Elliptic Hypergeometric Summations by Taylor Series Expansion and
Interpolation

Elliptic Hypergeometric Summations
by Taylor Series Expansion and Interpolation\star\star$\star$This paper is a
contribution to the Special Issue on Orthogonal Polynomials, Special Functions
and Applications. The full collection is available at
http://www.emis.de/journals/SIGMA/OPSFA2015.html

Michael J. SCHLOSSER and Meesue YOO M.J. Schlosser and M. Yoo Fakultät für
Mathematik, Universität Wien,
Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
<EMAIL_ADDRESS><EMAIL_ADDRESS>http://www.mat.univie.ac.at/~schlosse/

Received March 01, 2016, in final form April 13, 2016; Published online April
19, 2016

We use elliptic Taylor series expansions and interpolation to deduce a number
of summations for elliptic hypergeometric series. We extend to the well-poised
elliptic case results that in the $q$-case have previously been obtained by
Cooper and by Ismail and Stanton. We also provide identities involving S.
Bhargava’s cubic theta functions.

elliptic hypergeometric series; summations; Taylor series expansion;
interpolation

30E05; 33D15; 33D70; 33E05; 33E20

##  1 Introduction

Previously, one of us [21] established an elliptic Taylor expansion theorem
which extends Ismail’s [11] expansion for functions symmetric in $z$ and $1/z$
in terms of the Askey–Wilson monomial basis. The expansion theorem in [21]
involves a special case of Rains’ [17] elliptic extension of the Askey–Wilson
divided difference operator. As applications, new simple proofs were given for
Frenkel and Turaev’s [8] elliptic extensions of Jackson’s ${}_{8}\phi_{7}$
summation and of Bailey’s ${}_{10}\phi_{9}$ transformation. A further
application concerned the computation of the connection coefficients of
Spiridonov’s [23] elliptic extension of Rahman’s biorthogonal rational
functions.

Here we take a closer look at elliptic Taylor expansions. In particular, we
describe the action of the $m$-th elliptic divided difference on a function,
expressed in terms of the function. In the ordinary case, if $\delta_{h}$
denotes the central difference operator, defined by
$\delta_{h}\,f(x)=f(x+\frac{h}{2})-f(x-\frac{h}{2})$, the $m$-th difference is
given by

$\displaystyle\delta_{h}^{m}f(x)=\sum_{k=0}^{m}(-1)^{k}\binom{m}{k}f\left(x+\left(\frac{m}{2}-k\right)h\right).$

For the $q$-case, where $\delta_{h}$ is replaced by the Askey–Wilson operator
$\mathcal{D}_{q}$, acting on functions $f(z)$ symmetric in $z$ and $1/z$, an
explicit formula for $\mathcal{D}_{q}^{m}f(z)$ was established by Cooper [5].
One of the results of our paper concerns an extension of Cooper’s formula to
the elliptic setting. We remark that Ismail, Rains and Stanton [12]
independently have also proved an elliptic extension of Cooper’s formula which
turns out to be equivalent to our result by a multiplication of operators. In
[14], Ismail and Stanton have used Cooper’s explicit formula to work out an
explicit interpolation formula for polynomials symmetric in $z$ and $1/z$.
Likewise, we use our elliptic extension of Cooper’s formula to find an
elliptic interpolation formula. Application of this formula yields single and
multivariable identities of Karlsson–Minton type.

Ismail and Stanton [13] not only considered Taylor expansions in terms of the
Askey–Wilson monomial basis $\\{(az,a/z;q)_{n},\,n\geq 0\\}$ (see the
subsequent subsection for the $q$-shifted factorial notation), but also in
terms of the basis
$\big{\\{}\big{(}q^{\frac{1}{4}}z,q^{\frac{1}{4}}/z;q^{\frac{1}{2}}\big{)}_{n},\,n\geq
0\big{\\}}$, for which they deduced quadratic summations as applications. We
are able to extend Ismail and Stanton’s analysis and provide, in particular, a
Taylor expansion for an elliptic extension of this other basis. We note that
in addition, Ismail and Stanton [13, Theorem 2.2] gave a Taylor expansion
theorem for the basis
$\big{\\{}(1+z^{2})\big{(}{-}q^{2-n}z^{2};q^{2}\big{)}_{n-1}z^{-n},\,n\geq
0\big{\\}}$, however this result (which involves an evaluation at $z=0$)
appears not to extend to the elliptic setting.

Finally, we consider series partially involving products of S. Bhargava’s [3]
cubic theta functions. Such series have not been considered before. We
introduce two different cubic theta extensions of shifted factorials which are
designed in such forms that they behave well under the iterated action of the
elliptic Askey–Wilson operator. Applications of Taylor expansion yield cubic
theta extensions of Jackson’s ${}_{8}\phi_{7}$ summation formula and of a
quadratic summation of Gessel and Stanton.

Before we present our new results, to make this paper more self-contained, we
briefly review some important material from the theory of elliptic
hypergeometric series. Afterwards we turn to the Askey–Wilson operator and its
elliptic extension, and then we provide our new results.

###  1.1 Elliptic hypergeometric series

For basic hypergeometric series, see Gasper and Rahman’s textbook [9].
Elliptic hypergeometric series are treated there in Chapter 11.

By definition, a function is elliptic if it is meromorphic and doubly
periodic. It is well known (cf., e.g., [25]) that elliptic functions can be
built from quotients of theta functions.

As building blocks we will use the _modified Jacobi theta function_ with
argument $x$ and nome $p$, defined (in multiplicative notation) by

$\displaystyle\theta(x;p)=\prod_{j\geq
0}\big{(}\big{(}1-p^{j}x\big{)}\big{(}1-p^{j+1}/x\big{)}\big{)},\qquad\theta(x_{1},\dots,x_{m};p)=\prod_{k=1}^{m}\theta(x_{k};p),$

where $x,x_{1},\dots,x_{m}\neq 0$, $|p|<1$.

The modified Jacobi theta functions satisfy the following basic properties
which are essential in the theory of elliptic hypergeometric series:

$\displaystyle\theta(x;p)=-x\theta(1/x;p),$ (1.1a)
$\displaystyle\theta(px;p)=-\frac{1}{x}\theta(x;p),$ (1.1b) and the addition
formula
$\displaystyle\theta(xy,x/y,uv,u/v;p)-\theta(xv,x/v,uy,u/y;p)=\frac{u}{y}\,\theta(yv,y/v,xu,x/u;p)$
(1.1c)

(cf. [26, p. 451, Example 5]).

Note that in the theta function $\theta(x;p)$ we cannot let $x\to 0$ (unless
we first let $p\to 0$) for $x$ is a pole of infinite order.

Further, we define the theta shifted factorial (or $q,p$-shifted factorial) by

$\displaystyle(a;q,p)_{n}=\begin{cases}\displaystyle\prod^{n-1}_{k=0}\theta\big{(}aq^{k};p\big{)},&\quad
n=1,2,\ldots,\\\ 1,&\quad n=0,\\\ \displaystyle
1/\prod^{-n-1}_{k=0}\theta\big{(}aq^{n+k};p\big{)},&\quad
n=-1,-2,\ldots,\end{cases}$

together with

$\displaystyle(a_{1},a_{2},\ldots,a_{m};q,p)_{n}=\prod^{m}_{k=1}(a_{k};q,p)_{n},$

for compact notation. For $p=0$ we have $\theta(x;0)=1-x$ and, hence,
$(a;q,0)_{n}=(a;q)_{n}$ is a $q$-shifted factorial in base $q$. The parameters
$q$ and $p$ in $(a;q,p)_{n}$ are called the base and nome, respectively.
Observe that

$\displaystyle(pa;q,p)_{n}=(-1)^{n}a^{-n}q^{-\binom{n}{2}}(a;q,p)_{n},$

which follows from (1.1b). A list of other useful identities for manipulating
the $q,p$-shifted factorials is given in [9, Section 11.2].

A series $\sum c_{n}$ is called an elliptic hypergeometric series if
$g(n)=c_{n+1}/c_{n}$ is an elliptic function of $n$ with $n$ considered as a
complex variable, i.e., the function $g(x)$ is a doubly periodic meromorphic
function of the complex variable $x$. Without loss of generality, by the
theory of theta functions, one may assume that

$\displaystyle
g(x)=\frac{\theta\big{(}a_{1}q^{x},a_{2}q^{x},\dots,a_{s+1}q^{x};p\big{)}}{\theta\big{(}q^{1+x},b_{1}q^{x},\dots,b_{s}q^{x};p\big{)}}z,$

where the elliptic balancing condition, namely

$\displaystyle a_{1}a_{2}\cdots a_{s+1}=qb_{1}b_{2}\cdots b_{s},$

holds. If we write $q=e^{2\pi i\sigma}$, $p=e^{2\pi i\tau}$, with complex
$\sigma$, $\tau$, then $g(x)$ is indeed periodic in $x$ with periods
$\sigma^{-1}$ and $\tau\sigma^{-1}$.

