4.27)
$\displaystyle\Lambda(x+M)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c(N,M)\prod_{k=1}^{M}(2x+k+d)_{k}}{(x-N,x+d)_{M}},$
(4.28)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}\check{P}_{n}](x)$
$\displaystyle=\frac{(-1)^{M}c(M)\prod_{k=1}^{M-2}(2x+2+k+d)_{k}}{(x+1,x+N+1+d)_{M}}\sum_{j=0}^{M}(-1)^{j}\binom{M}{j}\mathcal{T}(x,M,j,d)$
$\displaystyle\qquad\times(x+1+j,x+1+j+N+d)_{M-j}(x-N,x+d)_{j}\check{P}_{n}(x+j),$
(4.29)
$\displaystyle\mathcal{T}(x,M,j,d)\stackrel{{\scriptstyle\text{def}}}{{=}}(2x+M+1+j+d)_{M-j-1}(2x+1+d)_{j-1}$
$\displaystyle\phantom{\mathcal{T}(x,M,j,d)\stackrel{{\scriptstyle\text{def}}}{{=}}}\times\left\\{\begin{array}[]{cl}1&:j=0,M\\\
(2x+2j+d)&:\text{otherwise}\end{array}\right.,$ (4.32)

Family (iii) :

$\displaystyle\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1}](x)=c_{q}(M)\,q^{\frac{1}{2}M(M-1)x}q^{\frac{1}{2}M(M-1)^{2}},$
(4.33)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c_{q}(N,M)\,q^{\frac{1}{2}M(M+1)x}q^{\frac{1}{2}M(M^{2}-2N-1)}}{(q^{x+1}\,;q)_{M}},$
(4.34)
$\displaystyle\Lambda(x+M)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c_{q}(N,M)\,q^{\frac{1}{2}M(M+1)x}q^{\frac{1}{2}M(M^{2}-2N-1)}}{(q^{x-N}\,;q)_{M}},$
(4.35)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}\check{P}_{n}](x)$
$\displaystyle=\frac{(-1)^{M}c_{q}(M)q^{\frac{1}{2}M(M-1)x}q^{\frac{1}{2}M^{2}(M-1)-MN}}{(q^{x+1}\,q)_{M}}\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+M(N-j)}$
$\displaystyle\hskip
227.62204pt\times(q^{x+1+j}\,;q)_{M-j}(q^{x-N}\,;q)_{j}\check{P}_{n}(x+j),$
(4.36)

Family (iv) :

$\displaystyle\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1}](x)=c_{q}(M)\,q^{-\frac{1}{2}M(M-1)x},$
(4.37)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c_{q}(N,M)\,q^{-\frac{1}{2}M(M-1)x}}{(q^{x+1}\,;q)_{M}},$
(4.38)
$\displaystyle\Lambda(x+M)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c_{q}(N,M)\,q^{-\frac{1}{2}M(M-1)x}}{q^{M(N+1)}(q^{x-N}\,;q)_{M}},$
(4.39)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}\check{P}_{n}](x)$
$\displaystyle=\frac{(-1)^{M}c_{q}(M)\,q^{-\frac{1}{2}M(M-1)x}q^{-\frac{1}{2}M(M-1)}}{(q^{x+1}\,;q)_{M}}\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+Nj}$
$\displaystyle\hskip
213.39566pt\times(q^{x+1+j}\,;q)_{M-j}(q^{x-N}\,;q)_{j}\check{P}_{n}(x+j),$
(4.40)

Family (v) :

