# Bispectrality and biorthogonality of the rational functions of $q$-Hahn type

Ismaël Bussière<EMAIL_ADDRESS>Julien
Gaboriaud<EMAIL_ADDRESS>Luc Vinet<EMAIL_ADDRESS>Alexei Zhedanov<EMAIL_ADDRESS>
###### Abstract

We introduce families of rational functions that are biorthogonal with respect
to the $q$-hypergeometric distribution. A triplet of $q$-difference operators
$X$, $Y$, $Z$ is shown to play a role analogous to the pair of bispectral
operators of orthogonal polynomials. The recurrence relation and difference
equation take the form of generalized eigenvalue problems involving the three
operators. The algebra generated by $X$, $Y$, $Z$ is akin to the algebras of
Askey–Wilson type in the case of orthogonal polynomials. The actions of these
operators in three different basis are presented. Connections with Wilson’s
${}_{10}\phi_{9}$ biorthogonal rational functions are also discussed.
Keywords: Biorthogonal rational functions, bispectrality, $q$-Hahn
polynomials, generalized eigenvalue problems.

##  1 Introduction

The Askey scheme of orthogonal polynomials [1] presents the classical basic
hypergeometric orthogonal polynomials and their $q\to 1$ limits in a
hierarchical way, with the most general polynomials (the Askey–Wilson and
Wilson polynomials [2]) sitting at the top. These families of polynomials
possess various properties of interest such as a three term recurrence
relation, a differential/difference equation, a Rodriguez formula, etc. They
also have interpretations and applications in algebra and group theory as
matrix elements and overlaps in particular. The algebraic study of their
bispectrality through the algebras of Askey–Wilson type [3] has shed
remarkable light on these orthogonal polynomials (OP) and led to connections
with various fields.

Put simply, the bispectrality of polynomials [4] corresponds to the fact that
they obey (i) a three term recurrence relation and (ii) a
differential/difference equation that amount to eigenvalue equations for two
operators: (i) one that acts on the degree $n$, with eigenvalue depending on
the variable $x$, and (ii) another one acting on the variable $x$ with
eigenvalue involving the degree $n$.

It is quite natural to look for extensions of the $q$-Askey scheme. One
possible direction is the multivariable generalization of the various families
of the $q$-Askey scheme (see for example [5, 6]). This paper looks at another
direction. We are interested in biorthogonal rational functions (BRF) that are
bispectral with the aim of enlarging systematically the Askey scheme to BRF.

The biorthogonal rational functions to be considered will be in correspondence
with the classical hypergeometric orthogonal polynomials. Their
biorthogonality relations will involve the same weights as those for which the
polynomials of the Askey scheme are orthogonal. The key feature in the case of
BRF is that bispectrality will arise through generalized eigenvalue problems
(GEVP) involving three operators.

The simplest case of BRF of the ${}_{3}F_{2}$ type was studied in [7]. These
functions which possess the same orthogonality weight as the Hahn polynomials
were called “BRF of the Hahn type”. Here we shall focus on the
$q$-generalization of the Hahn case. We will obtain and characterize a family
of BRF that are orthogonal with respect to the $q$-hypergeometric distribution
and shall call this family the “BRF of $q$-Hahn type”.

The study of BRF is not new [8] and various connections with Padé approximants
[9, 10], continued fractions [11, 12], Gram determinants [13] and GEVP [14,
15] have been established before. Pioneering work was done by Wilson in [13],
where he identified BRF expressed as ${}_{10}\phi_{9}$ basic hypergeometric
series. These functions are very general and possess a large number of
parameters. One can recover from them the full $q$-Askey scheme, its $q\to\pm
1$ limits and the associated BRF through appropriate limits and contractions
[16]. As will be shown below, both the rational functions of Hahn and $q$-Hahn
type can indeed be obtained from Wilson’s BRF through appropriate limits.

As is well appreciated already, it is most useful to have detailed explicit
characterizations for the various families that descend through limits or
specializations from say the Askey–Wilson polynomials. This view is adopted
here as we study the rational functions of ${}_{3}\phi_{2}$ type with an eye
to working our way up to those of type ${}_{4}F_{3}$ and ${}_{4}\phi_{3}$. Let
us stress moreover that our focus is on the bispectral properties as well as
their algebraic desription.

Here is the outline for the rest of the paper. In Section 2, the three
operators $X$, $Y$, $Z$ that entail the desired bispectrality will be
introduced. The solutions $U_{n}$ of the GEVP $YU_{n}=\lambda_{n}XU_{n}$ are
derived in Section 3 and identified as the BRF of $q$-Hahn type. The
biorthgonality of these functions is the object of Section 4 and the precise
connection between the BRF of $q$-Hahn/Hahn type and Wilson’s
${}_{10}\phi_{9}$ BRF is detailed in Section 5. The tridiagonal actions of
$X$, $Y$, $Z$ on the basis of these functions is presented in Section 6. How
the bispectrality of these rational functions is encoded in two GEVP is
explained in Section 7 and the bispectrality algebra realized by the three
operators is examined in Section 8. Final remarks conclude the paper.

##  2 A triplet of q-difference operators

In this paper we assume $q$ to be generic and we use the following notation
for $q$-numbers:

$\displaystyle[x]_{q}=\frac{1-q^{x}}{1-q}.$ (2.1)

It is directly seen that in the limit $q\to 1$, the $q$-number $[x]_{q}$ goes
to the ordinary number $x$.

Let us now introduce the following three $q$-difference operators that will
play a central role in the paper:

$\displaystyle X^{(\alpha,\beta)}$
$\displaystyle=[x-\alpha]_{q}\mathds{1}-q^{-\alpha}[x]_{q}T^{-},$ (2.2)
$\displaystyle Y^{(\alpha,\beta)}$
$\displaystyle=A_{1}(x)T^{+}+A_{2}(x)T^{-}+A_{0}(x)\mathds{1},$ (2.3)
$\displaystyle Z^{(\alpha,\beta)}$
$\displaystyle=\frac{[-x]_{q}}{[\alpha-x]_{q}}T^{-}-\mathds{1},$ (2.4)

where

$\displaystyle\begin{aligned}
A_{1}(x)&=q^{\beta-x}[x-N]_{q}[x+1-\alpha]_{q}[x-\alpha]_{q},\\\
A_{2}(x)&=q^{-x}[x]_{q}[x-N-\alpha+\beta]_{q}[x-\alpha]_{q},\\\
A_{0}(x)&=-(A_{1}(x)+A_{2}(x)).\end{aligned}$ (2.5)

Note that

$\displaystyle Z^{(\alpha,\beta)}=-\frac{1}{[x-\alpha]_{q}}X^{(\alpha,\beta)}$
(2.6)

and that $Y^{(\alpha,\beta)}$ has $[x-\alpha]_{q}$ as a global factor. We use
the notation

$\displaystyle T^{\pm}f(x)=f(x\pm 1),\qquad\mathds{1}f(x)=f(x)$ (2.7)

to denote the shift operators $T^{\pm}$ and the identity operator
$\mathds{1}$.

The three operators $X^{(\alpha,\beta)}$, $Y^{(\alpha,\beta)}$,
$Z^{(\alpha,\beta)}$ depend on the parameters $\alpha$ and $\beta$. In the
remainder of the paper, we will generally omit this dependency and write $X$,
$Y$, $Z$.

These operators act on the space $M$ of real functions $f(x)$ defined on the
linear grid $x=0,\pm 1,\pm 2,\dots$; that is
$M=\\{f:\mathbb{Z}\to\mathbb{R}\\}$. In what follows we will restrict $M$ to
be the space $M_{N}$ of functions defined on the finite set of points
$x=0,1,\dots,N$. $M_{N}$ is hence of dimension $N+1$ and one has the following
standard basis on this space:

$\displaystyle e_{k}(x)=\delta_{k,x},\qquad k,x\in\\{0,1,2,\dots,N\\}.$ (2.8)

In this basis, the operators $X$, $Y$, $Z$ act as matrices of size
$(N+1)\times(N+1)$. More precisely, the matrices for $X$ and $Z$ are lower bi-
diagonal while the matrix for $Y$ is tridiagonal:

$\displaystyle X$
$\displaystyle=\left[\begin{matrix}[-\alpha]_{q}&0&0&\dots&0\\\
-q^{-\alpha}[1]_{q}&[1-\alpha]_{q}&0&&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&&-q^{-\alpha}[N-1]_{q}&[N-1-\alpha]_{q}&0\\\
0&\dots&0&-q^{-\alpha}[N]_{q}&[N-\alpha]_{q}\end{matrix}\right],$ (2.9)
$\displaystyle Y$
$\displaystyle=\left[\begin{matrix}A_{0}(0)&A_{1}(0)&0&\dots&0\\\
A_{2}(1)&A_{0}(1)&A_{1}(1)&&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&&A_{2}(N-1)&A_{0}(N-1)&A_{1}(N-1)\\\
0&\dots&0&A_{2}(N)&A_{0}(N)\end{matrix}\right],$ (2.10) $\displaystyle Z$
$\displaystyle=\left[\begin{matrix}-1&0&0&\dots&0\\\
\tfrac{[-1]_{q}}{[\alpha-1]_{q}}&-1&0&&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&&\tfrac{[1-N]_{q}}{[\alpha-N+1]_{q}}&-1&0\\\
0&\dots&0&\tfrac{[-N]_{q}}{[\alpha-N]_{q}}&-1\end{matrix}\right].$ (2.11)

##  3 The rational functions of the ${}_{3}\phi_{2}$ type as solutions of a
generalized eigenvalue problem

In this section we will solve the GEVP built from the pair of operators $X$
and $Y$

$\displaystyle YU=\lambda XU.$ (3.1)

Let us introduce the notation for the basic hypergeometric series
${}_{r+1}\phi_{r}$:

$\displaystyle{}_{r+1}\phi_{r}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a_{0}\mskip
8.0mua_{1}\mskip 8.0mu\dots\mskip 8.0mua_{r}}{b_{1}\mskip 8.0mu\dots\mskip
8.0mub_{r}};{q},{z}\biggr{)}=\sum_{k=0}^{\infty}\frac{(a_{0};q)_{k}(a_{1};q)_{k}\dots(a_{r};q)_{k}}{(q;q)_{k}(b_{1};q)_{k}\dots(b_{r};q)_{k}}z^{k},$
(3.2)

where $(a;q)_{k}=(1-a)(1-qa)\dots(1-q^{k-1}a)$ is the $q$-Pochammer symbol.
The main results are summarized in the following statement:

######  Proposition 1.

The normalized solutions $U_{n}$ of the GEVP

$\displaystyle YU_{n}=\lambda_{n}XU_{n}$ (3.3)

are given by

$\displaystyle
U_{n}(x;\alpha,\beta,N;q)=\frac{(q^{-N};q)_{n}}{(q^{-n-\beta};q)_{n}}{}_{3}\phi_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-n}\mskip
8.0muq^{n+\beta-N}\mskip 8.0muq^{-x}}{q^{-N}\mskip
8.0muq^{\alpha-x}};{q},{q^{\alpha-\beta}}\biggr{)},$ (3.4)

with the spectrum $\lambda_{n}$ given by

$\displaystyle\lambda_{n}=[-n]_{q}[n+\beta-N]_{q}.$ (3.5)

The functions $U_{n}$ will be called “biorthogonal rational functions of the
$q$-Hahn type”.