For convergence reasons, one usually requires $a_{s+1}=q^{-n}$ ($n$ being a
nonnegative integer), so that the sum of an elliptic hypergeometric series is
in fact finite.

Very-well-poised elliptic hypergeometric series are defined as

$\displaystyle{}_{s+1}V_{s}(a_{1};a_{6},\dots,a_{s+1};q,p):=\sum_{k=0}^{\infty}\frac{\theta\big{(}a_{1}q^{2k};p\big{)}}{\theta(a_{1};p)}\frac{(a_{1},a_{6},\dots,a_{s+1};q,p)_{k}}{(q,a_{1}q/a_{6},\dots,a_{1}q/a_{s+1};q,p)_{k}}(qz)^{k},$

where

$\displaystyle q^{2}a_{6}^{2}a_{7}^{2}\cdots a_{s+1}^{2}=(a_{1}q)^{s-5}.$

Note that in the elliptic case the number of pairs of numerator and
denominator parameters involved in the construction of the very-well-poised
term $\theta(a_{1}q^{2k};p)/\theta(a_{1};p)$ is four (whereas in the basic
case this number is two, in the ordinary case only one). See Spiridonov [23]
or Gasper and Rahman [9, Chapter 11] for details.

In their study of elliptic $6j$ symbols (which are elliptic solutions of the
Yang–Baxter equation found by Baxter [2] and Date et al. [7]), Frenkel and
Turaev [8] discovered the following ${}_{10}V_{9}$ summation formula (as a
result of a more general ${}_{12}V_{11}$ transformation, being a consequence
of the tetrahedral symmetry of the elliptic $6j$ symbols):

$\displaystyle{}_{10}V_{9}\big{(}a;b,c,d,e,q^{-n};q,p\big{)}=\frac{(aq,aq/bc,aq/bd,aq/cd;q,p)_{n}}{(aq/b,aq/c,aq/d,aq/bcd;q,p)_{n}},$
(1.2)

where $a^{2}q^{n+1}=bcde$. The ${}_{10}V_{9}$ summation is an elliptic
analogue of Jackson’s ${}_{8}\phi_{7}$ summation formula (cf. [9, equation
(2.6.2)])

$\displaystyle\sum_{k=0}^{n}\frac{(1-aq^{2k})(a,b,c,d,e,q^{-n};q)_{k}}{(1-a)(q,aq/b,aq/c,aq/d,aq/e,aq^{n+1};q)_{k}}q^{k}=\frac{(aq,aq/bc,aq/bd,aq/cd;q)_{n}}{(aq/b,aq/c,aq/d,aq/bcd;q)_{n}},$
(1.3)

where $a^{2}q^{n+1}=bcde$, which in turn is a $q$-analogue of Dougall’s
${}_{7}F_{6}$ summation formula.

###  1.2 The Askey–Wilson operator

The Askey–Wilson operator $\mathcal{D}_{q}$ was first defined in [1]. We
consider meromorphic functions $f(z)$ symmetric in $z$ and $1/z$. Writing
$z=e^{i\theta}$ (note that $\theta$ need not to be real), we may consider $f$
to be a function in $x=\cos\theta=(z+1/z)/2$ and write $f[x]:=f(z)$. (I.e.,
$f$ can be considered as a function in $z$, or equivalently, as a function in
$x$, where the two different notations specify the dependency to be
considered.)

The Askey–Wilson operator acts on functions of $x=\cos\theta$. It is defined
as follows:

$\displaystyle\mathcal{D}_{q}f[x]=\frac{f\big{(}q^{\frac{1}{2}}z\big{)}-f\big{(}q^{-\frac{1}{2}}z\big{)}}{\iota\big{(}q^{\frac{1}{2}}z\big{)}-\iota\big{(}q^{-\frac{1}{2}}z\big{)}},$
(1.4)

where $\iota[x]=x$ (i.e., $\iota(z)=(z+1/z)/2$). Equation (1.4) can also be
written as

$\displaystyle\mathcal{D}_{q}f[x]=\frac{f\big{(}q^{\frac{1}{2}}z\big{)}-f\big{(}q^{-\frac{1}{2}}z\big{)}}{i\big{(}q^{\frac{1}{2}}-q^{-\frac{1}{2}}\big{)}\sin\theta}.$

The operator $\mathcal{D}_{q}$ is a $q$-analogue of the differentiation
operator (which is different to Jackson’s $q$-difference operator). In
particular, since

$\displaystyle\mathcal{D}_{q}T_{n}[x]=\frac{q^{\frac{n}{2}}-q^{-\frac{n}{2}}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}U_{n-1}[x],$

where $T_{n}[\cos\theta]=\cos n\theta$ and
$U_{n}[\cos\theta]=\sin(n+1)\theta/\sin\theta$ are the Chebyshev polynomials
of the first and second kind, one easily sees that $\mathcal{D}_{q}$ maps
polynomials to polynomials, lowering the degree by one.

In the calculus of the Askey–Wilson operator the so-called “Askey–Wilson
monomials”$\phi_{n}(x;a)=(az,a/z;q)_{n}$ form a natural basis for polynomials
or power series in $x$. One readily computes

$\displaystyle\mathcal{D}_{q}(az,a/z;q)_{n}=-\frac{2a(1-q^{n})}{(1-q)}\big{(}aq^{\frac{1}{2}}z,aq^{\frac{1}{2}}/z;q\big{)}_{n-1}.$

Ismail [11] proved the following Taylor theorem for polynomials $f[x]$.

######  Theorem 1.1.

If $f[x]$ is a polynomial in $x$ of degree $n$, then

$\displaystyle f[x]=\sum_{k=0}^{n}f_{k}\phi_{k}(x;a),$

where

$\displaystyle
f_{k}=\frac{(q-1)^{k}}{(2a)^{k}(q;q)_{k}}q^{-k(k-1)/4}\big{[}\mathcal{D}_{q}^{k}f[x]\big{]}_{x=x_{k}},\qquad
x_{k}:=\frac{1}{2}\big{(}aq^{\frac{k}{2}}+q^{-\frac{k}{2}}/a\big{)}.$

As it was shown in [11], the application of Theorem 1.1 to
$f(z)=(bz,b/z;q)_{n}$ immediately gives the $q$-Pfaff–Saalschütz summation
(cf. [9, equation (1.7.2)]), in the form

$\displaystyle\frac{(bz,b/z;q)_{n}}{(ba,b/a;q)_{n}}={}_{3}\phi_{2}\\!\left[\begin{matrix}az,a/z,q^{-n}\\\
ab,q^{1-n}a/b\end{matrix};q,q\right],$

and its application to the Askey–Wilson polynomials,

$\displaystyle\omega_{n}(x;a,b,c,d;q):={}_{4}\phi_{3}\\!\left[\begin{matrix}az,a/z,abcdq^{n-1},q^{-n}\\\
ab,ac,ad\end{matrix};q,q\right],$

gives a connection coefficient identity which, by specialization, can be
reduced to the Sears transformation (cf. [9, equation (3.2.1)]), in the form

$\displaystyle\omega_{n}(x;a,b,c,d;q)=\frac{a^{n}(bc,bd;q)_{n}}{b^{n}(ac,ad;q)_{n}}\,\omega_{n}(x;b,a,c,d;q).$

Ismail and Stanton [13] extended the above polynomial Taylor theorem to hold
for entire functions of exponential growth, resulting in infinite Taylor
expansions. Marco and Parcet [15] extended this yet further to hold for
arbitrary $q$-differentiable functions, resulting in infinite Taylor
expansions with explicit remainder term. Among other results they were able to
recover the nonterminating $q$-Pfaff–Saalschütz summation (cf. [9, Appendix
(II.24)]).