$\displaystyle\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1}](x)=c_{q}(M)\,q^{-\frac{1}{2}M(M-1)x}\prod_{k=1}^{M-1}(dq^{2x+k}\,;q)_{k},$
(4.41)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c_{q}(N,M)\,q^{-\frac{1}{2}M(M-1)x}\prod_{k=1}^{M}(dq^{2x+k}\,;q)_{k}}{(q^{x+1},dq^{x+1+N}\,;q)_{M}},$
(4.42)
$\displaystyle\Lambda(x+M)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}](x)=\frac{c_{q}(N,M)\,q^{-\frac{1}{2}M(M-1)x}\prod_{k=1}^{M}(dq^{2x+k}\,;q)_{k}}{q^{M(N+1)}(q^{x-N},dq^{x}\,;q)_{M}},$
(4.43)
$\displaystyle\Lambda(x)\text{W}_{\text{C}}[1,\eta,\eta^{2},\ldots,\eta^{M-1},\Lambda^{-1}\check{P}_{n}](x)$
$\displaystyle=\frac{(-1)^{M}c_{q}(M)\,q^{-\frac{1}{2}M(M-1)x}q^{-\frac{1}{2}M(M-1)}\prod_{k=1}^{M-2}(dq^{2x+2+k}\,;q)_{k}}{(q^{x+1},dq^{x+N+1}\,;q)_{M}}\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+Nj}$
$\displaystyle\qquad\times\mathcal{T}_{q}(x,M,j,d)(q^{x+1+j},dq^{x+1+j+N}\,;q)_{M-j}(q^{x-N},dq^{x}\,;q)_{j}\check{P}_{n}(x+j),$
(4.44)
$\displaystyle\mathcal{T}_{q}(x,M,j,d)\stackrel{{\scriptstyle\text{def}}}{{=}}(dq^{2x+M+1+j}\,;q)_{M-j-1}(dq^{2x+1}\,;q)_{j-1}\,\times\left\\{\begin{array}[]{cl}1&:j=0,M\\\
(1-dq^{2x+2j})&:\text{otherwise}\end{array}\right.\\!.$ (4.47)

Based on these results, the transformed functions $\bar{B}$ (3.14) and
$\bar{D}$ (3.15) are calculated for each family:

$\displaystyle\text{Family (i)}:\ \ \bar{B}(x)$
$\displaystyle=B(x+M)\,\frac{(x-N)_{M}}{(x+1-N)_{M}}=B(x+M)\,\frac{N-x}{N-x-M},$
(4.48) $\displaystyle\bar{D}(x)$
$\displaystyle=D(x)\,\frac{(x+1)_{M}}{(x)_{M}}=D(x)\,\frac{x+M}{x},$ (4.49)
$\displaystyle\text{Family (ii)}:\ \ \bar{B}(x)$
$\displaystyle=B(x+M)\,\frac{(N-x)(x+d)(2x+2M+d)_{2}}{(N-x-M)(x+d+M)(2x+M+d)_{2}},$
(4.50) $\displaystyle\bar{D}(x)$
$\displaystyle=D(x)\,\frac{(x+M)(x+N+M+d)(2x-1+d)_{2}}{x(x+N+d)(2x-1+M+d)_{2}},$
(4.51) $\displaystyle\text{Family (iii)}:\ \ \bar{B}(x)$
$\displaystyle=B(x+M)\,\frac{q^{M}(q^{x-N}\,;q)_{M}}{(q^{x+1-N}\,;q)_{M}}=B(x+M)\,\frac{q^{M}(1-q^{x-N})}{1-q^{x+M-N}},$
(4.52) $\displaystyle\bar{D}(x)$
$\displaystyle=D(x)\,\frac{q^{-M}(q^{x+1}\,;q)_{M}}{(q^{x}\,;q)_{M}}=D(x)\,\frac{q^{-M}(1-q^{x+M})}{1-q^{x}},$
(4.53) $\displaystyle\text{Family (iv)}:\ \ \bar{B}(x)$
$\displaystyle=B(x+M)\,\frac{(q^{x-N}\,;q)_{M}}{(q^{x+1-N}\,;q)_{M}}=B(x+M)\,\frac{1-q^{x-N}}{1-q^{x+M-N}},$
(4.54) $\displaystyle\bar{D}(x)$
$\displaystyle=D(x)\,\frac{(q^{x+1}\,;q)_{M}}{(q^{x}\,;q)_{M}}=D(x)\,\frac{1-q^{x+M}}{1-q^{x}},$
(4.55) $\displaystyle\text{Family (v)}:\ \ \bar{B}(x)$
$\displaystyle=B(x+M)\,\frac{(1-q^{x-N})(1-dq^{x})(dq^{2x+2M}\,;q)_{2}}{(1-q^{x+M-N})(1-dq^{x+M})(dq^{2x+M}\,;q)_{2}},$
(4.56) $\displaystyle\bar{D}(x)$
$\displaystyle=D(x)\,\frac{(1-q^{x+M})(1-dq^{x+M+N})(dq^{2x-1}\,;q)_{2}}{(1-q^{x})(1-dq^{x+N})(dq^{2x+M-1}\,;q)_{2}}.$
(4.57)