###### Proof.

With hindsight, we introduce the following basis on the space $M_{N}$ of
functions:

$\displaystyle\varphi_{n}(x;\alpha;q)=\frac{(q^{-x};q)_{n}}{(q^{\alpha-x};q)_{n}},\qquad
n=0,1,\dots,N.$ (3.6)

Enforcing the restriction to $M_{N}$, it is verified that the operators $X$,
$Y$, $Z$ act in a tridiagonal fashion in this basis:

$\displaystyle X\,\varphi_{n}(x;\alpha;q)$
$\displaystyle=-[n]_{q}\varphi_{n+1}(x;\alpha;q)+[n-\alpha]_{q}\varphi_{n}(x;\alpha;q),$
(3.7) $\displaystyle Y\,\varphi_{n}(x;\alpha;q)$
$\displaystyle=\nu_{n}^{(1)}\varphi_{n+1}(x;\alpha;q)+\nu_{n}^{(2)}\varphi_{n}(x;\alpha;q)+\nu_{n}^{(3)}\varphi_{n-1}(x;\alpha;q),$
(3.8) $\displaystyle Z\,\varphi_{n}(x;\alpha;q)$
$\displaystyle=\varphi_{n+1}(x;\alpha;q)-\varphi_{n}(x;\alpha;q),$ (3.9)

where

$\displaystyle\nu_{n}^{(1)}$ $\displaystyle=-[-n]_{q}[n]_{q}[n+\beta-N]_{q},$
(3.10) $\displaystyle\nu_{n}^{(2)}$
$\displaystyle=[-n]_{q}[n+\beta-N]_{q}[n-\alpha]_{q}+q^{\beta-\alpha}[n]_{q}[n-N-1]_{q}[1-n]_{q},$
(3.11) $\displaystyle\nu_{n}^{(3)}$
$\displaystyle=-q^{\beta-2\alpha}[n]_{q}[n-N-1]_{q}[1+\alpha-n]_{q},$ (3.12)

and the boundary specializations are given by:

$\displaystyle X\,\varphi_{N}(x;\alpha;q)$
$\displaystyle=[N-\alpha]_{q}\varphi_{N}(x;\alpha;q),$ (3.13) $\displaystyle
Y\,\varphi_{0}(x;\alpha;q)$
$\displaystyle=\nu_{0}^{(1)}\varphi_{1}(x;\alpha;q)+\nu_{0}^{(2)}\varphi_{0}(x;\alpha;q),$
(3.14) $\displaystyle Y\,\varphi_{N}(x;\alpha;q)$
$\displaystyle=\nu_{N}^{(2)}\varphi_{N}(x;\alpha;q)+\nu_{N}^{(3)}\varphi_{N-1}(x;\alpha;q),$
(3.15) $\displaystyle Z\,\varphi_{N}(x;\alpha;q)$
$\displaystyle=-\varphi_{N}(x;\alpha;q).$ (3.16)

Equivalently, the actions of the operators $X$, $Y$, $Z$ on the basis
$\varphi_{n}(x;\alpha;q)$ correspond to the following matrices:

$\displaystyle X$
$\displaystyle=\left[\begin{matrix}[-\alpha]_{q}&0&0&\dots&0\\\
0&[1-\alpha]_{q}&0&&0\\\ 0&-[1]_{q}&[2-\alpha]_{q}&\ddots&\vdots\\\
&\ddots&\ddots&\ddots&0\\\ \vdots&&-[N-2]_{q}&[N-1-\alpha]_{q}&0\\\
0&\dots&0&-[N-1]_{q}&[N-\alpha]_{q}\end{matrix}\right],$ (3.17) $\displaystyle
Y$ $\displaystyle=\left[\begin{matrix}\nu_{0}^{(2)}&\nu_{1}^{(3)}&0&\dots&0\\\
\nu_{0}^{(1)}&\nu_{1}^{(2)}&\nu_{2}^{(3)}&&\vdots\\\
0&\ddots&\ddots&\ddots&0\\\
\vdots&&\nu_{N-2}^{(1)}&\nu_{N-1}^{(2)}&\nu_{N}^{(3)}\\\
0&\dots&0&\nu_{N-1}^{(1)}&\nu_{N}^{(2)}\end{matrix}\right],$ (3.18)
$\displaystyle Z$ $\displaystyle=\left[\begin{matrix}-1&0&0&\dots&0\\\
1&-1&0&&\vdots\\\ 0&\ddots&\ddots&\ddots&0\\\ \vdots&&1&-1&0\\\
0&\dots&0&1&-1\end{matrix}\right].$ (3.19)

Remarkably, in the basis of the functions $\varphi_{n}(x;\alpha;q)$, one
readily observes that the operator $Y$ factorizes as follows:

$\displaystyle Y^{(\alpha,\beta)}=X^{(\alpha,\beta)}V^{(\alpha,\beta)},$
(3.20)

where $V\equiv V^{(\alpha,\beta)}$ acts as a lower bi-diagonal matrix

$\displaystyle V\,\varphi_{0}(x;\alpha;q)$ $\displaystyle=0,$ (3.21)
$\displaystyle V\,\varphi_{n}(x;\alpha;q)$
$\displaystyle=[-n]_{q}[n+\beta-N]_{q}\varphi_{n}(x;\alpha;q)+q^{\beta-\alpha+1-n}[n]_{q}[n-N-1]_{q}\varphi_{n-1}(x;\alpha;q).$
(3.22)

This factorization leads to a remarkable simplification of the GEVP and allows
to find the an explicit basic hypergeometric solution to the GEVP. This goes
as follows: Expand the solutions to the GEVP $U_{n}(x)$ over the set of
functions $\varphi_{k}(x;\alpha;q)$, $k\in\\{0,1,\dots,n\\}$. Owing to the
factorization (3.20), the generalized eigenvalue equation (3.3) can be
converted into the usual eigenvalue equation

$\displaystyle VU_{n}=\lambda_{n}U_{n}.$ (3.23)

In view of (3.22), the eigenvalues $\lambda_{n}$ are identified to be

$\displaystyle\lambda_{n}=[-n]_{q}[n+\beta-N]_{q}.$ (3.24)

One can now expand $U_{n}$ in the basis $\varphi_{k}(x;\alpha;q)$

$\displaystyle U_{n}=\sum_{k=0}^{n}C_{n,k}\,\varphi_{k}(x;\alpha;q)$ (3.25)

and it is readily seen using the action (3.22) of $V$ on the basis functions
$\varphi_{k}(x;\alpha;q)$ that (3.23) implies the following two-term
recurrence relation for the coefficients $C_{n,k}$:

$\displaystyle
q^{\beta-\alpha-k}[k+1]_{q}[k-N]_{q}C_{n,k+1}=\left(-q^{-n}[n]_{q}[n+\beta-N]_{q}+q^{-k}[k]_{q}[k+\beta-N]_{q}\right)C_{n,k}.$
(3.26)

Its solution is found to be

$\displaystyle
C_{n,k}=C_{n,0}\,q^{k(\alpha-\beta)}\frac{(q^{-n};q)_{k}(q^{n+\beta-N};q)_{k}}{(q;q)_{k}(q^{-N};q)_{k}}.$
(3.27)

It then follows from (3.25) that

$\displaystyle
U_{n}(x;\alpha,\beta,N;q)=\frac{(q^{-N};q)_{n}}{(q^{-n-\beta};q)_{n}}{}_{3}\phi_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-n}\mskip
8.0muq^{n+\beta-N}\mskip 8.0muq^{-x}}{q^{-N}\mskip
8.0muq^{\alpha-x}};{q},{q^{\alpha-\beta}}\biggr{)}.$ (3.28)

The normalization constant $C_{n,0}$ has been chosen so that

$\displaystyle\lim_{x\to\infty}U_{n}=1.$ (3.29)

∎

######  Remark 2.

It is manifest that the presence of $q$-Pochhammer symbols
$\left(q^{\alpha-x};q\right)_{n}$ in the denominator implies that the function
has poles when $x\in\\{\alpha,\alpha+1,\dots,\alpha+n-1\\}$. Hence, one could
alternatively express the function $U_{n}(x)$ as

$\displaystyle
U_{n}(x;\alpha,\beta,N;q)=1+\sum_{k=0}^{n-1}\frac{\eta_{n,k}}{[x-\alpha-k]_{q}}$
(3.30)

for some coefficients $\eta_{n,k}$.

Note that Askey and Suslov have also studied ${}_{3}\phi_{2}$ BRF [17] but
they do not appear to be related to the ones discussed here.

The functions $U_{n}(x;\alpha,\beta,N;q)$ will be called “biorthogonal
rational functions of $q$-Hahn type” (see Section 4 for a justification of the
name). From their hypergeometric expression and contiguity relations, it is
seen that the operators $X$, $Y$, $Z$ shift the parameter $\alpha$ by one:

$\displaystyle X\,U_{n}(x;\alpha,\beta,N;q)$
$\displaystyle=[-\alpha]_{q}\,U_{n}(x;\alpha+1,\beta,N;q),$ (3.31)
$\displaystyle Y\,U_{n}(x;\alpha,\beta,N;q)$
$\displaystyle=[-\alpha]_{q}[-n]_{q}[n+\beta-N]_{q}\,U_{n}(x;\alpha+1,\beta,N;q),$
(3.32) $\displaystyle Z\,U_{n}(x;\alpha,\beta,N;q)$
$\displaystyle=-\frac{[-\alpha]_{q}}{[x-\alpha]_{q}}\,U_{n}(x;\alpha+1,\beta,N;q).$
(3.33)

Another way to write this is to introduce the operator
$P:\alpha\mapsto\alpha+1$. Then, the operators $X$, $Y$, $Z$ act as

$\displaystyle X$ $\displaystyle=[-\alpha]_{q}\,P,$ (3.34) $\displaystyle Y$
$\displaystyle=[-\alpha]_{q}[-n]_{q}[n+\beta-N]_{q}\,P,$ (3.35) $\displaystyle
Z$ $\displaystyle=-\frac{[-\alpha]_{q}}{[x-\alpha]_{q}}\,P,$ (3.36)

on the solutions $U_{n}$ of the GEVP. This will prove useful in the
examination of the biorthogonality properties of these rational functions of
Hahn type.

##  4 Biorthogonality

Let us introduce the following $q$-hypergeometric distribution:

$\displaystyle
w_{x}^{(\alpha,\beta)}=q^{(1-\alpha)N}\frac{(q^{\alpha-\beta-1};q)_{N}}{(q^{-\beta};q)_{N}}q^{(\beta+1)x}\frac{(q^{-N};q)_{x}(q^{1-\alpha};q)_{x}}{(q;q)_{x}(q^{\beta-\alpha-N+2})_{x}},\qquad\sum_{x=0}^{N}w_{x}^{(\alpha,\beta)}=1,$
(4.1)

whose normalization is checked with help of the $q$-analog of the
Chu–Vandermonde formula [1]:

$\displaystyle{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-N}\mskip
8.0mua}{c};{q},{\frac{cq^{N}}{a}}\biggr{)}=\frac{(c/a;q)_{N}}{(c;q)_{N}}.$
(4.2)

This weight is the one with respect to which the $q$-Hahn polynomials are
orthogonal [1].