###  1.3 The well-poised and elliptic Askey–Wilson operator

Since

$\displaystyle\mathcal{D}_{q}\frac{(az,a/z;q)_{n}}{(cz,c/z;q)_{n}}=\frac{2}{\big{(}q^{\frac{1}{2}}-q^{-\frac{1}{2}}\big{)}(z-1/z)}\left[\frac{\big{(}aq^{\frac{1}{2}}z,aq^{-\frac{1}{2}}/z;q\big{)}_{n}}{\big{(}cq^{\frac{1}{2}}z,cq^{-\frac{1}{2}}/z;q\big{)}_{n}}-\frac{\big{(}aq^{-\frac{1}{2}}z,aq^{\frac{1}{2}}/z;q\big{)}_{n}}{\big{(}cq^{-\frac{1}{2}}z,cq^{\frac{1}{2}}/z;q\big{)}_{n}}\right]$
$\displaystyle\qquad{}=\frac{2}{\big{(}q^{\frac{1}{2}}-q^{-\frac{1}{2}}\big{)}(z-1/z)}\frac{\big{(}aq^{\frac{1}{2}}z,aq^{\frac{1}{2}}/z;q\big{)}_{n-1}}{\big{(}cq^{\frac{1}{2}}z,cq^{\frac{1}{2}}/z;q\big{)}_{n-1}}$
$\displaystyle\qquad\quad{}\times\left[\frac{\big{(}1-azq^{n-\frac{1}{2}}\big{)}\big{(}1-aq^{-\frac{1}{2}}/z\big{)}}{\big{(}1-czq^{n-\frac{1}{2}}\big{)}\big{(}1-cq^{-\frac{1}{2}}/z\big{)}}-\frac{\big{(}1-azq^{-\frac{1}{2}}\big{)}\big{(}1-aq^{n-\frac{1}{2}}/z\big{)}}{\big{(}1-czq^{-\frac{1}{2}}\big{)}\big{(}1-cq^{n-\frac{1}{2}}/z\big{)}}\right]$
$\displaystyle\qquad{}=\frac{(-1)2a(1-c/a)\big{(}1-acq^{n-1}\big{)}(1-q^{n})}{\big{(}1-czq^{-\frac{1}{2}}\big{)}\big{(}1-czq^{\frac{1}{2}}\big{)}\big{(}1-cq^{-\frac{1}{2}}/z\big{)}\big{(}1-cq^{\frac{1}{2}}/z\big{)}(1-q)}\frac{\big{(}aq^{\frac{1}{2}}z,aq^{\frac{1}{2}}/z;q\big{)}_{n-1}}{\big{(}cq^{\frac{3}{2}}z,cq^{\frac{3}{2}}/z;q\big{)}_{n-1}},$

we were led in [21] to define a $c$-generalized well-poised Askey–Wilson
operator acting on $x$ (or $z$) by

$\displaystyle\mathcal{D}_{c,q}=\big{(}1-czq^{-\frac{1}{2}}\big{)}\big{(}1-czq^{\frac{1}{2}}\big{)}\big{(}1-cq^{-\frac{1}{2}}/z\big{)}\big{(}1-cq^{\frac{1}{2}}/z\big{)}\mathcal{D}_{q},$

which acts “degree-lowering” on the “rational monomials” (or “well-poised
monomials”)

$\displaystyle\frac{(az,a/z;q)_{n}}{(cz,c/z;q)_{n}}$

in the form

$\displaystyle\mathcal{D}_{c,q}\frac{(az,a/z;q)_{n}}{(cz,c/z;q)_{n}}=\frac{(-1)2a(1-c/a)\big{(}1-acq^{n-1}\big{)}(1-q^{n})}{(1-q)}\frac{\big{(}aq^{\frac{1}{2}}z,aq^{\frac{1}{2}}/z;q\big{)}_{n-1}}{\big{(}cq^{\frac{3}{2}}z,cq^{\frac{3}{2}}/z;q\big{)}_{n-1}}.$

Clearly, $\mathcal{D}_{0,q}=\mathcal{D}_{q}$.

More generally, for parameters $c$, $q$, $p$ with $|q|,|p|<1$, we defined an
elliptic extension of the Askey–Wilson operator, acting on functions symmetric
in $z^{\pm 1}$, by

$\displaystyle\mathcal{D}_{c,q,p}f(z)=2q^{\frac{1}{2}}z\frac{\theta\big{(}czq^{-\frac{1}{2}},czq^{\frac{1}{2}},cq^{-\frac{1}{2}}/z,cq^{\frac{1}{2}}/z;p\big{)}}{\theta(q,z^{2};p)}\big{(}f\big{(}q^{\frac{1}{2}}z\big{)}-f\big{(}q^{-\frac{1}{2}}z\big{)}\big{)}.$
(1.5)

Note that $\mathcal{D}_{c,q,0}=\mathcal{D}_{c,q}$.

In particular, using (1.1c), we have

$\displaystyle\mathcal{D}_{c,q,p}\frac{(az,a/z;q,p)_{n}}{(cz,c/z;q,p)_{n}}=\frac{(-1)2a\theta\big{(}c/a,acq^{n-1},q^{n};p\big{)}}{\theta(q;p)}\frac{\big{(}aq^{\frac{1}{2}}z,aq^{\frac{1}{2}}/z;q,p\big{)}_{n-1}}{\big{(}cq^{\frac{3}{2}}z,cq^{\frac{3}{2}}/z;q,p\big{)}_{n-1}}.$
(1.6)

######  Remark 1.2.

The operator $\mathcal{D}_{c,q,p}$ happens to be a special case of a
multivariable difference operator introduced by Rains in [16]. Already in the
single variable case Rains’ operator involves two more parameters than
$\mathcal{D}_{c,q,p}$. (Rains’ difference operators generate a representation
of the Sklyanin algebra, as observed in [16] and made explicit in [18] and
[19, Section 6].) Rains’ operator can be specialized to act as degree-lowering
(as the above $\mathcal{D}_{c,q,p}$ does), degree-preserving or degree-raising
on abelian functions. Rains used his multivariable difference operators in
[16] to construct $BC_{n}$-symmetric biorthogonal abelian functions which
generalize Koornwinder’s orthogonal polynomials. He further used his operator
in [17] to derive $BC_{n}$-symmetric extensions of Frenkel and Turaev’s
${}_{10}V_{9}$ summation and ${}_{12}V_{11}$ transformation.

##  2 Elliptic Taylor expansions and interpolation

We work in the following space of abelian functions.

For a complex number $c$, let

$\displaystyle
W_{c}^{n}:=\text{span}_{\mathbb{C}}\left\\{\frac{g_{k}(z)}{(cz,c/z;q,p)_{k}},\,0\leq
k\leq n\right\\},$

where $g_{k}(z)$ runs over all functions being holomorphic for $z\neq 0$ with
$g_{k}(z)=g_{k}(1/z)$ and

$\displaystyle g_{k}(pz)=\frac{1}{p^{k}z^{2k}}g_{k}(z).$

In classical terminology, $g_{k}(z)$ is an even theta function of order $2k$
and zero characteristic. Rains [17] refers to such functions as $BC_{1}$ theta
functions of degree $k$, whereas in Rosengren and Schlosser [20] they are
referred to as $D_{k}$ theta functions. It is well-known that the space
$V^{k}$ of even theta functions of order $2k$ and zero characteristic has
dimension $k+1$ (see, e.g., Weber [25, p. 49]).

Note that $W_{c}^{n}$ consists of certain abelian functions. (For $p\to 0$
these degenerate to certain rational functions which we may call “well-
poised”.)

######  Lemma 2.1 ([21, Lemma 4.1]).

For any arbitrary but fixed complex number $a$ $($satisfying $a\neq
cq^{j}p^{k}$, for $j=0,\dots,n-1$, and $k\in\mathbb{Z}$, and $a\neq
q^{j}p^{k}/c$, for $j=2-2n,\dots,1-n$, and $k\in\mathbb{Z})$, the set

$\displaystyle\left\\{\frac{(az,a/z;q,p)_{k}}{(cz,c/z;q,p)_{k}},\,0\leq k\leq
n\right\\}$

forms a basis for $W_{c}^{n}$.

Note that, in view of (1.6), the elliptic Askey–Wilson operator maps functions
in $W_{c}^{n}$ to functions in $W_{cq^{\frac{3}{2}}}^{n-1}$.

We now define

$\displaystyle\mathcal{D}_{c,q,p}^{(k)}=\mathcal{D}_{cq^{\frac{3}{2}},q,p}^{(k-1)}\mathcal{D}_{c,q,p},$

with $\mathcal{D}_{c,q,p}^{(0)}=\varepsilon$, the identity operator. We have
the following elliptic Taylor expansion theorem which extends Theorem 1.1 of
Ismail.