It is straightforward to verify Theorem 4.1 for the transformations of $B(x)$
and $D(x)$ for the families (i)–(v) by consulting the explicit expressions of
$B(x)$ and $D(x)$ of each polynomial listed in § 2.2. The transformations
rules of the polynomials (4.4), (4.7), (4.10), and (4.13) are the direct
consequences of those for $\bar{B}$ and $\bar{D}$. We list the explicit forms
of the transformations of the polynomials of degree $n\in\mathcal{X}$ in the
following:

###### Theorem 4.2

$\displaystyle\text{Family {\rm(i)}}:\ \ $
$\displaystyle\sum_{j=0}^{M}(-1)^{j}\binom{M}{j}(x+1+j)_{M-j}(x-N)_{j}\check{P}_{n}(x+j;N,{\boldsymbol{\lambda}})$
$\displaystyle=(N+1)_{M}\,\check{P}_{n}(x+M;N+M,{\boldsymbol{\lambda}}),$
(4.58) $\displaystyle\text{Family {\rm(ii)}}:\ \ $
$\displaystyle\sum_{j=0}^{M}(-1)^{j}\binom{M}{j}\mathcal{T}(x,M,j,d)(x+1+j,x+1+j+N+d)_{M-j}$
$\displaystyle\quad\times(x-N,x+d)_{j}\,\check{P}_{n}(x+j;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle=(N+1)_{M}(2x+1+d)_{2M-1}\,\check{P}_{n}(x+M;N+M,d-M,\bar{\boldsymbol{\lambda}}),$
(4.59) $\displaystyle\text{Family {\rm(iii)}}:\ \ $
$\displaystyle\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+M(N-j)}(q^{x+1+j}\,;q)_{M-j}(q^{x-N}\,;q)_{j}\check{P}_{n}(x+j;N,p)$
$\displaystyle=(q^{N+1}\,;q)_{M}q^{Mx}\,\check{P}_{n}(x+M;N+M,pq^{-M}),$
(4.60) $\displaystyle\text{Family {\rm(iv)}}:\ \ $
$\displaystyle\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+Nj}(q^{x+1+j}\,;q)_{M-j}(q^{x-N}\,;q)_{j}\check{P}_{n}(x+j;N,{\boldsymbol{\lambda}})$
$\displaystyle=(q^{N+1}\,;q)_{M}\,\check{P}_{n}(x+M;N+M,{\boldsymbol{\lambda}}),$
(4.61) $\displaystyle\text{Family {\rm(v)}}:\ \ $
$\displaystyle\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+Nj}\mathcal{T}_{q}(x,M,j,d)(q^{x+1+j},dq^{x+1+j+N}\,;q)_{M-j}$
$\displaystyle\quad\times(q^{x-N},dq^{x}\,;q)_{j}\,\check{P}_{n}(x+j;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle=(q^{N+1}\,;q)_{M}(dq^{2x+1}\,;q)_{2M-1}\,\check{P}_{n}(x+M;N+M,dq^{-M},\bar{\boldsymbol{\lambda}}).$
(4.62)

The summations in l.h.s. are taken from (4.25), (4.29), (4.36), (4.40) and
(4.44). The factors in r.h.s. are determined by setting $n=0$. A consistency
check of the formulas can be done by setting $x=-M$. Again, it is
straightforward to verify these transformation formulas for the families
(i)–(v) by consulting the explicit expressions of the polynomials. Since the
size parameter of the matrix is now $N+M$, there are additional members of the
orthogonal polynomials. They are simply
$\\{\check{P}_{N+1+m}(x+M,N+M,{\boldsymbol{\lambda}})\\}$ $(0\leq m\leq M)$,
for family (i), etc. This concludes the proof of Theorem 4.1.