######  Remark 3.

Under the involution $S$

$\displaystyle\begin{aligned} S:\alpha&\mapsto\beta-\alpha+2,\\\ x&\mapsto
N-x,\\\ q&\mapsto q^{-1},\end{aligned}$ (4.3)

the orthogonality weight is unchanged:

$\displaystyle S\,w_{x}^{(\alpha,\beta)}=w_{x}^{(\alpha,\beta)}.$ (4.4)

We use this $q$-hypergeometric weight to define a scalar product on the space
$\mathcal{L}^{(\alpha,\beta)}$ of rational functions with parameters $\alpha$
and $\beta$ and defined on the grid $x=0,1,\dots,N$:

$\displaystyle\left(f(x),g(x)\right)_{(\alpha,\beta)}=\sum_{x=0}^{N}w_{x}^{(\alpha,\beta)}f(x)g(x),\qquad
f,g\in\mathcal{L}^{(\alpha,\beta)}.$ (4.5)

In the following, the parameters $(\alpha,\beta)$ will be omitted from the
scalar products.

Our goal is now to find biorthogonal partners for the functions $U_{n}(x)$,
that is, functions ${\mathcal{U}}_{m}(x)$ orthogonal to $U_{n}(x)$ under the
scalar product (4.5):

$\displaystyle\left(U_{n},{\mathcal{U}}_{m}\right)=\delta_{n,m}H_{n}.$ (4.6)

The essential ingredient is the fact that the $U_{n}$ are solutions to a GEVP.
Given the scalar product, we can define the adjoint GEVP:

$\displaystyle Y^{*}U^{*}_{m}=\lambda_{m}X^{*}U^{*}_{m},$ (4.7)

where the asterisk denotes the adjoint of operators and $U^{*}_{m}$ denotes
the solution of the adjoint GEVP.

We know that [14]:

######  Proposition 4.

The biorthogonal partner of $U_{n}$ is given by
${\mathcal{U}}_{m}=X^{*}U^{*}_{m}$.

###### Proof.

Consider the scalar product $(YU_{n},U^{*}_{m})$. By definition of the adjoint
operators, one has

$\displaystyle(YU_{n},U^{*}_{m})=(U_{n},Y^{*}U^{*}_{m}).$ (4.8)

Using the GEVP (3.3) and the adjoint GEVP (4.7) leads to

$\displaystyle\lambda_{n}(XU_{n},U^{*}_{m})=\lambda_{m}(U_{n},X^{*}U^{*}_{m}),$
(4.9)

from where we have

$\displaystyle(\lambda_{n}-\lambda_{m})(U_{n},X^{*}U^{*}_{m})=0.$ (4.10)

In view of formula (3.24) for $\lambda_{n}$, it is clear that first factor is
zero only if $n=m$. Hence the second factor must be zero for $n\neq m$ and
this shows the biorthogonality between $U_{n}$ and
${\mathcal{U}}_{m}=X^{*}U^{*}_{m}$. ∎

######  Proposition 5.

Up to a normalization constant $c_{\alpha,\beta}$, the biorthogonal partners
${\mathcal{U}}_{m}(x;\alpha,\beta,N;q)$ of $U_{n}(x;\alpha,\beta,N;q)$ are
given by

$\displaystyle{\mathcal{U}}_{m}(x;\alpha,\beta,N;q)=c_{\alpha,\beta}\,U_{m}(N-x;\beta-\alpha+2,\beta,N;q^{-1}).$
(4.11)

###### Proof.

We shall obtain ${\mathcal{U}}_{m}=X^{*}U_{m}^{*}$ using the fact that $U_{m}$
satisfies the GEVP $YU_{m}=\lambda_{m}XU_{m}$ described in Proposition 1 and
the properties of the adjoints of $X$ and $Y$. The adjoints of the operators
involved are computed using the expression for the weight (4.1) and the
definition of the scalar product (4.5). A straightforward calculation yields

$\displaystyle(T^{-})^{*}$
$\displaystyle=q^{\beta+1}\frac{[x-N]_{q}[x-\alpha+1]_{q}}{[x+1]_{q}[x-N+\beta-\alpha+2]_{q}}T^{+},$
(4.12) $\displaystyle(T^{+})^{*}$
$\displaystyle=q^{-\beta-1}\frac{[x]_{q}[x-N+\beta-\alpha+1]_{q}}{[x-N-1]_{q}[x-\alpha]_{q}}T^{-}.$
(4.13)

It follows that the adjoints of $X$, $Y$, $Z$ are given by

$\displaystyle X^{*}$
$\displaystyle=[x-\alpha]_{q}\mathds{1}-q^{\beta-\alpha+1}\frac{[x-N]_{q}[x-\alpha+1]_{q}}{[x-N+\beta-\alpha+2]_{q}}T^{+},$
(4.14) $\displaystyle Z^{*}$
$\displaystyle=q^{\beta-\alpha+1}\frac{[x-N]_{q}}{[x-N+\beta-\alpha+2]_{q}}T^{+}-\mathds{1},$
(4.15) $\displaystyle Y^{*}$
$\displaystyle=q^{-\beta-1}\frac{[x]_{q}[x-N+\beta-\alpha+1]_{q}}{[x-N-1]_{q}[x-\alpha]_{q}}A_{1}(x-1)T^{-}$
$\displaystyle+q^{\beta+1}\frac{[x-N]_{q}[x-\alpha+1]_{q}}{[x+1]_{q}[x-N+\beta-\alpha+2]_{q}}A_{2}(x+1)T^{+}$
$\displaystyle+A_{0}(x)\mathds{1},$ (4.16)

with $A_{1}(x)$, $A_{2}(x)$, $A_{0}(x)$ given by (2.5).

We now introduce the transformation $\tau$

$\displaystyle\begin{aligned} \tau:\,&\alpha\mapsto\beta-\alpha+1,\\\
&x\mapsto N-x,\\\ &q\mapsto q^{-1},\\\ &T^{\pm}\mapsto T^{\mp}.\end{aligned}$
(4.17)

to define the operators

$\displaystyle\tilde{X}=\tau(X),\qquad\tilde{Y}=\tau(Y).$ (4.18)

Note that $\tau$ was chosen in order to reproduce the action of $P\circ S$ on
the parameters $\alpha$, $x$, $q$.

A simple calculation shows that

$\displaystyle\tilde{X}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}f(x)\right)$
$\displaystyle=-qX^{*}\left(f(x)\right),$ (4.19)
$\displaystyle\tilde{Y}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}f(x)\right)$
$\displaystyle=-q^{N-\beta+3}Y^{*}\left(f(x)\right).$ (4.20)

Taking the transformation $\tau$ on both sides of (3.31), we get the following
action of $\tilde{X}$:

$\displaystyle\tilde{X}U_{n}(N-x;\beta-\alpha+1,\beta,N;q^{-1})=[\alpha-\beta-1]_{q}U_{n}(N-x;\beta-\alpha+2,\beta,N;q^{-1}).$
(4.21)

Using (4.19) and (4.20), we can also rewrite the adjoint GEVP (4.7) in terms
of the transformed operators $\tilde{X}$ and $\tilde{Y}$:

$\displaystyle-q^{-N+\beta-3}\tilde{Y}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U_{m}^{*}\right)=-q^{-1}\lambda_{m}\,\tilde{X}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U^{*}_{m}\right),$
(4.22)

or equivalently

$\displaystyle\tilde{Y}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U^{*}_{m}\right)=q^{N-\beta+2}\lambda_{m}\,\tilde{X}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U^{*}_{m}\right).$
(4.23)

We readily find that

$\displaystyle\tau(\lambda_{m})=\tau([-m]_{q}[m+\beta-N]_{q})=[-m]_{q^{-1}}[m+\beta-N]_{q^{-1}}=q^{N-\beta+2}\lambda_{m}$
(4.24)

and from (4.23) and definition (4.18), we get

$\displaystyle\tau(Y)\\!\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U^{*}_{m}\right)=\tau(\lambda_{m})\,\,\tau(X)\\!\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U^{*}_{m}\right).$
(4.25)

Comparing with the GEVP (3.3) $YU_{m}=\lambda_{m}XU_{m}$, we see that

$\displaystyle\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U^{*}_{m}=\tau(U_{n}(x;\alpha,\beta,N;q))=U_{n}(N-x;\beta-\alpha+1,\beta,N;q^{-1}).$
(4.26)

We can now obtain the biorthogonal partner ${\mathcal{U}}_{m}=X^{*}U_{m}^{*}$.
From this last relation and (4.19), we have

$\displaystyle\mathcal{U}_{m}$ $\displaystyle=X^{*}U^{*}_{m}$
$\displaystyle=-q^{-1}\tilde{X}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}U^{*}_{m}\right)$
$\displaystyle=-q^{-1}\tilde{X}\left(\frac{[x-\alpha]_{q}}{[x-N+\beta-\alpha+1]_{q}}\frac{[x-N+\beta-\alpha+1]_{q}}{[x-\alpha]_{q}}U_{m}(N-x;\beta-\alpha+1,\beta,N;q^{-1})\right).$

Finally, using (4.21), we get

$\displaystyle{\mathcal{U}}_{m}(x;\alpha,\beta,N;q)$
$\displaystyle=-q^{-1}[\alpha-\beta-1]_{q}\,U_{m}(N-x;\beta-\alpha+2,\beta,N;q^{-1}).$
(4.27)

∎

######  Remark 6.

Since $Y$ factorizes as $XV$, the adjoint problem (4.7) takes the form
$V^{*}X^{*}U^{*}_{m}=\lambda_{m}X^{*}U^{*}_{m}$. It thus follows that the
biorthogonal partners ${\mathcal{U}}_{m}=X^{*}U_{m}^{*}$ satisfy the
eigenvalue equation

$\displaystyle V^{*}\mathcal{U}_{m}=\lambda_{m}\mathcal{U}_{m}.$ (4.28)

To find the norm $H_{n}$ in the biorthogonality relation (4.6), we will make
use of the results in Wilson’s paper [13] and connect them to ours through an
appropriate limiting procedure. This will be detailed in the next section.

##  5 Connection with Wilson’s ${}_{10}\phi_{9}$ BRF

In his 1991 paper [13], Wilson introduced biorthogonal rational functions
expressed as ${}_{10}\phi_{9}$ basic hypergeometric series. These functions
obey a biorthogonality relation on the $N+1$ points of a discrete grid
$z+z^{-1}$ where $z\in\\{q^{a},q^{a+1},\dots,q^{a+N}\\}$. The biorthogonality
relation can be written as follows ***We have corrected two typos from
Wilson’s formulas: we added a missing $q^{-n}$ and changed $q^{a+1}$ to
$q^{2a+1}$ in the expression of $h_{n}$. These typos were also observed in
[18].:

$\displaystyle\sum_{x=0}^{N}w_{x}(a,b,c,d,e,f)u_{n}(x+a;a,b,c,d,e,f)v_{n}(x+a;a,b,c,d,e,f)=\delta_{nm}h_{n}(a,b,c,d,e,f),$
(5.1)

with

$\displaystyle w_{x}(a,b,c,d,e,f)$
$\displaystyle=q^{x}\frac{(q^{2a};q)_{x}}{(q;q)_{x}}\frac{1-q^{2a+2x}}{1-q^{2a}}$
$\displaystyle\quad\times\frac{(q^{a+b};q)_{x}(q^{a+c};q)_{x}(q^{a+d};q)_{x}(q^{a+e};q)_{x}(q^{a+f};q)_{x}}{(q^{1+a-b};q)_{x}(q^{1+a-c};q)_{x}(q^{1+a-d};q)_{x}(q^{1+a-e};q)_{x}(q^{1+a-f};q)_{x}},$
(5.2) $\displaystyle u_{n}(x+a;a,b,c,d,e,f)$
$\displaystyle={}_{10}\phi_{9}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-n}\mskip
8.0muq^{\frac{a-e}{2}+1}\mskip 8.0mu-q^{\frac{a-e}{2}+1}\mskip
8.0muq^{a-e}\mskip 8.0muq^{1-c-e}\mskip 8.0muq^{1-d-e}\mskip
8.0muq^{1-e-b}\mskip 8.0muq^{n-e-f}\mskip 8.0muq^{2a+z}\mskip
8.0muq^{-z}}{q^{\frac{a-e}{2}}\mskip 8.0mu-q^{\frac{a-e}{2}}\mskip
8.0muq^{a+b}\mskip 8.0muq^{a+c}\mskip 8.0muq^{a+d}\mskip
8.0muq^{1-e+a+n}\mskip 8.0muq^{1-n+a+f}\mskip 8.0muq^{1-e-z-a}\mskip
8.0muq^{1-e+z+a}};{q},{q}\biggr{)},$ (5.3) $\displaystyle
v_{n}(x+a;a,b,c,d,e,f)$ $\displaystyle=u_{n}(x+a;b,a,c,d,f,e),$ (5.4)
$\displaystyle h_{n}(a,b,c,d,e,f)$
$\displaystyle=\frac{(q^{2a+1};q)_{N}(q^{1-c-d};q)_{N}(q^{1-c-e};q)_{N}(q^{1-d-e})_{N}}{(q^{b+f};q)_{N}(q^{1+a-c};q)_{N}(q^{1+a-d};q)_{N}(q^{1+a-e};q)_{N}}$
$\displaystyle\quad\times
q^{-n}(q;q)_{n}\frac{(q^{n-e-f};q)_{n}}{(q^{1-e-f};q)_{2n}}\frac{(q^{c+d};q)_{n}(q^{1-e+c};q)_{n}(q^{1-f+b};q)_{n}}{(q^{a+b};q)_{n}(q^{-b-e};q)_{n}(q^{-a-f};q)_{n}},$
(5.5)

with the constraints

$\displaystyle q^{a+b+c+d+e+f}=q,\qquad q^{a+b}=q^{-N}.$ (5.6)

Note that here we have absorbed normalization factors in the $h_{n}$ so the
functions $u_{n}$ are no longer symmetric in $a$, $b$, $c$, $d$ whereas it is
the case for the functions $r_{n}$ in Wilson’s paper. Moreover, we have
denoted his parameters $z$, $a$, $b$, $c$, $d$, $e$, $f$ by $q^{x}$, $q^{a}$,
$q^{b}$, $q^{c}$, $q^{d}$, $q^{e}$, $q^{f}$ respectively so as to make the
correspondence with our functions more direct.

The limits can now be taken individually for each of the $w_{x}$, $u_{n}$,
$v_{n}$, $h_{n}$.

First, let us account for the constraints (5.6) and replace $b$ and $f$ using

$\displaystyle b=-N-a,\qquad f=N+1-c-d-e.$ (5.7a) Now, we redefine the
remaining parameters as $\displaystyle a+e=1-\alpha,\qquad c+d=1+\beta$ and
replace the parameters $d$ and $e$ in the expressions using $\displaystyle
d=1+\beta-c,\qquad e=1-\alpha-a.$ (5.7b)

If we now take the limit $q^{a}\to\infty$, we recover the biorthogonality
relation for the BRF of $q$-Hahn type. Indeed, under the parametrization
(5.7), we obtain

$\displaystyle\lim_{q^{a}\to\infty}w_{x}(a,b,c,d,e,f)$
$\displaystyle=q^{(1+\beta)x}\frac{(q^{-N};q)_{x}(q^{1-\alpha};q)_{x}}{(q;q)_{x}(q^{\beta-\alpha-N+2};q)_{x}}=\bar{w}_{x}(\alpha,\beta,N;q),$
(5.8a) $\displaystyle\lim_{q^{a}\to\infty}u_{n}(x+a;a,b,c,d,e,f)$
$\displaystyle={}_{3}\phi_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-n}\mskip
8.0muq^{n+\beta-N}\mskip 8.0muq^{-x}}{q^{-N}\mskip
8.0muq^{\alpha-x}};{q},{q^{\alpha-\beta}}\biggr{)}=\bar{u}_{n}(x;\alpha,\beta,N;q),$
(5.8b) $\displaystyle\lim_{q^{a}\to\infty}v_{n}(x+a;a,b,c,d,e,f)$
$\displaystyle={}_{3}\phi_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-n}\mskip
8.0muq^{n+\beta-N}\mskip 8.0muq^{x-N}}{q^{-N}\mskip
8.0muq^{x-N+\beta-\alpha+2}};{q},{q}\biggr{)}=\bar{v}_{n}(x;\alpha,\beta,N;q),$
(5.8c) $\displaystyle\lim_{q^{a}\to\infty}h_{n}(a,b,c,d,e,f)$
$\displaystyle=q^{N(\alpha-1-n)}(q;q)_{n}\frac{(q^{-\beta};q)_{N}}{(q^{\alpha-\beta-1};q)_{N}}\frac{(q^{\beta+1};q)_{n}}{(q^{-N};q)_{n}}\frac{(q^{n+\beta-N};q)_{n}}{(q^{1+\beta-N};q)_{2n}}$
$\displaystyle=\bar{h}_{n}(\alpha,\beta,N;q)$ (5.8d)

and the biorthogonality relation is written

$\displaystyle\sum_{x=0}^{N}\bar{w}_{x}(\alpha,\beta,N;q)\,\bar{u}_{n}(x;\alpha,\beta,N;q)\,\bar{v}_{n}(x;\alpha,\beta,N;q)=\delta_{nm}\,\bar{h}_{n}(\alpha,\beta,N;q).$
(5.9)

Note that

$\displaystyle\bar{v}_{n}(x;\alpha,\beta,N;q)=\bar{u}_{n}(N-x;\beta-\alpha+2,\beta,N;q^{-1}).$
(5.10)

Using (5.8)-(5.9) and comparing with (4.1), (3.28), (4.27), one can now
extract $H_{n}$ in the biorthogonality relation (4.6).

###  5.1 The $q\to 1$ limit

The $q=1$ limiting case of the present paper was previously investigated in
[7]. There, the authors presented the biorthogonality relation of the BRF of
Hahn type†††Note that the proof of the biorthogonality given in [7] is not
correct although the end result is right. The proof in Section 4 of the
present paper can be adapted to the $q\to 1$ case. but they did not include
the normalization. We present it here. It can be obtained straightforwardly as
a $q\to 1$ limit of (5.8)-(5.9). To that end, write $q=e^{h}$ and take the
limit $h\to 0$. This leads to the biorthogonality relation

$\displaystyle\sum_{x=0}^{N}\underline{w}_{x}(\alpha,\beta,N)\,\underline{u}_{n}(x;\alpha,\beta,N)\,\underline{v}_{n}(x;\alpha,\beta,N)=\delta_{nm}\,\underline{h}_{n}(\alpha,\beta,N)$
(5.11)

with

$\displaystyle\lim_{h\to 0}\bar{w}_{x}(\alpha,\beta,N;e^{h})$
$\displaystyle=\frac{(-N)_{x}(1-\alpha)_{x}}{x!(\beta-\alpha-N+2)_{x}}=\underline{w}_{x}(\alpha,\beta,N),$
(5.12a) $\displaystyle\lim_{h\to 0}\bar{u}_{n}(x;\alpha,\beta,N;e^{h})$
$\displaystyle={}_{3}F_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{-n\mskip
8.0mun+\beta-N\mskip 8.0mu-x}{-N\mskip
8.0mu\alpha-x};{1}\biggr{)}=\underline{u}_{n}(x;\alpha,\beta,N),$ (5.12b)
$\displaystyle\lim_{h\to 0}\bar{v}_{n}(x;\alpha,\beta,N;e^{h})$
$\displaystyle={}_{3}F_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{-n\mskip
8.0mun+\beta-N\mskip 8.0mux-N}{-N\mskip
8.0mux-N+\beta-\alpha+2};{1}\biggr{)}=\underline{v}_{n}(x;\alpha,\beta,N),$
(5.12c) $\displaystyle\lim_{h\to 0}\bar{h}_{n}(\alpha,\beta,N;e^{h})$
$\displaystyle=n!\frac{(-\beta)_{N}}{(\alpha-\beta-1)_{N}}\frac{(\beta+1)_{n}}{(-N)_{n}}\frac{(n+\beta-N)_{n}}{(1+\beta-N)_{2n}}=\underline{h}_{n}(\alpha,\beta,N).$
(5.12d)

######  Remark 7.

We note that Wilson also provided $q=1$ analogues of his ${}_{10}\phi_{9}$ BRF
in [13], expressed as truncating hypergeometric series of ${}_{9}F_{8}$ type.
It is also possible to recover (5.11)-(5.12) starting instead from these
functions, taking the parametrization in (5.7) and then taking the limit
$a\to\infty$.

##  6 Tridiagonal actions of the triplet $X$, $Y$, $Z$

We have already shown how the operators $X$, $Y$, $Z$ act as matrices on the
basis $e_{n}(x)$ defined on the finite uniform grid $x=0,1,\dots,N$. The
operator $Y$ is tridiagonal, while $X$ and $Z$ are lower bi-diagonal (the
upper off-diagonal is zero).

Alternatively, in the basis $\varphi_{n}(x;\alpha;q)$ the operator $Y$ is
again a tridiagonal matrix while $X$ and $Z$ are upper bi-diagonal.

Now consider the action of these operators on the rational functions
$U_{n}(x;\alpha,\beta,N;q)$. The functions $U_{n}$ form another basis on the
finite-dimensional space $M_{N}$. The actions of $X$, $Y$, $Z$ on this basis
are given as follows:

$\displaystyle X\,U_{n}(x;\alpha,\beta,N;q)$
$\displaystyle=\mu_{n}^{(1)}U_{n+1}(x;\alpha,\beta,N;q)+\mu_{n}^{(2)}U_{n}(x;\alpha,\beta,N;q)+\mu_{n}^{(3)}U_{n-1}(x;\alpha,\beta,N;q),$
(6.1a) $\displaystyle Y\,U_{n}(x;\alpha,\beta,N;q)$
$\displaystyle=\mu_{n}^{(4)}U_{n+1}(x;\alpha,\beta,N;q)+\mu_{n}^{(5)}U_{n}(x;\alpha,\beta,N;q)+\mu_{n}^{(6)}U_{n-1}(x;\alpha,\beta,N;q),$
(6.1b) $\displaystyle Z\,U_{n}(x;\alpha,\beta,N;q)$
$\displaystyle=\mu_{n}^{(7)}U_{n+1}(x;\alpha,\beta,N;q)+\mu_{n}^{(8)}U_{n}(x;\alpha,\beta,N;q)+\mu_{n}^{(9)}U_{n-1}(x;\alpha,\beta,N;q),$
(6.1c) where $\displaystyle\begin{aligned}
\mu_{n}^{(1)}&=-q^{-\alpha-n}\frac{[n]_{q}[\beta+n+1]_{q}[N-\beta-n]_{q}}{[N-\beta-2n]_{q}[2n+1+\beta-N]_{q}},\\\
\mu_{n}^{(2)}&=q^{-\alpha}\left(-[\alpha]_{q}+[-n]_{q}+\frac{[n]_{q}[1-n]_{q}[\beta+n]_{q}}{[2n-1+\beta-N]_{q}}-\frac{[-n]_{q}[n+1]_{q}[\beta+n+1]_{q}}{[2n+1+\beta-N]_{q}}\right),\\\
\mu_{n}^{(3)}&=-q^{-\alpha}\frac{[-n]_{q}[N-\beta-n]_{q}[N-n+1]_{q}}{[N-\beta-2n]_{q}[N-\beta-2n+1]_{q}},\\\
\mu_{n}^{(7)}&=q^{-\alpha-n}\frac{[\beta+n+1]_{q}[N-\beta-n]_{q}}{[N-\beta-2n]_{q}[2n+1+\beta-N]_{q}},\\\
\mu_{n}^{(8)}&=q^{-\alpha+1}\left(\frac{[-n-1]_{q}[N-n]_{q}}{[2n+1+\beta-N]_{q}}-\frac{[-n]_{q}[N-n+1]_{q}}{[2n-1+\beta-N]_{q}}\right)\\\
&\quad+q^{-\alpha-n}\left((1-q^{N+1})-q^{N-n}(1-q)\right)-1,\\\
\mu_{n}^{(9)}&=q^{-\alpha}\frac{[-n]_{q}[N-n+1]_{q}}{[N-\beta-2n]_{q}[N-\beta-2n+1]_{q}}.\end{aligned}$
(6.1d) As for the coefficients $\mu_{n}^{(4)}$, $\mu_{n}^{(5)}$,
$\mu_{n}^{(6)}$, they are proportional to the coefficients $\mu_{n}^{(1)}$,
$\mu_{n}^{(2)}$, $\mu_{n}^{(3)}$
$\displaystyle\mu_{n}^{(3+i)}=[-n]_{q}[n+\beta-N]_{q}\,\mu_{n}^{(i)},\qquad
i\in\\{1,2,3\\}.$ (6.1e) The formulas above are valid for $n=0,\dots,N$ except
for $\ell\in\\{1,4,7\\}$, for which the expressions for $n=N$ are given by
$\displaystyle\mu_{N}^{(1)}=\mu_{N}^{(4)}=\mu_{N}^{(7)}=0.$ (6.1f)