######  Theorem 2.2 ([21, Theorem 4.2]).

If $f$ is in $W_{c}^{n}$, then

$\displaystyle
f(z)=\sum_{k=0}^{n}f_{k}\frac{(az,a/z;q,p)_{k}}{(cz,c/z;q,p)_{k}},$

where

$\displaystyle
f_{k}=\frac{(-1)^{k}q^{-k(k-1)/4}\theta(q;p)^{k}}{(2a)^{k}(q,c/a,acq^{k-1};q,p)_{k}}\big{[}\mathcal{D}_{c,q,p}^{(k)}f(z)\big{]}_{z=aq^{\frac{k}{2}}}.$

######  Example 2.3.

Let

$\displaystyle f(z)=\frac{(bz,b/z;q,p)_{n}}{(cz,c/z;q,p)_{n}}.$

Application of Theorem 2.2 in conjunction with (1.6) gives

$\displaystyle
f_{k}=\frac{(-1)^{k}q^{-k(k-1)/4}\theta(q;p)^{k}}{(2a)^{k}(q,c/a,acq^{k-1};q,p)_{k}}$
$\displaystyle\hphantom{f_{k}=}{}\times(-1)^{k}(2b)^{k}q^{k(k-1)/4}\frac{(q;q,p)_{n}\big{(}c/b,bcq^{n-1};q,p\big{)}_{k}}{(q;q,p)_{n-k}\theta(q;p)^{k}}\frac{\big{(}abq^{k},b/a;q,p\big{)}_{n-k}}{\big{(}acq^{2k},cq^{k}/a;q,p\big{)}_{n-k}}$
$\displaystyle\hphantom{f_{k}}{}=\frac{(ab,b/a;q,p)_{n}}{(ac,c/a;q,p)_{n}}\frac{\theta\big{(}acq^{2k-1};p\big{)}}{\theta\big{(}acq^{-1};p\big{)}}\frac{\big{(}acq^{-1},c/b,bcq^{n-1},q^{-n};q,p\big{)}_{k}}{\big{(}q,ab,aq^{1-n}/b,acq^{n};q,p\big{)}_{k}}q^{k},$

thus yielding Frenkel and Turaev’s ${}_{10}V_{9}$ summation (1.2), in the form

$\displaystyle\frac{(ac,c/a,bz,b/z;q,p)_{n}}{(ab,b/a,cz,c/z;q,p)_{n}}={}_{10}V_{9}\big{(}acq^{-1};az,a/z,c/b,bcq^{n-1},q^{-n};q,p\big{)}.$

We now prove an elliptic extension of a theorem of S. Cooper [5] which
explicitly describes the action of the $m$-iterated Askey–Wilson operator.

######  Theorem 2.4.

The action of $\mathcal{D}_{c,q,p}^{(m)}$ on a function $f\in W_{c}^{n}$ is
given by

$\displaystyle\mathcal{D}_{c,q,p}^{(m)}f(z)=(-2z)^{m}q^{\frac{m(3-m)}{4}}\frac{\big{(}cq^{\frac{m}{2}-1}z,cq^{\frac{m}{2}-1}/z;q,p\big{)}_{m+1}}{(\theta(q;p))^{m}}$
(2.1)
$\displaystyle\hphantom{\mathcal{D}_{c,q,p}^{(m)}f(z)=}{}\times\sum_{k=0}^{m}q^{k(m-k)}\begin{bmatrix}m\\\
k\end{bmatrix}_{p,q}\frac{z^{2(k-m)}\big{(}cq^{\frac{m}{2}-k}z,cq^{-\frac{m}{2}+k}/z;q,p\big{)}_{m-1}}{\big{(}q^{m-2k+1}z^{2};q,p\big{)}_{k}\big{(}q^{2k-m+1}z^{-2};q,p\big{)}_{m-k}}f\big{(}q^{\frac{m}{2}-k}z\big{)},$

where

$\displaystyle\begin{bmatrix}m\\\
k\end{bmatrix}_{p,q}=\frac{\big{(}q^{1+k};q,p\big{)}_{m-k}}{(q;q,p)_{m-k}}.$

###### Proof.

We prove this by induction. If $m=1$, then (2.1) just reduces to the
definition of $\mathcal{D}_{c,q,p}f(z)$ in (1.5). Now say (2.1) holds up to
some $m$. Then if we let $f^{(m)}(z):=\mathcal{D}_{c,q,p}^{(m)}f(z)$,