It is interesting and instructive to scrutinise Theorem 4.2 from a different
perspective. The formulas (4.58)–(4.62) contain $\sum_{j=0}^{M}$,
$\binom{M}{j}$ or $\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}$ and subscripts $M-j$ and
$j$, showing a semblance to binomial expansions. Let us introduce operators
$\tilde{\mathcal{F}}$ to realise one step transformation for each family. For
$M=1$, Theorem 4.2 tells

$\displaystyle\text{(i)}:$ $\displaystyle\ \
\tilde{\mathcal{F}}(x,N)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{N+1}\bigl{(}x+1+(N-x)e^{\partial}\bigr{)},$
(4.63)
$\displaystyle\qquad\tilde{\mathcal{F}}(x,N)\check{P}_{n}(x;N,{\boldsymbol{\lambda}})=\check{P}_{n}(x+1;N+1,{\boldsymbol{\lambda}}),$
$\displaystyle\text{(ii)}:$ $\displaystyle\ \
\tilde{\mathcal{F}}(x,N,d)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{(N+1)(2x+1+d)}\bigl{(}(x+1)(x+1+N+d)+(N-x)(x+d)e^{\partial}\bigr{)},$
(4.64)
$\displaystyle\qquad\tilde{\mathcal{F}}(x,N,d)\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}})=\check{P}_{n}(x+1;N+1,d-1,\bar{\boldsymbol{\lambda}}),$
$\displaystyle\text{(iii)}:$ $\displaystyle\ \
\tilde{\mathcal{F}}(x,N)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{q^{N-x}}{1-q^{N+1}}\bigl{(}1-q^{x+1}+(q^{x-N}-1)e^{\partial}\bigr{)},$
(4.65)
$\displaystyle\qquad\tilde{\mathcal{F}}(x,N)\check{P}_{n}(x;N,p)=\check{P}_{n}(x+1;N+1,pq^{-1}),$
$\displaystyle\text{(iv)}:$ $\displaystyle\ \
\tilde{\mathcal{F}}(x,N)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{1-q^{N+1}}\bigl{(}1-q^{x+1}+q^{N+1}(q^{x-N}-1)e^{\partial}\bigr{)},$
(4.66)
$\displaystyle\qquad\tilde{\mathcal{F}}(x,N)\check{P}_{n}(x;N,{\boldsymbol{\lambda}})=\check{P}_{n}(x+1;N+1,{\boldsymbol{\lambda}}),$
$\displaystyle\text{(v)}:$ $\displaystyle\ \
\tilde{\mathcal{F}}(x,N,d)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{(1-q^{N+1})(1-dq^{2x+1})}$
$\displaystyle\phantom{\ \
\tilde{\mathcal{F}}(x,N,d)\stackrel{{\scriptstyle\text{def}}}{{=}}}\times\bigl{(}(1-q^{x+1})(1-dq^{x+1+N})+q^{N+1}(q^{x-N}-1)(1-dq^{x})e^{\partial}\bigr{)},$
(4.67)
$\displaystyle\qquad\tilde{\mathcal{F}}(x,N,d)\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}})=\check{P}_{n}(x+1;N+1,dq^{-1},\bar{\boldsymbol{\lambda}}).$

Here $e^{\partial}$ is the finite shift operator (3.8). Let us tentatively
call $\tilde{\mathcal{F}}$ a forward $x$-shift operator. Each
$\tilde{\mathcal{F}}$ has two terms, one is an ordinary function and the other
is a function times $e^{\partial}$. Since the polynomial is mapped from
$\check{P}_{n}(x;N,{\boldsymbol{\lambda}})$ to
$\check{P}_{n}(x+1;N+1,{\boldsymbol{\lambda}})$, for family (i), the next
steps are obviously

$\displaystyle\text{(i)}:\ \
\tilde{\mathcal{F}}(x+1,N+1)\tilde{\mathcal{F}}(x,N)\check{P}_{n}(x;N,{\boldsymbol{\lambda}})=\check{P}_{n}(x+2;N+2,{\boldsymbol{\lambda}}),$
$\displaystyle\phantom{\text{(i)}:\ \
}\tilde{\mathcal{F}}(x+2,N+2)\tilde{\mathcal{F}}(x+1,N+1)\tilde{\mathcal{F}}(x,N)\check{P}_{n}(x;N,{\boldsymbol{\lambda}})=\check{P}_{n}(x+3;N+3,\boldsymbol{\lambda}),$
$\displaystyle\hskip 113.81102pt\vdots\ ,$