##  7 Bispectrality

Collecting our observations, let us now record how the bispectrality of the
BRF of $q$-Hahn type is expressed through two generalized eigenvalue
equations: a difference equation in the variable $x$ with eigenvalue depending
on the degree $n$, and a recurrence relation in the variable $n$ with
eigenvalue depending on $x$.

###  7.1 The difference equation

Return to the GEVP (3.3) that the operators $X$ and $Y$ define. We have
established in Proposition 1 that the $U_{n}(x;\alpha,\beta,N;q)$ are its
solutions, with eigenvalues ${\lambda_{n}=[-n]_{q}[n+\beta-N]_{q}}$.

From the expressions of $X$ and $Y$ given in (2.2) and (2.3)-(2.5) we see that
this amounts to the equation

$\displaystyle\begin{aligned}
&A_{1}(x)U_{n}(x+1;\alpha,\beta,N;q)+A_{0}(x)U_{n}(x;\alpha,\beta,N;q)+A_{2}(x)U_{n}(x-1;\alpha,\beta,N;q)\\\
&\quad=\lambda_{n}\left([x-\alpha]_{q}U_{n}(x;\alpha,\beta,N;q)-q^{-\alpha}[x]_{q}U_{n}(x-1;\alpha,\beta,N;q)\right),\end{aligned}$
(7.1)

with $A_{0}(x)$, $A_{1}(x)$, $A_{2}(x)$ given in (2.5). This is the difference
equation for the $U_{n}(x;\alpha,\beta,N;q)$ BRF.

###  7.2 The recurrence relation

Next, we look at the pair of operators $X$ and $Z$. As per (2.6), these define
the GEVP

$\displaystyle X\,U_{n}(x)=-[x-\alpha]_{q}Z\,U_{n}(x).$ (7.2)

From the actions on the $U_{n}$ basis given in the preceding section we see
that (7.2) amounts to the recurrence relation

$\displaystyle\mu_{n}^{(1)}U_{n+1}(x;\alpha,\beta,N;q)+\mu_{n}^{(2)}U_{n}(x;\alpha,\beta,N;q)+\mu_{n}^{(3)}U_{n-1}(x;\alpha,\beta,N;q)$
(7.3)
$\displaystyle\quad=-[x-\alpha]_{q}\left(\mu_{n}^{(7)}U_{n+1}(x;\alpha,\beta,N;q)+\mu_{n}^{(8)}U_{n}(x;\alpha,\beta,N;q)+\mu_{n}^{(9)}U_{n-1}(x;\alpha,\beta,N;q)\right)$

for the BRF $U_{n}(x;\alpha,\beta,N;q)$.

##  8 An algebra for the bispectrality of the $q$-Hahn rational functions

In the previous section, we have used the three operators $X$, $Y$, $Z$ to
obtain the bispectral equations of the rational functions. For orthogonal
polynomials of the Askey scheme, one would use two operators to encode the
bispectrality through eigenvalue equations. The algebraic relations obeyed by
these two operators lead to rich algebraic structures. This is precisely how
one obtains the Askey–Wilson algebra relations, starting from the operators
defining the difference and recurrence relations of the Askey–Wilson
polynomials [3]. To extend this construction, we now look at the algebraic
relations obeyed by the triplet of operators that express the bispectrality
properties of the biorthogonal rational functions $U_{n}$, the operators $X$,
$Y$, $Z$.

The relations realized by $X$, $Y$, $Z$ will be taken to represent the
following abstract algebra.

######  Definition 8.

The rational $q$-Hahn algebra $r\mathfrak{h}_{q}$ is the unital associative
algebra generated by $X$, $Y$, $Z$ obeying the relations

$\displaystyle XZ-qZX$ $\displaystyle=-Z^{2}-Z+\xi_{6}X,$ (8.1) $\displaystyle
ZY-qYZ$
$\displaystyle=\xi_{1}X^{2}+\xi_{3}\\{X,Z\\}+\xi_{4}Z^{2}+\xi_{5}X+\xi_{6}Y+\xi_{7}Z+\xi_{0},$
(8.2) $\displaystyle YX-qXY$
$\displaystyle=\xi_{3}X^{2}+\xi_{4}\\{X,Z\\}-\\{Y,Z\\}+\xi_{2}Z^{2}+\xi_{7}X-Y+\xi_{8}Z+\xi_{0},$
(8.3)

with

$\displaystyle\begin{gathered}\xi_{1}=-q^{\beta-N-1}[3]_{q},\qquad\xi_{2}=q^{-1}[\beta-N]_{q},\\\
\xi_{3}=[\beta-N-1]_{q}+q^{\beta-N},\qquad\xi_{4}=-q^{\beta-N-1},\\\
\xi_{5}=[\beta-\alpha]_{q}+[\beta-\alpha+1]_{q}+q^{\beta-N}\left(1+[-\alpha]_{q}+[-\alpha-1]_{q}\right),\\\
\xi_{6}=-(1-q),\qquad\xi_{7}=q^{\beta-N}[-\alpha-1]_{q}-[-\alpha]_{q}[\beta-N]_{q},\\\
\xi_{8}=q^{-1}[\beta-\alpha-N]_{q}-[-\alpha]_{q}[\beta-N]_{q},\qquad\xi_{0}=-[-\alpha]_{q}[\beta-\alpha]_{q}.\end{gathered}$
(8.9)

It can be observed that this algebra derives from a potential
$\Phi_{r\mathfrak{h}_{q}}$ [19]. Let $F=\mathbb{C}[x_{1},\dots,x_{n}]$ be a
free associative algebra with $n$ generators and $F_{cyc}=F/[F,F]$. The basis
of $F_{cyc}$ is spanned by the cyclic words $[x_{i_{1}}\dots x_{i_{k}}]$. One
can introduce the map $\tfrac{\partial}{\partial{x_{j}}}:F_{cyc}\to F$ that
acts as follows on basis elements:

$\displaystyle\frac{\partial[x_{i_{1}}\dots
x_{i_{r}}]}{\partial{x_{j}}}=\sum_{s|i_{s}=j}x_{i_{s+1}}x_{i_{s+2}}\dots
x_{i_{r}}x_{i_{1}}x_{i_{2}}\dots x_{i_{s-1}}.$ (8.10)

An algebra $F$ whose defining relations are given by

$\displaystyle\frac{\partial\Phi}{\partial{x_{j}}}=0,\qquad j=1,\dots,n$
(8.11)

for a $\Phi\in F_{cyc}$ is said to derive from a potential $\Phi$.

In the case of the rational $q$-Hahn algebra $r\mathfrak{h}_{q}$, it is seen
that the defining relations are obtained from the following potential:

$\displaystyle\begin{aligned}
\Phi_{r\mathfrak{h}_{q}}&=q[XYZ]-[YXZ]+\frac{\xi_{1}}{3}[X^{3}]+\frac{\xi_{2}}{3}[Z^{3}]+\xi_{3}[X^{2}Z]+\xi_{4}[XZ^{2}]-[YZ^{2}]\\\
&\quad+\frac{\xi_{5}}{2}[X^{2}]+\xi_{6}[XY]+\xi_{7}[XZ]-[YZ]+\frac{\xi_{8}}{2}[Z^{2}]+\xi_{0}\left([X]+[Z]\right).\end{aligned}$
(8.12)

More precisely, (8.1)-(8.3) are respectively equivalent to

$\displaystyle\frac{\partial\Phi_{r\mathfrak{h}_{q}}}{\partial{Y}}=0,\qquad\frac{\partial\Phi_{r\mathfrak{h}_{q}}}{\partial{X}}=0,\qquad\frac{\partial\Phi_{r\mathfrak{h}_{q}}}{\partial{Z}}=0.$
(8.13)

A Casimir element for this algebra is given by

$\displaystyle\begin{aligned}
Q_{r\mathfrak{h}_{q}}&=(1-q)XYZ+q^{\beta-N+1}X^{3}+\gamma_{1}X^{2}Z+\gamma_{2}XZ^{2}+qYZ^{2}\\\
&\quad+\gamma_{3}X^{2}+(1-q)XY+\gamma_{4}XZ-(1-2q)YZ+\gamma_{5}X-(1-q)Y\end{aligned}$
(8.14)

with

$\displaystyle\begin{gathered}\gamma_{1}=-q^{\beta-N}-[\beta-N+2]_{q},\qquad\gamma_{2}=[\beta-N+1]_{q},\\\
\gamma_{3}=-q^{\beta-N}[-\alpha]_{q}-[\beta-\alpha+1]_{q}-q^{\beta-N+1},\\\
\gamma_{4}=q^{\beta-N}+[\beta-\alpha]_{q}+[\beta-\alpha+1]_{q}+[-\alpha]_{q}[\beta-N+1]_{q},\\\
\gamma_{5}=[\beta-\alpha+1]_{q}+q^{\beta-N}[-\alpha]_{q}+[-\alpha]_{q}[\beta-\alpha]_{q}.\end{gathered}$
(8.19)

Next, let us bring to the fore the factorization of $Y$ as $XV$. In the
$q$-shift representation, the operator $V$ takes the form

$\displaystyle\begin{aligned} V&=q^{-x+\beta}[x-N]_{q}[x-\alpha+1]_{q}T^{+}\\\
&\quad+q^{-\alpha+\beta+1}[x]_{q}[x-N-1]-q^{-x+\beta}[x-N]_{q}[x-\alpha+1]_{q}-q^{-x}[x]_{q}[x-N-\alpha+\beta]_{q}\\\
&\quad+q^{\beta+1-\alpha-x}[\alpha-\beta-1]_{q}[\alpha-1]_{q}\sum_{k=1}^{N}q^{k}\frac{(q^{-x};q)_{k}}{(q^{\alpha-x};q)_{k}}\left(T^{-}\right)^{k},\end{aligned}$
(8.20)

with $(T^{-})^{N+1}$ acting as zero in our truncated grid. Based on the
nomenclature introduced in the $q=1$ case [20], we introduce what we will call
the meta-$q$-Hahn algebra $m\mathfrak{h}_{q}$.