$\displaystyle\mathcal{D}_{c,q,p}^{(m+1)}f(z)=\mathcal{D}_{cq^{\frac{3}{2}m},q,p}f^{(m)}(z)$
$\displaystyle{}=2q^{\frac{1}{2}}z\frac{\big{(}czq^{\frac{3}{2}m-\frac{1}{2}},cq^{\frac{3}{2}m-\frac{1}{2}}/z;q,p\big{)}_{2}}{\theta(q,z^{2};p)}\big{(}f^{(m)}\big{(}q^{\frac{1}{2}}z\big{)}-f^{(m)}\big{(}q^{-\frac{1}{2}}z\big{)}\big{)}$
$\displaystyle{}=2q^{\frac{1}{2}}z\frac{\big{(}czq^{\frac{3}{2}m-\frac{1}{2}},cq^{\frac{3}{2}m-\frac{1}{2}}/z;q,p\big{)}_{2}}{\theta(q,z^{2};p)}\frac{(-2z)^{m}q^{\frac{m(3-m)}{4}}}{(\theta(q;p))^{m}}\Bigg{\\{}q^{\frac{m}{2}}\big{(}cq^{\frac{m}{2}-\frac{1}{2}}z,cq^{\frac{m}{2}-\frac{3}{2}}/z;q,p\big{)}_{m+1}$
$\displaystyle\quad{}\times\sum_{k=0}^{m}q^{(k-1)(m-k)}\begin{bmatrix}m\\\
k\end{bmatrix}_{p,q}\frac{z^{2(k-m)}\big{(}cq^{\frac{m}{2}-k+\frac{1}{2}}z,cq^{-\frac{m}{2}+k-\frac{1}{2}}/z;q,p\big{)}_{m-1}}{\big{(}q^{m-2k+2}z^{2};q,p\big{)}_{k}\big{(}q^{2k-m}z^{-2};q,p\big{)}_{m-k}}f\big{(}q^{\frac{m}{2}-k+\frac{1}{2}}z\big{)}$
$\displaystyle\quad{}-q^{-\frac{m}{2}}\big{(}cq^{\frac{m}{2}-\frac{3}{2}}z,cq^{\frac{m}{2}-\frac{1}{2}}/z;q,p\big{)}_{m+1}$
$\displaystyle\quad{}\times\sum_{k=0}^{m}q^{(k+1)(m-k)}\begin{bmatrix}m\\\
k\end{bmatrix}_{p,q}\frac{z^{2(k-m)}\big{(}cq^{\frac{m}{2}-k-\frac{1}{2}}z,cq^{-\frac{m}{2}+k+\frac{1}{2}}/z;q,p\big{)}_{m-1}}{\big{(}q^{m-2k}z^{2};q,p\big{)}_{k}\big{(}q^{2k-m+2}z^{-2};q,p\big{)}_{m-k}}f\big{(}q^{\frac{m}{2}-k-\frac{1}{2}}z\big{)}\Bigg{\\}}$
$\displaystyle{}=(2z)^{m+1}q^{\frac{m(3-m)}{4}+\frac{m+1}{2}}(-1)^{m}\frac{\big{(}czq^{\frac{3}{2}m-\frac{1}{2}},cq^{\frac{3}{2}m-\frac{1}{2}}/z;q,p\big{)}_{2}}{(\theta(q;p))^{m+1}\theta(z^{2};p)}\big{(}cq^{\frac{m}{2}-\frac{1}{2}}z,cq^{\frac{m}{2}-\frac{1}{2}}/z;q,p\big{)}_{m}$
$\displaystyle\quad{}\times\left\\{\theta\big{(}cq^{\frac{3}{2}m-\frac{1}{2}}z,cq^{\frac{m}{2}-\frac{3}{2}}/z;p\big{)}\right.$
$\displaystyle\quad{}\times\sum_{k=0}^{m}q^{(k-1)(m-k)}\begin{bmatrix}m\\\
k\end{bmatrix}_{p,q}\frac{z^{2(k-m)}\big{(}cq^{\frac{m}{2}-k+\frac{1}{2}}z,cq^{-\frac{m}{2}+k-\frac{1}{2}}/z;q,p\big{)}_{m-1}}{\big{(}q^{m-2k+2}z^{2};q,p\big{)}_{k}\big{(}q^{2k-m}z^{-2};q,p\big{)}_{m-k}}f\big{(}q^{\frac{m}{2}-k+\frac{1}{2}}z\big{)}$
$\displaystyle\quad{}-q^{-m}\theta\big{(}cq^{\frac{m}{2}-\frac{3}{2}}z,cq^{\frac{3}{2}m-\frac{1}{2}}/z;q,p\big{)}$
$\displaystyle\left.\quad{}\times\sum_{k=1}^{m+1}q^{k(m-k+1)}\\!\begin{bmatrix}m\\\
k-1\end{bmatrix}_{p,q}\\!\frac{z^{2(k-m-1)}\big{(}cq^{\frac{m}{2}-k+\frac{1}{2}}z,cq^{-\frac{m}{2}+k-\frac{1}{2}}/z;q,p\big{)}_{m-1}}{\big{(}q^{m-2k+2}z^{2};q,p\big{)}_{k-1}\big{(}q^{2k-m}z^{-2};q,p\big{)}_{m-k+1}}f\big{(}q^{\frac{m}{2}-k+\frac{1}{2}}z\big{)}\right\\}$
$\displaystyle=(2z)^{m+1}q^{\frac{m(3-m)}{4}+\frac{m+1}{2}}(-1)^{m}\frac{\big{(}cq^{\frac{m}{2}-\frac{1}{2}}z,cq^{\frac{m}{2}-\frac{1}{2}}/z;q,p\big{)}_{m+2}}{(\theta(q;p))^{m+1}\theta\big{(}z^{2};p\big{)}}$
$\displaystyle\quad{}\times\left\\{\theta\big{(}cq^{\frac{3}{2}m-\frac{1}{2}}z,cq^{\frac{m}{2}-\frac{3}{2}}/z;p\big{)}\times\frac{q^{-m}z^{-2m}\big{(}cq^{\frac{m}{2}+\frac{1}{2}}z,cq^{-\frac{m}{2}-\frac{1}{2}}/z;q,p\big{)}_{m-1}}{\big{(}q^{-m}z^{-2};q,p\big{)}_{n}}f\big{(}q^{\frac{m}{2}+\frac{1}{2}}z\big{)}\right.$
$\displaystyle\quad{}-q^{-m}\theta\big{(}cq^{\frac{m}{2}-\frac{3}{2}}z,cq^{\frac{3}{2}m-\frac{1}{2}}/z;p\big{)}\times\frac{\big{(}cq^{-\frac{m}{2}-\frac{1}{2}}z,cq^{\frac{m}{2}+\frac{1}{2}}/z;q,p\big{)}_{m-1}}{\big{(}q^{-m}z^{2};q,p\big{)}_{m}}f\big{(}q^{-\frac{m}{2}-\frac{1}{2}}z\big{)}$
$\displaystyle\quad{}+q^{-m}\sum_{k=1}^{m}q^{k(m-k+1)}\frac{\big{(}q^{1+k};q,p\big{)}_{m-k}}{(q;q,p)_{m-k+1}}\frac{\big{(}cq^{\frac{m}{2}-k+\frac{1}{2}}z,cq^{-\frac{m}{2}+k-\frac{1}{2}}/z;q,p\big{)}_{m-1}f\big{(}q^{\frac{m}{2}-k+\frac{1}{2}}z\big{)}}{\big{(}q^{m-2k+2}z^{2};q,p\big{)}_{k}\big{(}q^{2k-m}z^{-2};q,p\big{)}_{m-k+1}}$
$\displaystyle\quad{}\times
z^{2k-2m}\left[\theta\big{(}q^{m-k+1},cq^{\frac{3}{2}m-\frac{1}{2}}z,cq^{\frac{m}{2}-\frac{3}{2}}/z,q^{k}z^{-2};p\big{)}\right.$
$\displaystyle\left.\left.\quad{}-z^{-2}\theta\big{(}q^{k},cq^{\frac{m}{2}-\frac{3}{2}}z,cq^{\frac{3}{2}m-\frac{1}{2}}/z,q^{m-k+1}z^{2};p\big{)}\right]\right\\}$
$\displaystyle{}=(-2z)^{m+1}q^{\frac{(m+1)(2-m)}{4}}\frac{\big{(}cq^{\frac{m}{2}-\frac{1}{2}}z,cq^{\frac{m}{2}-\frac{1}{2}}/z;q,p\big{)}_{m+2}}{(\theta(q;p))^{m+1}}$
$\displaystyle\quad{}\times\sum_{k=0}^{m+1}q^{k(m+1-k)}\begin{bmatrix}m+1\\\
k\end{bmatrix}_{p,q}\frac{z^{2(k-m-1)}\big{(}cq^{\frac{m+1}{2}-k}z,cq^{-\frac{m+1}{2}+k}/z;q,p\big{)}_{m}}{\big{(}q^{m-2k+2}z^{2};q,p\big{)}_{k}\big{(}q^{2k-m}z^{-2};q,p\big{)}_{m+1-k}}f\big{(}q^{\frac{m+1}{2}-k}z\big{)}.$

Hence the theorem is proved. Notice that in the last step we used the addition
formula (1.1c) with the substitutions

$\displaystyle(x,y,u,v)\mapsto\big{(}cq^{m-1},q^{\frac{m+1}{2}}z,q^{\frac{m+1}{2}}/z,q^{\frac{m+1}{2}-k}z\big{)}$

to simplify the summand. ∎

######  Remark 2.5.

Our formula in Theorem 2.4 involves the $c$-dependent elliptic Askey–Wilson
operator ${\cal D}^{(m)}_{c,q,p}$. Independently, Ismail, Rains and Stanton
[12, Proposition 8.1] also gave an elliptic extension of Cooper’s result.
Although at first glance Ismail, Rains and Stanton’s result, which involves a
slightly different operator (without the denominator variable $c$), looks
different from ours in Theorem 2.4, the two results are indeed equivalent up
to a multiplication of operators. (We would like to thank Eric Rains for
helping us to clarify this.) In fact, the operator

$\displaystyle\big{(}q^{\frac{m}{2}-1}cz,q^{\frac{m}{2}-1}c/z;q,p\big{)}_{m+1}^{-1}{\cal
D}^{(m)}_{c,q,p}(cz,c/z;q,p)_{m-1}^{-1}$

(which acts on functions $h(z)=g(z)/(czq^{m-1},cq^{m-1}/z;q,p)_{n-m+1}$ with
$g(z)=g(1/z)$ and $g(pz)=p^{-n}z^{-2n}g(z)$) is independent of $c$, thus our
operator ${\cal D}^{(m)}_{c,q,p}$ is proportional to

$\displaystyle
v^{2m}\theta\big{(}q^{\frac{m}{2}}vz,q^{\frac{m}{2}}v/z;q\big{)}\big{(}q^{\frac{m}{2}-1}cz,q^{\frac{m}{2}-1}c/z;q,p\big{)}_{m+1}{\cal
D}_{m}(q;p)(cz,c/z;q,p)_{m-1}\theta(vz,v/z;q)^{-1}\\!,$

with ${\cal D}_{m}(q;p)$ from [12, Section 8], where the constant of
proportionality is independent of $v$.

We are now able to obtain an elliptic extension of Ismail and Stanton’s [14,
Theorem 3.4] interpolation formula. In particular, for any $a\in\mathbb{C}$,
the function $f\in W_{c}^{n}$ is uniquely determined by its evaluation at the
$n+1$ interpolation points $a,aq,\dots,aq^{n}$, with closed form coefficients.