leading to

$\displaystyle\text{(i),\,(iv)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k)\cdot\check{P}_{n}(x;N,\boldsymbol{\lambda})$
$\displaystyle=\check{P}_{n}(x+M;N+M,\boldsymbol{\lambda}),$ (4.68)
$\displaystyle\text{(ii)}:$ $\displaystyle\
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k,d-k)\cdot\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle=\check{P}_{n}(x+M;N+M,d-M,\bar{\boldsymbol{\lambda}}),\\!$
(4.69) $\displaystyle\text{(iii)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k)\cdot\check{P}_{n}(x;N,p)$
$\displaystyle=\check{P}_{n}(x+M;N+M,pq^{-M}),$ (4.70)
$\displaystyle\text{(v)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k,dq^{-k})\cdot\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}})$
$\displaystyle=\check{P}_{n}(x+M;N+M,dq^{-M},\bar{\boldsymbol{\lambda}}).$
(4.71)

Here we use the convention to express the ordered product

$\prod_{j=1}^{\stackrel{{\scriptstyle
n}}{{\longleftarrow}}}a_{j}\stackrel{{\scriptstyle\text{def}}}{{=}}a_{n}\cdots
a_{2}a_{1}.$ (4.72)

Thus we arrive at the following

###### Theorem 4.3

The formulas representing the special cases of the multi-indexed polynomials
(4.58)–(4.62) are factorised,

$\displaystyle{\rm(i)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k)=\frac{1}{(N+1)_{M}}\sum_{j=0}^{M}(-1)^{j}\binom{M}{j}(x+1+j)_{M-j}(x-N)_{j}\,e^{j\partial},$
(4.73) $\displaystyle{\rm(ii)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k,d-k)$
$\displaystyle=\frac{1}{(N+1)_{M}(2x+1+d)_{2M-1}}\sum_{j=0}^{M}(-1)^{j}\binom{M}{j}\mathcal{T}(x,M,j,d)$
$\displaystyle\qquad\qquad\times(x+1+j,x+1+j+N+d)_{M-j}(x-N,x+d)_{j}\,e^{j\partial},$
(4.74) $\displaystyle{\rm(iii)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k)$
$\displaystyle=\frac{q^{-Mx}}{(q^{N+1}\,;q)_{M}}\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+M(N-j)}(q^{x+1+j}\,;q)_{M-j}(q^{x-N}\,;q)_{j}\,e^{j\partial},$
(4.75) $\displaystyle{\rm(iv)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k)$
$\displaystyle=\frac{1}{(q^{N+1}\,;q)_{M}}\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+Nj}(q^{x+1+j}\,;q)_{M-j}(q^{x-N}\,;q)_{j}\,e^{j\partial},$
(4.76) $\displaystyle{\rm(v)}:$ $\displaystyle\ \
\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k,dq^{-k})$
$\displaystyle=\frac{1}{(q^{N+1}\,;q)_{M}(dq^{2x+1}\,;q)_{2M-1}}\sum_{j=0}^{M}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{\frac{1}{2}j(j+1)+Nj}\mathcal{T}_{q}(x,M,j,d)$
$\displaystyle\qquad\qquad\times(q^{x+1+j},dq^{x+1+j+N}\,;q)_{M-j}(q^{x-N},dq^{x}\,;q)_{j}\,e^{j\partial}.$
(4.77)

By expanding the l.h.s and collecting terms containing $e^{j\partial}$ gives
r.h.s.

These formulas can be demonstrated by straightforward induction. For example,
for (i),

$\displaystyle\quad\prod_{k=0}^{\stackrel{{\scriptstyle
M}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k)=\tilde{\mathcal{F}}(x+M,N+M)\prod_{k=0}^{\stackrel{{\scriptstyle
M-1}}{{\longleftarrow}}}\tilde{\mathcal{F}}(x+k,N+k)$
$\displaystyle=\frac{1}{(N+1)_{M+1}}\sum_{j=0}^{M}\Bigl{(}(-1)^{j}\binom{M}{j}(x+1+j)_{M+1-j}(x-N)_{j}\,e^{j\partial}$
$\displaystyle\phantom{=\frac{1}{(N+1)_{M+1}}\sum_{j=0}^{M}\Bigl{(}}-(-1)^{j}\binom{M}{j}(x+2+j)_{M-j}(x-N)_{j+1}\,e^{(j+1)\partial}\Bigr{)}$
$\displaystyle=\frac{1}{(N+1)_{M+1}}\sum_{j=0}^{M+1}(-1)^{j}\binom{M+1}{j}(x+1+j)_{M+1-j}(x-N)_{j}\,e^{j\partial}.$