######  Definition 9.

The meta-$q$-Hahn algebra $m\mathfrak{h}_{q}$ is the unital associative
algebra generated by $X$, $V$, $Z$ obeying the relations

$\displaystyle XZ-qZX$ $\displaystyle=-Z^{2}-Z-(1-q)X,$ (8.21) $\displaystyle
VX-qXV$ $\displaystyle=-\\{V,Z\\}+\eta_{1}X-V-q^{-1}\eta_{1}Z-\eta_{0},$
(8.22) $\displaystyle ZV-qVZ$
$\displaystyle=-\eta_{2}X-(1-q)V+\eta_{1}Z+\eta_{3},$ (8.23)

with

$\displaystyle\begin{aligned}
\eta_{0}&=[-\alpha]_{q}[\beta-N]_{q}+q^{-1}[\beta-\alpha+1]_{q},\\\
\eta_{1}&=[\beta-N+1]_{q},\\\ \eta_{2}&=-q^{\beta-N}[2]_{q},\\\
\eta_{3}&=[\beta-\alpha+1]_{q}+q^{\beta-N}[-\alpha]_{q}.\end{aligned}$ (8.24)

We can compute a Casimir element for this algebra which is cubic:

$\displaystyle\begin{aligned}
Q_{m\mathfrak{h}_{q}}&=(1-q)XVZ+qVZ^{2}+q^{\beta-N+1}X^{2}+(1-q)XV+\gamma_{1}XZ-(1-2q)VZ\\\
&\quad+\eta_{1}Z^{2}+\gamma_{3}X-(1-q)V+\gamma_{4}Z.\end{aligned}$ (8.25)

One can see that the $q$-commutators between $V$ and other generators are
linear on the rhs up to a single quadratic term; they thus depart minimally
from pure linear relations. Such a characterization, and the fact that the
algebra is connected to rational functions of $q$-Hahn type, hints that the
meta-$q$-Hahn algebra is bound to have interesting properties. As an example,
let us mention that it derives from the potential $\Phi_{m\mathfrak{h}_{q}}$:

$\displaystyle\begin{aligned}
\Phi_{m\mathfrak{h}_{q}}&=q[XVZ]-[VXZ]-[VZ^{2}]-(1-q)[XV]-[VZ]+\eta_{1}[XZ]-\frac{q^{-1}\eta_{1}}{2}[Z^{2}]-\frac{\eta_{2}}{2}[X^{2}]\\\
&\quad+\eta_{0}[Z]+\eta_{3}[X].\end{aligned}$ (8.26)

Its representation theory will then be intimately tied to the rational
functions of $q$-Hahn type.

##  9 Conclusion

Let us repeat the main points of this paper. We have introduced a triplet of
$q$-difference operators $X$, $Y$, $Z$ that play a role analoguous to the
bispectral operators of orthogonal polynomials in the case of biorthogonal
rational functions. We have shown how these three operators provide a
characterization of the bispectrality properties of the BRF of $q$-Hahn type
through GEVP, and we have also proven the biorthogonality of these functions.
The underlying algebra was obtained and the connection with Wilson’s
${}_{10}\phi_{9}$ BRF was made explicit. A remarkable factorization of the
operator $Y$ as $XV$ was exhibited.

In future work we plan to explore the representations of the meta-$q$-Hahn
algebra introduced in the last section. It is expected that this algebra will
provide a unified algebraic interpretation of the bispectrality properties of
both the $q$-Hahn orthogonal polynomials and the biorthogonal rational
functions of $q$-Hahn type.

This work adds to the systematic exploration and characterization of the
bispectrality of BRF. The ${}_{3}\phi_{2}$ and ${}_{3}F_{2}$ Hahn type BRF
have now been well studied [21, 7, 20] and the next step would be to explore
BRF of $q$-Racah type. These should be expressable as ${}_{4}\phi_{3}$ and
their $q\to 1$ limits should provide BRF of Racah type. Note that the Askey
biorthogonal polynomials on the unit circle have also been shown in [22] to be
amenable to a similar treatment.

Lastly, let us mention that Wilson’s BRF have appeared in the study of the so-
called rational Heun operators [23]. These rational Heun operators were
defined by extending the construction of Heun operators to rational functions.
Sklyanin–Heun operators have been recently introduced as building blocks of
the Heun operators and have led to an algebraic characterization of various
families of para-polynomials [24, 25, 26]. One can expect a similar story to
unfold for the rational functions. Rational analogues of Sklyanin–Heun
operators should prove interesting objects of study and should lead to the
definition of para families of rational functions.

### Acknowledgments

I. Bussière benefitted from a CRM-ISM intern scholarship. J. Gaboriaud
received financial support from the ESP of the Université de Montréal. The
research of L. Vinet is supported in part by a Discovery Grant from NSERC. A.
Zhedanov who is funded by the National Foundation of China (Grant No.11771015)
gratefully acknowledges the hospitality of the CRM over an extended period.

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# Exploring the Correlation between Human and Machine Evaluation of
Simultaneous Speech Translation

Xiaoman Wang
University of Leeds
<EMAIL_ADDRESS>Claudio Fantinuoli
KUDO
University of Mainz
<EMAIL_ADDRESS>
###### Abstract

Assessing the performance of interpreting services is a complex task, given
the nuanced nature of spoken language translation, the strategies that
interpreters apply, and the diverse expectations of users. The complexity of
this task become even more pronounced when automated evaluation methods are
applied. This is particularly true because interpreted texts exhibit less
linearity between the source and target languages due to the strategies
employed by the interpreter.

This study aims to assess the reliability of automatic metrics in evaluating
simultaneous interpretations by analyzing their correlation with human
evaluations. We focus on a particular feature of interpretation quality,
namely translation accuracy or faithfulness. As a benchmark we use human
assessments performed by language experts, and evaluate how well sentence
embeddings and Large Language Models correlate with them. We quantify semantic
similarity between the source and translated texts without relying on a
reference translation. The results suggest GPT models, particularly GPT-3.5
with direct prompting, demonstrate the strongest correlation with human
judgment in terms of semantic similarity between source and target texts, even
when evaluating short textual segments. Additionally, the study reveals that
the size of the context window has a notable impact on this correlation.

##  1 Introduction

The assessment of interpreting quality is a common practice in both
professional interpreting and academic contexts. The results of these
evaluations offer valuable insights for a wide range of stakeholders,
including interpreter’s clients, users, practitioners, educators,
certification bodies, and researchers [Han, 2022].

Assessing quality in interpretation is a complex endeavor. Quality is not only
challenging to measure, but it manifests also an “elusive nature” [Becerra et
al., 2013, p. 7] making it difficult to define. The notion of quality in fact
may vary from one user to another, introducing a substantial degree of
subjectivity in determining what constitutes a good translation of speech.
Furthermore, the criteria for quality are contingent upon the type of
interpretation involved. For instance, in conference interpreting, the
emphasis is generally on the quality of the interpreter’s output, encompassing
aspects such as content, language, and delivery. In contrast, within community
settings like social and healthcare interpreting, interactional competencies
and discourse management play a crucial role in determining what quality is
[Kalina, 2012].

Traditionally, the assessment of interpreting performances has been carried
out manually, a methodology that comes with its own set of pros and cons. On
the positive side, human evaluations offer a holistic view of quality by
taking into account various facets of the communication process, thereby
delivering a more nuanced understanding of interpreting performance
[Pöchhacker, 2002, Becerra et al., 2013]. Conversely, manual assessment comes
with its own set of challenges, including being labor-intensive, time-
consuming and costly [Wu, 2011]. Furthermore, the results often have limited
generalizability due to either the restricted scope of the data sampled or the
inherent complexities associated with evaluating spoken translation.

In light of the limitations, there has been a growing interest to apply
automatic metrics to the evaluation of interpreting performances. While
traditional statistical metrics like BLEU have shown limited efficacy in
capturing translation quality from a user’s perspective, the emergence of
semantic vectors and pre-trained, large-scale generative language models has
yielded promising results, especially in the domain of written translation
[Kocmi and Federmann, 2023]. The application of these metrics is gradually
extending to the field of spoken translation as well [Han and Lu, 2021b].
However, it must be mentioned that orally translated texts possess certain
characteristics that might restrict the efficacy of employing metrics designed
for written texts. Interpreters, especially in the simultaneous modality, tend
to alter the text more extensively than translators, modifying the structure
and omitting parts deliberately as a strategy rather than a deficiency. For
example, interpreters may omit part of the original when experiencing
cognitive overload, when they cannot comprehend the original message, to name
just a few [Korpal, 2012]. This non-linearity between source and target texts
renders the task of automatic evaluation even more challenging.

The adoption of easy accessible and robust automatic evaluation in
interpreting offers several potential applications that could benefit a wide
range of stakeholders. Firstly, the ability to provide instant feedback to
trainees and practitioners would enable them to quickly assess their
performance and pinpoint areas for improvement, also in real-time, creating a
faster feedback loop that could substantially accelerate autonomous skill
development. Secondly, automatic evaluation might aid organizations in
consistently and objectively monitoring the quality of their multilingual
services. Thirdly, automatic metrics that correlate with human judgments might
serve as a useful tool for the continuous evaluation of machine
interpretation.

This study addresses two primary questions: First, is there an automatic
metric that aligns closely with human judgment and can thus be used to
automate the accuracy evaluation of spoken language translation? Second, do
these metrics evaluate human-generated translations, machine-generated
translations, or both more effectively?

The rest of the paper is organised as follows. In Section 2 we present an
overview of research in the field of automatic evaluation of interpreting
performances. In Section 3, we illustrate our research methodology, our data
and the experimental design. Section 3.1 describes the dataset created for
this task. Section 3.2 describes the process for human evaluation of the
translations while Section 3.3 delves on the process followed for the
automatic evaluation. Section 4 presents the results. Section 5 introduces
some ethical implications. Finally, Section 6 concludes the paper with some
discussion and remarks.

##  2 Related work

The evaluation of translation quality and in particular of accuracy or
information fidelity, i.e. the correspondence between source-language and
target-language renditions, has traditionally differed between computer
science, with its tradition of abundant use of automatic metrics, and
translation and interpreting community, with its focus on manual evaluation as
perceived by experts and users.

In computer science, evaluation metrics such as Bilingual Evaluation
Understudy (BLEU) [Papineni et al., 2002], National Institute of Standards and
Technology (NIST) [Doddington, 2002], Metric for Evaluation of Translation
with Explicit Ordering (METEOR) [Banerjee and Lavie, 2005], and Translation
Edit Rate (TER) have been foundational in establishing benchmarks for Machine
Translation Quality Estimation (MTQE). BLEU and NIST emphasise n-gram
precision, with NIST uniquely weighting distinct n-grams. METEOR integrates
both recall and precision, while TER quantifies requisite edits for optimal
translation alignment. However, recent scholarly discourse have suggested that
these metrics, while valuable, may possess intrinsic limitations in
encapsulating the multifaceted subtleties and overarching context of
linguistic structures [Fernandes et al., 2023]. This acknowledgement has
precipitated the exploration of advanced, data-driven methodologies for MTQE
without references. Neural networks, characterized by their bio-inspired
architectures, emerge as a compelling alternative. These computational
structures excel in managing voluminous datasets, discerning intricate
patterns, and, crucially, accounting for the inherent complexities associated
with linguistic phenomena.