######  Theorem 2.6.

If $f$ is in $W_{c}^{n}$, then

$\displaystyle\frac{\big{(}a^{2}q,q,cz,c/z;q,p\big{)}_{n}}{(ac,c/a,aqz,aq/z;q,p)_{n}}f(z)$
$\displaystyle\qquad{}=\sum_{k=0}^{n}q^{k}\frac{\theta\big{(}a^{2}q^{2k};p\big{)}}{\theta(a^{2};p)}\frac{\big{(}q^{-n},a^{2},aq/c,acq^{n},az,a/z;q,p\big{)}_{k}}{\big{(}q,a^{2}q^{n+1},ac,aq^{1-n}/c,aqz,aq/z;q,p\big{)}_{k}}f\big{(}aq^{k}\big{)}.$

###### Proof.

By combining Theorem 2.2 and Theorem 2.4, we find that

$\displaystyle
f(z)=\sum_{k=0}^{n}q^{k}\theta\big{(}cq^{-1}/a,acq^{2k-1};p\big{)}\frac{(az,a/z;q,p)_{k}}{(cz,c/z;q,p)_{k}}$
$\displaystyle\hphantom{f(z)=}{}\times\sum_{j=0}^{k}q^{-(k-j)^{2}}a^{2(j-k)}\frac{\big{(}acq^{k-j},cq^{-k+j}/a;q,p\big{)}_{k-1}}{\big{(}q,a^{2}q^{2k-2j+1};q,p\big{)}_{j}\big{(}q,a^{-2}q^{2j-2k+1};q,p\big{)}_{k-j}}f\big{(}aq^{k-j}\big{)}.$

By shifting the index $k\mapsto k+j$, we get

$\displaystyle
f(z)=\sum_{k=0}^{n}\sum_{j=0}^{n-k}q^{k+j}\theta\big{(}cq^{-1}/a,acq^{2k+2j-1};p\big{)}\frac{(az,a/z;q,p)_{k+j}}{(cz,c/z;q,p)_{k+j}}$
$\displaystyle\hphantom{f(z)=}{}\times
q^{-k^{2}}a^{-2k}\frac{\big{(}acq^{k},cq^{-k}/a;q,p\big{)}_{k+j-1}}{\big{(}q,a^{2}q^{2k+1};q,p\big{)}_{j}\big{(}q,a^{-2}q^{-2k+1};q,p\big{)}_{k}}f\big{(}aq^{k}\big{)}$
$\displaystyle\hphantom{f(z)}{}=\sum_{k=0}^{n}q^{k(1-k)}a^{-2k}\frac{\big{(}acq^{k},cq^{-k}/a,az,a/z;q,p\big{)}_{k}}{\big{(}q,a^{-2}q^{-2k+1},cz,c/z;q,p\big{)}_{k}}f\big{(}aq^{k}\big{)}$
$\displaystyle\hphantom{f(z)=}{}\times\sum_{j=0}^{n-k}q^{j}\frac{\theta\big{(}acq^{2k+2j-1};p\big{)}}{\theta\big{(}acq^{2k-1};p\big{)}}\frac{\big{(}acq^{2k-1},cq^{-1}/a,aq^{k}z,aq^{k}/z,q^{-n+k},acq^{n+k};q,p\big{)}_{j}}{\big{(}q,a^{2}q^{2k+1},cq^{k}/z,cq^{k}z,q^{-n+k},acq^{n+k};q,p\big{)}_{j}}$
$\displaystyle\hphantom{f(z)}{}=\sum_{k=0}^{n}q^{k(1-k)}a^{-2k}$
$\displaystyle\hphantom{f(z)=}{}\times\frac{\big{(}acq^{k},cq^{-k}/a,az,a/z;q,p\big{)}_{k}\big{(}acq^{2k},aq^{k+1}/z,aq^{k+1}z,c/a;q,p\big{)}_{n-k}}{\big{(}q,a^{-2}q^{-2k+1},cz,c/z;q,p\big{)}_{k}\big{(}q,a^{2}q^{2k+1},cq^{k}/z,cq^{k}z;q,p\big{)}_{n-k}}f\big{(}aq^{k}\big{)}.$

The last sum was obtained by virtue of the Frenkel and Turaev summation
formula (1.2). The theorem then follows by elementary manipulations. ∎

######  Corollary 2.7.

We have the elliptic Karlsson–Minton type identity

$\displaystyle\frac{\big{(}q,a^{2}q;q,p\big{)}_{n}}{(aqz,aq/z;q,p)_{n}}\frac{(bz,b/z;q,p)_{s}(dz,d/z;q,p)_{n-s}}{(ab,b/a;q,p)_{s}(ad,d/a;q,p)_{n-s}}$
$\displaystyle\qquad{}={}_{12}V_{11}\big{(}a^{2},q^{-n},az,a/z,aq/b,aq/d,abq^{s},adq^{n-s};q,p\big{)}.$

###### Proof.

We apply Theorem 2.6 to

$\displaystyle
f(z)=\frac{(bz,b/z;q,p)_{s}(dz,d/z;q,p)_{n-s}}{(cz,c/z;q,p)_{n}}.$ ∎

More generally, we have the following result.

######  Corollary 2.8.

We have the elliptic Karlsson–Minton type identity

$\displaystyle\frac{\big{(}a^{2}q,q;q,p\big{)}_{n}}{(aqz,aq/z;q,p)_{n}}\prod_{j=1}^{n}\theta(b_{j}z,b_{j}/z;p)$
$\displaystyle\qquad{}=\sum_{k=0}^{n}q^{k(n+1)}\frac{\theta\big{(}a^{2}q^{2k};p\big{)}}{\theta\big{(}a^{2};p\big{)}}\frac{\big{(}q^{-n},a^{2},az,a/z;q,p\big{)}_{k}}{\big{(}q,a^{2}q^{n+1},aqz,aq/z;q,p\big{)}_{k}}\prod_{j=1}^{n}\theta\big{(}ab_{k}q^{k},b_{j}q^{-k}/a;p\big{)}.$

###### Proof.

We take

$\displaystyle
f(z)=\frac{\prod\limits_{j=1}^{n}\theta(b_{j}z,b_{j}/z;p)}{(cz,c/z;q,p)_{n}}$
(2.2)

and apply Theorem 2.6. ∎

######  Remark 2.9.

It should be noted that if in the proof of Corollary 2.8 we instead would have
taken

$\displaystyle
f(z)=\frac{\prod\limits_{j=1}^{t}\theta(b_{j}z,b_{j}/z;p)}{(cz,c/z;q,p)_{t}}$

for $0\leq t\leq n$, we would have just obtained the special case of Corollary
2.8 with $b_{j}\mapsto cq^{j-1}$ for $t+1\leq n$, which is clear carrying out
those specializations in (2.2).

We now consider a multivariate version of Theorem 2.6. Let us consider the
space of functions

$\displaystyle
W_{c_{1},\dots,c_{m}}^{n_{1},\dots,n_{m}}:=\operatorname{Span}_{\mathbb{C}}\Bigg{\\{}\frac{g_{k_{1},\dots,k_{m}}(z_{1},\dots,z_{m})}{\prod\limits_{i=1}^{m}(c_{i}z_{i},c_{i}/z_{i};q,p)_{k_{i}}},\,0\leq
k_{i}\leq n_{i},\,i=1,\dots,m\Bigg{\\}},$

where $g_{k_{1},\dots,k_{m}}(z_{1},\dots,z_{m})$ runs over all functions being
holomorphic in $z_{1},z_{2},\dots,z_{m}\neq 0$ and symmetric in $z_{i}$ and
$1/z_{i}$, and

$\displaystyle
g_{k_{1},\dots,k_{m}}(z_{1},\dots,pz_{i},\dots,z_{m})=\frac{1}{p^{k_{i}}z_{i}^{2k_{i}}}g_{k_{1},\dots,k_{m}}(z_{1},\dots,z_{i},\dots,z_{m}),$

for all $i=1,\dots,m$.

We define a multivariate extension of the elliptic Askey–Wilson operator as
follows.

$\displaystyle\mathcal{D}_{c_{i},q,p;z_{i}}f(z_{1},\dots,z_{m})=2q^{\frac{1}{2}}z_{i}\frac{\theta\big{(}c_{i}z_{i}q^{-\frac{1}{2}},c_{i}z_{i}q^{\frac{1}{2}},c_{i}q^{-\frac{1}{2}}/z_{i},c_{i}q^{\frac{1}{2}}/z_{i};p\big{)}}{\theta\big{(}q,z_{i}^{2};p\big{)}}$
$\displaystyle\hphantom{\mathcal{D}_{c_{i},q,p;z_{i}}f(z_{1},\dots,z_{m})=}{}\times\Big{(}f\big{(}z_{1},\dots,q^{\frac{1}{2}}z_{i},\dots,z_{m}\big{)}-f\big{(}z_{1},\dots,q^{-\frac{1}{2}}z_{i},\dots,z_{m}\big{)}\Big{)},$
$\displaystyle\mathcal{D}_{c_{i},q,p;z_{i}}^{(k+1)}=\mathcal{D}_{c_{i}q^{\frac{3}{2}k},q,p;z_{i}}\mathcal{D}_{c_{i},q,p;z_{i}}^{(k)},$

and for $\mathbf{c}=(c_{1},\dots,c_{m})$, $\mathbf{k}=(k_{1},\dots,k_{m})$,
and $\mathbf{z}=(z_{1},\dots,z_{m})$,