Here we use Pascal’ triangle relation

$\binom{M}{j}+\binom{M}{j-1}=\binom{M+1}{j}.$ (4.78)

The $q$-analogues of the above relation

$\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}q^{j}+\genfrac{[}{]}{0.0pt}{}{\,M\,}{j-1}=\genfrac{[}{]}{0.0pt}{}{\,M+1\,}{j},\quad\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}+\genfrac{[}{]}{0.0pt}{}{\,M\,}{j-1}q^{M+1-j}=\genfrac{[}{]}{0.0pt}{}{\,M+1\,}{j}$
(4.79)

are used in the inductions for (iii)–(v) and the relations

$j\binom{M}{j}-(M+1-j)\binom{M}{j-1}=0,\quad(1-q^{j})\genfrac{[}{]}{0.0pt}{}{\,M\,}{j}-(1-q^{M+1-j})\genfrac{[}{]}{0.0pt}{}{\,M\,}{j-1}=0$
(4.80)

are used for (ii) and (v).

It is well known that the difference operator $\widetilde{\mathcal{H}}$ (2.5),
(2.6) is factorised into the forward $\mathcal{F}$ and backward $\mathcal{B}$
shift operators [5, 7],

$\displaystyle\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})=B(x;N,\boldsymbol{\lambda})(1-e^{\partial})+D(x;N,\boldsymbol{\lambda})(1-e^{-\partial})=\mathcal{B}(N,\boldsymbol{\lambda})\mathcal{F}(N,\boldsymbol{\lambda}),$
(4.81)

which is an expression of the shape invariance. For example, for the Racah,

$\displaystyle\text{Racah}:\ \
\mathcal{F}(N,\boldsymbol{\lambda})=\frac{Nbc}{2x+d+1}(1-e^{\partial}),$
$\displaystyle\phantom{\text{Racah}:\ \ }\
\mathcal{B}(N,\boldsymbol{\lambda})=\frac{1}{Nbc}\bigl{(}B(x;N,\boldsymbol{\lambda})-D(x;N,\boldsymbol{\lambda})e^{-\partial}\bigr{)}(2x+d+1),$
(4.82)
$\displaystyle\mathcal{F}(N,\boldsymbol{\lambda})\check{P}_{n}(x;N,b,c,d)=\mathcal{E}(n;\boldsymbol{\lambda})\check{P}_{n-1}(x;N-1,b+1,c+1,d+1),$
$\displaystyle\mathcal{B}(N,\boldsymbol{\lambda})\check{P}_{n-1}(x;N-1,b+1,c+1,d+1)=\check{P}_{n}(x;N,b,c,d).$
(4.83)

These $\mathcal{F}$ and $\mathcal{B}$ change the degree $n$ but keep $x$
unchanged, in contrast with $\tilde{\mathcal{F}}$, which changes $x$ and keeps
the degree $n$ unchanged. In a similar fashion as above (4.81), reflecting the
new type of shape invariance, $\widetilde{\mathcal{H}}-\mathcal{E}(N+1)$ is
factorised by the forward and backward $x$-shift operators,

$\widetilde{\mathcal{H}}(N,\boldsymbol{\lambda})-\mathcal{E}(N+1;\boldsymbol{\lambda})=-\tilde{\mathcal{B}}(x,N,\boldsymbol{\lambda})\tilde{\mathcal{F}}(x,N,\boldsymbol{\lambda}),$
(4.84)

as the zero norm solutions start at degree $N+1$. The explicit forms of the
operator $\tilde{\mathcal{B}}$ are