In the field of neural network architectures, the potential of Recurrent
Neural Networks (RNNs), Convolutional Neural Networks (CNNs), and the
groundbreaking Transformer for semantic similarity computations has been
explored. Of these, Transformer-based models like BERT and GPT have gained
considerable academic traction due to their outstanding performance across
numerous Natural Language Processing (NLP) tasks [Wang et al., 2023, Kocmi and
Federmann, 2023, Clark et al., 2020, Xenouleas et al., 2019, Yang et al.,
2019, Brown et al., 2020, Hendy et al., 2023, Clark et al., 2019, Vaswani et
al., 2017]. Central to these architectures is the concept of embeddings: dense
vector representations that capture the semantic essence of words or textual
segments. Within this high-dimensional space, vectors situated closely denote
semantic relatedness. In translation evaluation, embeddings offer a mediating
semantic layer, enabling comparisons between source and target linguistic
structures. However, the embeddings landscape is complex. Models from the
Universal Sentence Encoder Multilingual (USEM) to the Generative Pre-trained
Transformer (GPT) produce embeddings with varied purposes. For example, USEM
is geared towards retrieving semantically aligned entities, while GPT
emphasizes generating context-rich linguistic constructs. These nuances
highlight the need for researchers to thoughtfully choose models aligned with
their specific research goals.

In interpreting studies, a subdomain of translation studies dedicated to oral
translation, the traditional practice has been to assess accuracy manually,
with or without references. Many of the evaluation methodologies are derived
by written translation. For the assessment based on references, also known as
intra-lingua assessment, the notion of ”tertium comparisons” stands as a
pivotal benchmark within a particular language [Setton and Motta, 2007].
Tracing the historical evolution of this methodology, [Carroll, 1966] stands
out as a foundational contributor. He experimented with lay people to
ascertain accuracy in translations. Building on Carroll’s scale, [Tiselius,
2009] refined the process, integrating references to spoken language and
interpreting for intra-lingual assessment. These approaches, while receiving
considerable acceptance from seasoned interpreters, are not without
limitations. The impact of cognitive-linguistic factors can potentially alter
evaluation results [Han and Zhao, 2021]. Moreover, intra-lingual evaluations
face challenges in adapting to changing contexts and demographics and may lack
a universally acknowledged point of reference [Setton and Motta, 2007].

Academic discussions prioritise evaluation methods for gauging accuracy in
inter-language interpreting. These methods range from error analysis, as seen
in works by [Gerver, 1969] and [Gile, 1995] that identify translation
inaccuracies, to propositional analysis, endorsed by researchers like
[Mackintosh, 1983] and [Lee, 1999a, Lee, 1999b, Lee, 2002], which examines
textual accuracy. However, these methods present challenges in addressing
linguistic subtleties and differing interpretations. More recent research
emphasizes grading rubrics, tracing back to [Carroll, 1966], which outline
performance across competency tiers and are validated in multiple studies
[Han, 2016, Han, 2017, Nia and Modarresi, 2019, Wu et al., 2013]. Yet, even
with proven reliability, this rubric-based evaluation faces hurdles like the
development and validation of rubric descriptors and evaluator
inconsistencies.

A few studies have explored the effectiveness of various metrics in evaluating
the translation quality or interpreting performances. [Chung, 2020], for
instance, pinpointed the strong alignment between human evaluations and scores
determined by BLEU and METEOR for German-to-Korean translation. Subsequent
studies by [Han and Lu, 2021a] and [Lu and Han, 2022] reinforced the merit of
these automated tools. Han and Lu (2021) discerned that METEOR’s sentence-
level evaluations resonated more with human assessments than broader, text-
level evaluations. Conversely, Lu and Han’s (2022) exploration, fortified with
the integration of the BERT model [Devlin et al., 2019], showcased substantial
correlations between human and automated evaluations, underlining the
potential of these metrics in assessing interpreting performances. A recent
study by [Kocmi and Federmann, 2023] employed Large Language Models like GPT
to evaluate translation quality across three language pairs, concluding that
only models GPT3.5 and above possess the capability for such translation
quality assessment.

While initial studies have underscored positive and moderate-to-strong
correlations for MT metrics such as BLEU and METEOR, to the best of our
knowledge no research has been conducted so far on the use of language models
for reference-free interpreting assessment. Our study aims to fill this gap.

##  3 Data and methodology

###  3.1 Dataset

The dataset used for the study consists of 12 original speeches in English
translated into Spanish, each lasting approximately 5 minutes. These videos
were curated from a broader selection of real-life contexts, including
lectures, business presentations, live tutorials, and political
addresses.111The dataset is available under the Creative Commons 4.0 License
at https://github.com/renawang26/Information_fidelity

Although the corpus size is inherently limited, in order to allow high quality
human evaluation, the selection of videos was strategically designed to
capture a spectrum of speech features. The speeches were distributed equally
in terms of gender, with 6 from male speakers and 6 from female speakers. In
addition, the accent of the speakers comprises both native and non-native
speakers. The nature of the speeches is diversified into three categories: 4
corporate, 4 political, and 4 general presentations. The speeches comprise
3529 tokens.

For the evaluation purpose, each video was simultaneously interpreted in two
ways:

  * •

Translation H: Human professional interpreters were engaged. Three
interpreters, native Spanish speakers, were involved, each responsible for
translating four videos. Simultaneous interpreters were required to interpret
the entire video (approximately 5 minutes) to preserve the contextual
information essential for accurate interpretation. However, only 2 minutes of
the videos were randomly selected for the evaluation.

  * •

Translation M: Machine interpretation was carried out by the KUDO AI Speech
Translator, the only system available for simultaneous translation available
at the moment of writing 222www.kudo.ai.

All recordings were automatically transcribed, and the transcriptions were
post-edited for accuracy by expert linguists proficient in both English and
Spanish. The goal of this operation was to make sure that the transcripts did
not contain errors of transcription. The transcriptions were manually aligned
based on semantic units, a critical step due to the absence of formal
punctuation commonly found in written texts. These segments are roughly
comparable to sentences or smaller paragraphs. The average length of segments
is 29.41 tokens.

###  3.2 Human evaluation

The human evaluation process was guided by the methodology proposed by
[Fantinuoli and Prandi, 2021], which uses a Likert scale to assess two key
features of interpretation: accuracy (ability of the translation to convey the
meaning of the original) and intellegibility (ability of the translation of
being understandable). The two dimensions reflect the main criteria at the
core of the product-oriented approach to quality evaluation in Interpreting
Studies [Tiselius, 2009]. For the purposes of this study, however, the focus
was exclusively on the feature of informativeness, i.e. accuracy, leaving the
assessment of intellegibility and any other potential feature for future
research. One of the advantages of this framework lies in its being user-
centric and in line with the corpus-based evaluation already established in
Interpreting Studies to assess the quality of human interpretation.

The human evaluation was conducted using a diverse group of 18 evaluators.
This consisted of 9 professional interpreters and 9 bilingual individuals
without any translation or interpreting experience. The goal was to capture a
broad and unbiased evaluation of the translations, taking into account both
professional expertise and everyday bilingual proficiency. Each evaluator was
assigned 4 videos to evaluate. They were informed that the translations were
transcriptions of oral simultaneous interpretations.

For each speech, the raters were asked to assess on a six-point Likert scale
first the intelligibility of the output (without a comparison with the source
speech nor a comparison between the two outputs), then to evaluate the
accuracy of the renditions by comparing each one to the source speech.

An important feature of the evaluation process was the anonymity of the
translation sources. Evaluators were not informed whether the translations
were produced by a human or by a machine. This was a deliberate step to
prevent any evaluation bias, ensuring that the judgment was strictly based on
the quality of the translation, irrespective of the producer.

It is important to point out that with a value of 0.0964 the interater
agreement is low (“slight agreement” on a Fleiss’ Kappa scale). This aspect
showcases the intrinsic complexity of objectively assessing spoken language
translation due to different expectations by the evaluators about what
constitutes accuracy. This is an insidious limitation of the evaluation of
human interpretation (see Han 2022). While the low agreement between multiple
raters is expected, it also limits the generalizability of our findings.

###  3.3 Machine evaluation

Our approach to the machine assessment of spoken language translation is based
on the concept of semantic similarity leveraging sentence embeddings and large
language model prompting techniques. The rationale behind using embeddings to
measure semantic similarity is multifold, as it proffers a host of advantages,
such as the provision of a continuous representation [Mikolov et al., 2013,
Pennington et al., 2014], incorporation of contextual information [Devlin et
al., 2019, Peters et al., 1802], dimensionality reduction [Roweis and Saul,
2000], applicability of transfer learning [Howard and Ruder, 2018, Raffel et
al., 2020], multilingual support [Conneau et al., 2017, Wu et al., 2016],
interoperability [Ruder et al., 2019, Artetxe et al., 2018], ease of use
[Radford et al., 2018], and state-of-the-art performance [Vaswani et al.,
2017, Brown et al., 2020] in NLP tasks. The advantage of using sentence
embeddings over MTQE models for assessing interpreting performance lies in the
ability of embeddings to evaluate without the need for references. This
approach contrasts with some MTQE models, which typically depend on references
for quality assessments. Furthermore, sentence embeddings are adaptable across
a broad spectrum of languages and text genres, offering a versatile solution
for evaluating interpreting performance. This flexibility is beneficial across
different domains and contexts, whereas MTQE models often necessitate more
specific training data to achieve comparable effectiveness. The fundamental
operation of this methodology involves vectorising each sentence in both the
source and target texts. Essentially, this means mapping each sentence to its
corresponding vectors of real numbers, thereby projecting them into a shared
multi-dimensional space. This conversion of textual data into numerical format
empowers the machine to elaborate the semantics of the texts effectively. The
subsequent step involves the calculation of cosine similarity, which serves as
a measure of the similarity between each language pair.

We employed three neural network models to carry out sentence embedding: all-
MiniLM-L6-v2333https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2,
Generative Pre-trained Transformer (GPT), and in particular GPT-Ada.
model444https://platform.openai.com/docs/api-reference/embeddings, and
Universal Sentence Encoder Multilingual555https://tfhub.dev/google/universal-
sentence-encoder-multilingual-large/3 (USEM). By integrating the all-
MiniLM-L6-v2 for its efficient, compact design suitable for multi-language
applications, alongside GPT-Ada for their advanced generative capabilities and
adaptability, and USEM for its extensive language coverage and cross-lingual
semantic understanding, this strategy offers a balanced and comprehensive
approach. The embeddings obatained werer vectorised sentences in both English
and Spanish. The sentence embeddings computed with these models were later
used to calculate the Cosine Similarity between the source texts and the
translations.

In addition to the models for computing word vectors, we tested another method
to compute semantic similarity leveraging the prompt functionality of
GPT3.5666https://www.openai.com. The Large Language Model was assigned the
task of assessing the semantic similarity between pairs of sentences, one in
English and the other in Spanish, using a scale ranging from 1 to 5. An
example prompt provided was: ”Given the two sentences in English and Spanish,
rate from 1 to 5 their similarity, where 1 is not similar and 5 very similar.”

###  3.4 Computing correlations

The human and automatic assessments were put together in an evaluation matrix.
Pearson’s correlation coefficients were leveraged to explore the relationships
between human and machine evaluations. Specifically, the correlations between
human judgments and the cosine similarities derived from the embeddings of
segments from the source speech and their corresponding translations
(Translation H and Translation M) generated by three models. were calculated.
As stated before, the overarching aim was to probe the feasibility of
achieving machine-human parity in the evaluation results.