$\displaystyle\mathcal{D}_{\mathbf{c},q,p;\mathbf{z}}^{(\mathbf{k})}=\mathcal{D}_{c_{1},q,p;z_{1}}^{(k_{1})}\cdots\mathcal{D}_{c_{m},q,p;z_{m}}^{(k_{m})}.$

######  Theorem 2.10.

If $f(z_{1},\dots,z_{m})$ is in $W_{\mathbf{c}}^{\mathbf{n}}$, then

$\displaystyle
f(z_{1},\dots,z_{m})=\sum_{k_{1},\dots,k_{m}=0}^{n_{1},\dots,n_{m}}f_{k_{1},\dots,k_{m}}\prod_{i=1}^{m}\frac{(a_{i}z_{i},a_{i}/z_{i};q,p)_{k_{i}}}{(c_{i}z_{i},c_{i}/z_{i};q,p)_{k_{i}}},$
(2.3)

where

$\displaystyle
f_{k_{1},\dots,k_{m}}=\prod_{i=1}^{m}\frac{(-1)^{k_{i}}q^{-\frac{k_{i}(k_{i}-1)}{4}}(\theta(q;p))^{k_{i}}}{(2a_{i})^{k_{i}}\big{(}q,c_{i}/a_{i},a_{i}c_{i}q^{k_{i}-1};q,p\big{)}_{k_{i}}}\big{[}\mathcal{D}_{\mathbf{c},q,p;\mathbf{z}}^{(\mathbf{k})}f(z_{1},\dots,z_{m})\big{]}_{z_{i}=a_{i}q^{k_{i}/2}},$

where $\mathbf{k}=(k_{1},\dots,k_{m})$.

###### Proof.

Note that, for each $j=1,\dots,m$,

$\displaystyle\mathcal{D}_{c_{j},q,p;z_{j}}\prod_{i=1}^{m}\frac{(a_{i}z_{i},a_{i}/z_{i};q,p)_{n_{i}}}{(c_{i}z_{i},c_{i}/z_{i};q,p)_{n_{i}}}$
(2.4)
$\displaystyle\quad{}=\frac{(-1)2a_{j}\theta\big{(}c_{j}/a_{j},a_{j}c_{j}q^{n_{j}-1},q^{n_{j}};p\big{)}}{\theta(q;p)}\frac{\big{(}a_{j}q^{\frac{1}{2}}z_{j},a_{j}q^{\frac{1}{2}}/z_{j};q,p\big{)}_{n_{j}-1}}{\big{(}c_{j}q^{\frac{3}{2}}z_{j},c_{j}q^{\frac{3}{2}}/z_{j};q,p\big{)}_{n_{j}-1}}\prod_{\begin{subarray}{c}i=1,\\\
i\neq
j\end{subarray}}^{m}\frac{(a_{i}z_{i},a_{i}/z_{i};q,p)_{n_{i}}}{(c_{i}z_{i},c_{i}/z_{i};q,p)_{n_{i}}}.$

Iterating (2.4) gives

$\displaystyle\big{[}\mathcal{D}_{\mathbf{c},q,p;\mathbf{z}}^{(\mathbf{k})}f(z_{1},\dots,z_{m})\big{]}_{z_{i}=a_{i}q^{k_{i}/2}}$
$\displaystyle\qquad{}=\prod_{i=1}^{m}(-1)^{k_{i}}(2a_{i})^{k_{i}}q^{\frac{k_{i}(k_{i}-1)}{4}}\frac{(q;q,p)_{n_{i}}\big{(}c_{i}/a_{i},a_{i}c_{i}q^{n_{i}-1};q,p\big{)}_{k_{i}}}{(q;q,p)_{n_{i}-k_{i}}\theta(q;p)^{k_{i}}}$
$\displaystyle\qquad\quad{}\times\left[\frac{\big{(}a_{i}q^{\frac{k_{i}}{2}}z_{i},a_{i}q^{\frac{k_{i}}{2}}/z_{i};q,p\big{)}_{n_{i}-k_{i}}}{\big{(}c_{i}q^{\frac{3k_{i}}{2}}z_{i},c_{i}q^{\frac{3k_{i}}{2}}/z_{i};q,p\big{)}_{n_{i}-k_{i}}}\right]_{z_{i}=a_{i}q^{\frac{k_{i}}{2}}}$
$\displaystyle\qquad{}=\prod_{i=1}^{m}(-1)^{k_{i}}(2a_{i})^{k_{i}}q^{\frac{k_{i}(k_{i}-1)}{4}}\frac{\big{(}q,c_{i}/a_{i},a_{i}c_{i}q^{k_{i}-1};q,p\big{)}_{k_{i}}}{\theta(q;p)^{k_{i}}}\delta_{n_{i}k_{i}}.$

Then the theorem follows by applying
$\mathcal{D}_{\mathbf{c},q,p;\mathbf{z}}^{(\mathbf{j})}$ to both sides of
(2.3) and then setting $z_{i}=a_{i}q^{j_{i}/2}$, for $i=1,\dots,m$ and
$\mathbf{j}=(j_{1},\dots,j_{m})$. ∎

Now we provide a multivariate extension of Theorem 2.4.

######  Theorem 2.11.

For $\mathbf{n}=(n_{1},\dots,n_{m})$, $\mathbf{c}=(c_{1},\dots,c_{m})$,

$\displaystyle\mathcal{D}_{\mathbf{c},q,p;\mathbf{z}}^{(\mathbf{n})}f(z_{1},\dots,z_{m})=\prod_{i=1}^{m}\left[(-2z_{i})^{n_{i}}q^{\frac{n_{i}(3-n_{i})}{4}}\frac{\big{(}c_{i}q^{\frac{n_{i}}{2}-1}z_{i},c_{i}q^{\frac{n_{i}}{2}-1}/z_{i};q,p\big{)}_{n_{i}+1}}{(\theta(q;p))^{n_{i}}}\right]$
$\displaystyle\qquad{}\times\sum_{k_{m}=0}^{n_{m}}\cdots\sum_{k_{1}=0}^{n_{1}}\prod_{i=1}^{m}\Bigg{(}q^{k_{i}(n_{i}-k_{i})}\begin{bmatrix}n_{i}\\\
k_{i}\end{bmatrix}_{p,q}\frac{z_{i}^{2(k_{i}-n_{i})}\big{(}c_{i}q^{\frac{n_{i}}{2}-k_{i}}z_{i},c_{i}q^{-\frac{n_{i}}{2}+k_{i}}/z_{i};q,p\big{)}_{n_{i}-1}}{\big{(}q^{n_{i}-2k_{i}+1}z_{i}^{2};q,p\big{)}_{k_{i}}\big{(}q^{2k_{i}-n_{i}+1}z_{i}^{-2};q,p\big{)}_{n_{i}-k_{i}}}$
$\displaystyle\qquad\qquad{}\times
f\big{(}q^{\frac{n_{1}}{2}-k_{1}}z_{1},\dots,q^{\frac{n_{m}}{2}-k_{m}}z_{m}\big{)}\Bigg{)}.$

###### Proof.

The theorem follows by applying Theorem 2.4 successively for each
$i=1,\dots,m$. ∎

We combine Theorems 2.10 and 2.11 to obtain the following multivariable
elliptic interpolation formula.

######  Theorem 2.12.

For $f(z_{1},\dots,z_{m})$ in $W_{\mathbf{c}}^{\mathbf{n}}$, we have

$\displaystyle\prod_{i=1}^{m}\frac{\big{(}a_{i}^{2}q,q,c_{i}z_{i},c_{i}/z_{i};q,p\big{)}_{n_{i}}}{(a_{i}c_{i},c_{i}/a_{i},a_{i}qz_{i},a_{i}q/z_{i};q,p)_{n_{i}}}f(z_{1},\dots,z_{m})$
$\displaystyle\qquad{}=\sum_{k_{1},\dots,k_{m}=0}^{n_{1},\dots,n_{m}}\prod_{i=1}^{m}q^{k_{i}}\frac{\theta\big{(}a_{i}^{2}q^{2k_{i}};p\big{)}}{\theta\big{(}a_{i}^{2};p\big{)}}\frac{\big{(}q^{-n_{i}},a_{i}^{2},a_{i}q/c_{i},a_{i}c_{i}q^{n_{i}},a_{i}z_{i},a_{i}/z_{i};q,p\big{)}_{k_{i}}}{\big{(}q,a_{i}^{2}q^{n_{i}+1},a_{i}c_{i},a_{i}q^{1-n_{i}}/c_{i},a_{i}qz_{i},a_{i}q/z_{i};q,p\big{)}_{k_{i}}}$
$\displaystyle\qquad\qquad{}\times
f\big{(}a_{1}q^{k_{1}},\dots,a_{m}q^{k_{m}}\big{)}.$

This theorem extends a result given by Ismail and Stanton [14, Theorem 3.10],
which can be obtained by taking $m=2$, $p\to 0$, $c_{1}=c_{2}=0$ and
$n_{1}=n_{2}=n$.