$\displaystyle\text{K}:$ $\displaystyle\tilde{\mathcal{B}}(x,N,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(N+1)\bigl{(}p+(1-p)e^{-\partial}\bigr{)},$
(4.85) $\displaystyle\text{H}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,\boldsymbol{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(N+1)\bigl{(}x+a+(b+N-x)e^{-\partial}\bigr{)},$
(4.86) $\displaystyle\text{R}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,\boldsymbol{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(N+1)\Bigl{(}\frac{(x+b)(x+c)}{2x+d}+\frac{(b-d-x)(x+d-c)}{2x+d}e^{-\partial}\Bigr{)},$
(4.87) $\displaystyle\text{dH}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,\boldsymbol{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(N+1)\Bigl{(}\frac{x+a}{2x-1+a+b}+\frac{x+b-1}{2x-1+a+b}e^{-\partial}\Bigr{)},$
(4.88) $\displaystyle\text{dq$q$K}:$ $\displaystyle\tilde{\mathcal{B}}(x,N,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\bigl{(}p^{-1}q^{-x-N-1}+q^{-N-1}(1-p^{-1}q^{-x})e^{-\partial}\bigr{)},$
(4.89) $\displaystyle\text{$q$H}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,\boldsymbol{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\bigl{(}q^{-N-1}(1-aq^{x})+aq^{-1}(q^{x-N}-b)e^{-\partial}\bigr{)},$
(4.90) $\displaystyle\text{$q$K}:$ $\displaystyle\tilde{\mathcal{B}}(x,N,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\bigl{(}q^{-N-1}+p\,e^{-\partial}\bigr{)},$
(4.91) $\displaystyle\text{q$q$K}:$ $\displaystyle\tilde{\mathcal{B}}(x,N,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\bigl{(}p^{-1}q^{x-N-1}+(1-p^{-1}q^{x-N-1})e^{-\partial}\bigr{)},$
(4.92) $\displaystyle\text{a$q$K}:$ $\displaystyle\tilde{\mathcal{B}}(x,N,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\bigl{(}q^{-N-1}(1-pq^{x+1})+pq^{x-N}e^{-\partial}\bigr{)},$
(4.93) $\displaystyle\text{$q$R}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,\boldsymbol{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\Bigl{(}\frac{q^{-N-1}(1-bq^{x})(1-cq^{x})}{1-dq^{2x}}$
$\displaystyle\phantom{\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\Bigl{(}}+\tilde{d}\,\frac{(b^{-1}dq^{x}-1)(1-c^{-1}dq^{x})}{1-dq^{2x}}e^{-\partial}\Bigr{)},$
(4.94) $\displaystyle\text{d$q$H}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,\boldsymbol{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\Bigl{(}\frac{q^{-N-1}(1-aq^{x})}{1-abq^{2x-1}}+\frac{aq^{x-N-1}(1-bq^{x-1})}{1-abq^{2x-1}}e^{-\partial}\Bigr{)},$
(4.95) $\displaystyle\text{d$q$K}:$ $\displaystyle\tilde{\mathcal{B}}(x,N,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(1-q^{N+1})\Bigl{(}\frac{q^{-N-1}}{1+pq^{2x}}+\frac{pq^{2x-N-1}}{1+pq^{2x}}e^{-\partial}\Bigr{)}.$
(4.96)

The action of these operators on the polynomials is as follows:

$\displaystyle\text{(i),\,(iv)}:$ $\displaystyle\ \
\tilde{\mathcal{B}}(x,N)\check{P}_{n}(x+1;N+1,\boldsymbol{\lambda})$
$\displaystyle=\bigl{(}\mathcal{E}(N+1;\boldsymbol{\lambda})-\mathcal{E}(n;\boldsymbol{\lambda})\bigr{)}\check{P}_{n}(x;N,\boldsymbol{\lambda}),$
(4.97) $\displaystyle\text{(ii)}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,d,\bar{\boldsymbol{\lambda}})\check{P}_{n}(x+1;N+1,d-1,\bar{\boldsymbol{\lambda}})$
$\displaystyle=\bigl{(}\mathcal{E}(N+1;\boldsymbol{\lambda})-\mathcal{E}(n;\boldsymbol{\lambda})\bigr{)}\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}}),\\!\\!\\!$
(4.98) $\displaystyle\text{(iii)}:$ $\displaystyle\ \
\tilde{\mathcal{B}}(x,N,p)\check{P}_{n}(x+1;N+1,pq^{-1})$
$\displaystyle=\bigl{(}\mathcal{E}(N+1)-\mathcal{E}(n)\bigr{)}\check{P}_{n}(x;N,p),$
(4.99) $\displaystyle\text{(v)}:$
$\displaystyle\tilde{\mathcal{B}}(x,N,d,\bar{\boldsymbol{\lambda}})\check{P}_{n}(x+1;N+1,dq^{-1},\bar{\boldsymbol{\lambda}})$
$\displaystyle=\bigl{(}\mathcal{E}(N+1;\boldsymbol{\lambda})-\mathcal{E}(n;\boldsymbol{\lambda})\bigr{)}\check{P}_{n}(x;N,d,\bar{\boldsymbol{\lambda}}).\\!\\!\\!$
(4.100)