Since the neural network models chosen for this experiment have limited
reliably in computing semantic vectors for long texts, we opted to establish a
correlation between human and machine evaluations at the segment level. It is
essential to emphasize that similarities values obtained from isolated segment
pairs have intrinsic limitations since they are not able to consider accuracy
across segments. Thus, the quantity of tokens utilized as context for sentence
embeddings could potentially affect the model’s contextual comprehension and,
subsequently, the precision of semantic similarity assessments.

To take this into consideration, we examined the effect of “window size”, i.e.
the number of segments combined into a single one. For this purpose, we
computed similarities for window sizes up to five segments. The systematic
variation in window size aimed to shed light on how semantic similarity
between human and machine evaluations could potentially be influenced by the
availability of cross-segment context.

##  4 Results

To analyse the data, we devised charts from three perspectives, including the
comparison of correlation values among evaluation methods, a comparison
between Translation H and Translation M, and correlation values based on
window size.

In Figure 1 we compare the distribution of correlation values across all
machine evaluation methods, namely GPT-3.5, all-MiniLM-L6-v2, GPT-Ada, and
UMSE, for both Translation H and Translation M.

The correlation with GPT-3.5 displays the highest median correlation value.
The interquartile range (IQR) is also quite narrow, indicating that the
correlation values for this method are consistently high and well-aligned with
human evaluations. The correlation with all-MiniLM-L6-v2 has a wide IQR,
showcasing varied performance. The median value is close to zero, but there
are negative outliers, indicating instances where the machine evaluation is
inversely related to human judgment. The correlation values with GPT-Ada are
relatively consistent, with a narrow IQR. The median is slightly above 0.3,
which indicates moderate alignment with human judgments. UMSE’s performance
seems to be close to GPT-Ada with a median slightly above 0.3. The IQR is a
bit larger, suggesting a bit more variability in the correlation values.

Figure 1: Correlations among machine evaluation methods

For the comparison between Translation H and Translation M in Figure 2, paired
bar charts elucidate the average correlation disparities for each machine
evaluation method. GPT-3.5’s measurements exhibit robust correlation values
with human judgments for both translations, although Translation H marginally
outperforms Translation M. For all-MiniLM-L6-v2, the correlation of
Translation H gravitates towards zero, whereas Translation M registers a
negative value, implying a potential misalignment of the all-MiniLM-L6-v2’s
evaluations with human perspectives, predominantly for Translation M. GPT-Ada
embeddings yield nearly identical correlation values for both translations,
but with Translation H slightly edging out. Intriguingly, UMSE’s embeddings
produce a higher correlation value for Translation M compared to Translation
H.

Figure 2: Correlations for Translation H and Translation M

Turning to the third perspective, which examines the shift in correlation
values based on window size, line charts in Figure 3 and Figure 4 offer
insights into this dynamic for each machine evaluation method. For Translation
H (Human Translated) in Figure 3, GPT-Ada correlation with human ratings sees
a mild fluctuation across window sizes, initially decreasing from window
size-1 to size-2, then slightly rising in the following window size2-5. The
all-MiniLM-L6-v2 model, in contrast, exhibits a downward trend, indicating
reduced agreement with human evaluations as window size grows. UMSE
consistently maintains a stable correlation with human ratings, showing only
minor variations across different window sizes. GPT-3.5 presents a distinct
pattern; while its correlation initially drops from size-1 to size-2, it
surges notably in the subsequent window, outperforming the other models.

In observations for Translation M (Machine Translated) in Figure 4, GPT-Ada
begins with a positive correlation with human ratings with size-1, but this
declines as the window size expands, hinting at potential metric
inconsistencies for broader contexts. The all-MiniLM-L6-v2 model’s
correlation, on the other hand, commences positively by size-1 and
consistently rises with the window size, pointing to a more aligned evaluation
with human judgment for larger translation segments. UMSE’s performance
mirrors its evaluation with human translations, maintaining stability across
all window sizes and showcasing its consistent metric evaluation. In contrast,
GPT-3.5’s correlation fluctuates considerably across window sizes —
experiencing a drop, a subsequent rise, and then another decline — indicating
a variable level of concurrence with human assessments depending on the window
size.

Figure 3: Correlations for Translation H according to window size Figure 4:
Correlation for Translation M according to window size

##  5 Ethical considerations

The adoption of automatic evaluation also raises ethical concerns that warrant
careful consideration. One potential issue is the possibility to continuously
monitor interpreters, which might infringe on privacy rights and create a
sense of constant surveillance, negatively impacting job satisfaction and
professional autonomy. Additionally, decisions regarding the employability of
individuals could be blindly given to mechanical means, potentially leading to
unjust or biased outcomes if the algorithms fail to account for contextual
nuances or other essential aspects of human communication. As such, it is
crucial for the language industry to carefully weigh the benefits and
challenges of automatic evaluation, ensuring that ethical considerations are
addressed as advancements in AI technology continue to reshape the landscape.

##  6 Conclusions

This study aimed to analyze the correlation between automated and human
evaluations of translated content. The peculiarity of this experiment is that
we focused on a specific form of translation: the simultaneous interpretation
of English speeches into Spanish. This mode of translation introduces unique
challenges to assessment due to the nonlinear nature of the output (in spoken
translation, the differences between the source and target can be more
pronounced than in written translation) and varying user expectations
regarding interpretation quality. We evaluated both interpretations provided
by professional interpreters and those produced by a machine interpretation
system. The objective was to develop a metric reflecting interpreting quality
in a manner consistent with human judgment.

The direct prompting of GPT-3.5 for quality estimation on a Likert scale
exhibits the highest median correlation with human evaluation. This finding
establishes GPT-3.5 as the most promising tool among the evaluated methods to
gauge translation quality, both for interpretations produced by humans and
machines. GPT-3.5 benefits from a larger context, performing better with
larger segment windows. This suggests that the model can capture and evaluate
long dependencies more effectively.

Contrary to expectations, GPT-3.5’s correlation with human judgment is
somewhat stronger for translations produced by professional interpreters than
for machine-generated ones. This implies that GPT might be subtly more attuned
to the linguistic nuances of human translation, even though it remains adept
at evaluating speech translation. One possible explanation for this is that
human interpreters often introduce subtle contextual, tonal, and idiomatic
adjustments that are more aligned with GPT-3.5’s training on diverse data,
whereas machine translations might adhere more strictly to equivalences.
Looking forward, further research could explore deeper into the
characteristics of the datasets used for training such models and their
alignment with real-world interpretation tasks.

This study presents several limitations. The observed low interrater agreement
suggests potential inconsistencies in human evaluations, possibly affecting
correlation values, and generability of the results. Furthermore, the limited
scope of the sampled translations might not capture the full range of
linguistic complexities inherent to interpretation. Future research should
consider evaluations for higher window sizes. In light of GPT-3.5’s
performance in this study, future research might explore its ability to
delineate nuances of typologies of errors rather than merely providing
aggregate scores

This study is considered a preliminary attempt to test the feasibility of
applying automatic metrics to evaluate inputs from both human and machine
interpreters. Before these metrics can be used in production, more research
needs to be conducted.

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# A proof of Sylvester’s theorem

Saptak Bhattacharya  Indian Statistical Institute
New Delhi 110016
India<EMAIL_ADDRESS>
###### Abstract.

We give a new elementary proof of existence and uniqueness of a solution to
the Sylvester equation $AX-XB=Y$

###### Key words and phrases:

Sylvester’s equation, block matrices

###### 2010 Mathematics Subject Classification:

15A23, 15A30

##  1\. Introduction

Given finite dimensional Hilbert spaces $\mathcal{H}$, $\mathcal{K}$ and
operators $A\in\mathcal{L}(\mathcal{K})$, $B\in\mathcal{L}(\mathcal{H})$ and
$Y\in\mathcal{L}(\mathcal{H},{K})$ the Sylvester equation asks for solutions
$X\in\mathcal{L}(\mathcal{H},{K})$ to

$AX-XB=Y$

A particular case of interest is the Lyapunov equation

$A^{*}X+XA=Y$

which arises in stability theory (see [4]). Equation (1) was first studied by
Sylvester in [6], who showed that it has a unique solution if
$\sigma(A)\cap\sigma(B)=\emptyset$. This was generalized to infinite
dimensions by Rosenblum in [5].

The purpose of this note is to give a short proof of Sylvester’s theorem using
elementary block matrix arguments. Other different proofs are given in [1, 2,
3]. A thorough survey on equation (1) can be found in [1].

##  2\. Main result

######  Theorem 1.

Let $\mathcal{H}$ and $\mathcal{K}$ be finite dimensional Hilbert spaces and
let $A\in\mathcal{L}(\mathcal{K})$ and $B\in\mathcal{L}(\mathcal{H})$ with
$\sigma(A)\cap\sigma(B)=\emptyset$. Then for every
$Y\in\mathcal{L}(\mathcal{H},\mathcal{K})$ there exists a unique
$X\in\mathcal{L}(\mathcal{H},\mathcal{K})$ such that $AX-XB=Y$.

###### Proof.

Consider the map
$\Phi:\mathcal{L}(\mathcal{H},\mathcal{K})\to\mathcal{L}(\mathcal{H},\mathcal{K})$
given by $\Phi(X)=AX-XB$. It suffices to show that $\Phi$ is injective. If ker
$\Phi$ contains an invertible $X$, we have

$X^{-1}AX=B$

implying $\sigma(A)=\sigma(B)$, a contradiction. If not, we use a block matrix
argument to reduce to this case. Let $X\in$ ker $\Phi$ such that $X\neq O$.
Consider the direct sum decompositions

$\mathcal{H}=(\textrm{ker}\hskip 1.99168ptX)^{\perp}\oplus\textrm{ker}\hskip
1.99168ptX$

and

$\mathcal{K}=\textrm{im}\hskip 1.99168ptX\oplus\textrm{ker}\hskip
1.99168ptX^{*}.$

Note that $(\textrm{ker}\hskip 1.99168ptX)^{\perp}\neq\\{0\\}$. With respect
to these decompositions, we have the block matrices

$X=\begin{pmatrix}Y&O\\\ O&O\end{pmatrix}$ $A=\begin{pmatrix}E&F\\\
G&H\end{pmatrix}$

and

$B=\begin{pmatrix}P&Q\\\ R&S\end{pmatrix}.$

Observe that $Y$ is invertible. The condition $AX=XB$ now yields

$\begin{pmatrix}EY&O\\\ GY&O\end{pmatrix}=\begin{pmatrix}YP&YQ\\\
O&O\end{pmatrix}$

implying

$EY=YP,$ $G=O\hskip 5.69054pt\textrm{and}\hskip 5.69054ptQ=O.$

Now

$A=\begin{pmatrix}E&F\\\ O&H\end{pmatrix}$

and

$B=\begin{pmatrix}P&O\\\ R&S\end{pmatrix}.$

Thus, $\sigma(E)\subset\sigma(A)$ and
$\sigma(P)=\sigma(P^{t})\subset\sigma(B^{t})=\sigma(B)$ which implies
$\sigma(E)\cap\sigma(P)=\emptyset$. From (2), we have a contradiction due to
the invertibility of $Y$.

## References

  * [1] R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX - XB = Y, Bull. London Math. Soc. 29 (1997), no. 1, 1-21.
  * [2] N. Higham, Functions of Matrices : Theory and Computation, Society for Industrial and Applied Mathematics (2008).
  * [3] H. Dym, Linear Algebra in Action, Graduate Studies in Mathematics, American Mathematical Society (2013).
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