##  5 Summary and Comments

The reported “Diophantine” and factorisation properties of the Jacobi,
Laguerre etc [1], Wilson and Askey-Wilson polynomials etc [2] are very
interesting but rather puzzling, as these properties require the parameter
ranges in which the orthogonality does not hold. In this paper we show, after
the seminal work of [9], that these properties are shared by all the finite
polynomials in the Askey scheme in the conventional orthogonality parameter
ranges. All the monic higher degree polynomials $\\{\check{P}_{N+1+m}(x)\\}$
($m\in\mathbb{Z}_{\geq 0}$) in this group show the “Diophantine” and
factorisation properties Theorem 2.1 (2.48), (2.49). Here $N$ is the maximal
degree of the corresponding non-monic orthogonal polynomials. A simple and
intuitive explanation is that these higher degree monic polynomials are the
zero-norm solution of certain tri-diagonal real symmetric matrix $\mathcal{H}$
(2.10). The explicit expressions of the “Diophantine” and factorisation for
the twelve polynomials are presented in § 2.3. These higher degree monic
polynomials can be used as seed solutions for generating new types multi-
indexed polynomials based on the twelve finite orthogonal polynomials. In
order to obtain genuine multi-indexed orthogonal polynomials, correct choices
of the seed solutions and the parameter ranges are essential. A simplest
choice of $M$ contiguous lowest degree $\\{\check{P}_{N+1+m}(x)\\}$
($m=0,1,\ldots,M-1$) generate proper multi-indexed orthogonal polynomials as
shown in § 4 for each of the twelve polynomials listed in § 2.2.

In paper I [1] Calogero and coauthors reported “Diophantine” and factorisation
properties of some finite polynomials, the Racah, Hahn, dual Hahn and
Krawtchouk. In these cases, the phenomena occur only at the particular degree
with which the parameters are tuned. In ‘Remarks’ of paper II [2], Chen and
Ismail mentioned a possible explanation of the “Diophantine” and factorisation
properties of the Wilson polynomial as the occurrence of certain discrete
masses in the orthogonality measure when some of the parameters are negative.
It is a good challenge to provide a perspective in which their explanation and
ours could be unified, since the Wilson and Askey-Wilson polynomials are the
most general ones in the Askey scheme.

A few remarks on the results of Miki-Tsujimoto-Vinet paper [9]. They reported
four types of single-indexed exceptional Krawtchouk polynomials. Among the
them, the first one, using the polynomial itself as the seed solution, is the
well-known one called Krein-Adler polynomials, as briefly mentioned in § 2.4.
The second type is the $M=1$ case of the multi-indexed polynomials discussed
in § 3. The remaining two types exist only for the Krawtchouk polynomials.
They are obtained from the first and second types by using the mirror symmetry
(2.15) of the Krawtchouk. Another polynomial in the same $\eta(x)=x$ family,
the Hahn polynomial, has similar mirror symmetry (2.18). The forms of the
other $\eta(x)$, family (ii)–(v), are not compatible with the mirror symmetry.

## Acknowledgements

S. O. is supported by JSPS KAKENHI Grant Number JP19K03667.

## References

  * [1] M. Bruschi, F. Calogero and R. Droghei, “Additional recursion relations, factorizations, and Diophantine properties associated with the polynomials of the Askey scheme,” Adv. Math. Phys. 2009 (2009) 268134 (43pp).
  * [2] Y. Chen and M. Ismail, “Hypergeometric origins of Diophantine properties associated with the Askey scheme,” Proceedings of the American Mathematical Society 138 (2010) 943-951.
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