######  Corollary 2.13.

We have the following multivariable elliptic Karlsson–Minton type identity

$\displaystyle\prod_{i=1}^{m}\left(\frac{\big{(}a_{i}^{2}q,q;q,p\big{)}_{n_{i}}}{(a_{i}qz_{i},a_{i}q/z_{i};q,p)_{n_{i}}}\prod_{j=1}^{s_{i}}\frac{(b_{ij}z_{i},b_{ij}/z_{i};q,p)_{v_{ij}}}{(b_{ij}a_{i},b_{ij}/a_{i};q,p)_{v_{ij}}}\right)\prod_{1\leq
i<j\leq
m}\Bigg{(}a_{i}^{w_{ij}}z_{i}^{-w_{ij}}\theta(z_{i}z_{j},z_{i}/z_{j};p)^{w_{ij}}$
$\displaystyle\qquad{}\times\prod_{l_{ij}=1}^{r_{ij}}\frac{(\alpha_{l_{ij}}z_{i}z_{j},\alpha_{l_{ij}}z_{i}/z_{j},\alpha_{l_{ij}}z_{j}/z_{i},\alpha_{l_{ij}}/z_{i}z_{j};q,p)_{u_{lij}}}{(\alpha_{l_{ij}}a_{i}a_{j},\alpha_{l_{ij}}a_{i}/a_{j},\alpha_{l_{ij}}a_{j}/a_{i},\alpha_{l_{ij}}/a_{i}a_{j};q,p)_{u_{lij}}}\Bigg{)}$
$\displaystyle=\sum_{k_{1},\dots,k_{m}=0}^{n_{1},\dots,n_{m}}\prod_{i=1}^{m}\Bigg{(}q^{k_{i}}\frac{\theta\big{(}a_{i}^{2}q^{2k_{i}};p\big{)}}{\theta\big{(}a_{i}^{2};p\big{)}}\frac{\big{(}q^{-n_{i}},a_{i}^{2},a_{i}z_{i},a_{i}/z_{i};q,p\big{)}_{k_{i}}}{\big{(}q,a_{i}^{2}q^{n_{i}+1},a_{i}qz_{i},a_{i}q/z_{i};q,p\big{)}_{k_{i}}}$
$\displaystyle\qquad{}\times\prod_{j=1}^{s_{i}}\frac{\big{(}a_{i}b_{ij}q^{v_{ij}},a_{i}q/b_{ij};q,p\big{)}_{k_{i}}}{\big{(}a_{i}b_{ij},a_{i}q^{1-v_{ij}}/b_{ij};q,p\big{)}_{k_{i}}}\Bigg{)}$
$\displaystyle\qquad{}\times\prod_{1\leq i<j\leq
m}\prod_{l_{ij}=1}^{r_{ij}}q^{-2u_{l_{ij}}k_{i}}$
$\displaystyle\qquad{}\times\frac{\big{(}\alpha_{l_{ij}}a_{i}a_{j}q^{u_{l_{ij}}},qa_{i}a_{j}/\alpha_{l_{ij}};q,p\big{)}_{k_{i}+k_{j}}\big{(}\alpha_{l_{ij}}a_{i}q^{u_{l_{ij}}}/a_{j},qa_{i}/a_{j}\alpha_{l_{ij}};q,p\big{)}_{k_{i}-k_{j}}}{\big{(}\alpha_{l_{ij}}a_{i}a_{j},q^{1-u_{l_{ij}}}a_{i}a_{j}/\alpha_{l_{ij}};q,p\big{)}_{k_{i}+k_{j}}\big{(}\alpha_{l_{ij}}a_{i}/a_{j},q^{1-u_{l_{ij}}}a_{i}/a_{j}\alpha_{l_{ij}};q,p\big{)}_{k_{i}-k_{j}}}$
$\displaystyle\times\prod_{1\leq i<j\leq
m}q^{-w_{ij}k_{i}}\theta\big{(}a_{i}a_{j}q^{k_{i}+k_{j}},a_{i}q^{k_{i}-k_{j}}/a_{j};p\big{)}^{w_{ij}},$

where

$\displaystyle
n_{i}=\sum_{j=1}^{s_{i}}v_{ij}+\sum_{j=1}^{i-1}w_{ji}+\sum_{j=i+1}^{m}w_{ij}+2\sum_{l=1}^{r_{ij}}\left(\sum_{j=1}^{i-1}u_{l_{ji}}+\sum_{j=i+1}^{m}u_{l_{ij}}\right),$

for $i=1,\dots,m$.

###### Proof.

We apply Theorem 2.12 to

$\displaystyle
f(z_{1},\dots,z_{m})=\prod_{i=1}^{m}\frac{\prod\limits_{j=1}^{s_{i}}(b_{ij}z_{i},b_{ij}/z_{i};q,p)_{v_{ij}}}{(c_{i}z_{i},c_{i}/z_{i};q,p)_{n_{i}}}\prod_{1\leq
i<j\leq m}z_{i}^{-w_{ij}}\theta(z_{i}z_{j},z_{i}/z_{j};p)^{w_{ij}}$
$\displaystyle\hphantom{f(z_{1},\dots,z_{m})=}{}\times\prod_{1\leq i<j\leq
m}\prod_{l_{ij}=1}^{r_{ij}}(\alpha_{l_{ij}}z_{i}z_{j},\alpha_{l_{ij}}z_{i}/z_{j},\alpha_{l_{ij}}z_{j}/z_{i},\alpha_{l_{ij}}/z_{i}z_{j};q,p)_{u_{l_{ij}}},$

where

$\displaystyle
n_{i}=\sum_{j=1}^{s_{i}}v_{ij}+\sum_{j=1}^{i-1}w_{ji}+\sum_{j=i+1}^{m}w_{ij}+2\sum_{j=1}^{i-1}\sum_{l_{ji}=1}^{r_{ji}}u_{l_{ji}}+2\sum_{j=i+1}^{m}\sum_{l_{ij}=1}^{r_{ij}}u_{l_{ij}},$

for $i=1,\dots,m$. ∎

Corollary 2.13 extends a result by Ismail and Stanton (see [14, Corollary
3.11]), corresponding to a special case of its $m=2$ instance.

More generally, $f(z_{1},\dots,z_{m})$ could involve symmetrized products of
$2^{k}$ factors of the form $(\lambda z_{i_{1}}^{\pm}z_{i_{2}}^{\pm}\cdots
z_{i_{k}}^{\pm};q,p)_{y}$ (the notation $z_{i_{j}}^{\pm}$ means that the
resprective variable could appear as $z_{i_{j}}$ or $z_{i_{j}}^{-1}$, where
all possible combinations appear), where $\\{i_{1},\dots,i_{k}\\}$ is any
subset of $\\{1,\dots,n\\}$. (In the corollary, we only considered factors for
$k=1,2$.)

Corollary 2.13 can be easily seen to be equivalent to its $u_{l_{ij}}=1$ and
$v_{ij}=1$ case, for all $i$, $j$, in which case the respective factorials
reduce to simple theta functions. To recover the general case from this
special case one can suitably increase $r_{ij}$ and $s_{1},\dots,s_{m}$ and
choose the parameters partially in geometric progression to obtain shifted
factorials. In particular, we can replace $r_{ij}$ by $u_{1}+\dots+u_{r_{ij}}$
and relabel
$\alpha_{u_{1}+\dots+u_{l_{ij}-1}+h}\mapsto\alpha_{l_{ij}}q^{h-1}$, for all
$1\leq l_{ij}\leq r_{ij}$, $1\leq h\leq u_{l_{ij}}$, etc. (One could even add
extra bases, in addition to $q$. This feature is typical for series of
Karlsson–Minton type.)

For convenience, we restate the corollary in this equivalent form.

######  Corollary 2.14.

We have the following multivariable elliptic Karlsson–Minton type identity

$\displaystyle\prod_{i=1}^{m}\left(\frac{(a_{i}^{2}q,q;q,p)_{n_{i}}}{(a_{i}qz_{i},a_{i}q/z_{i};q,p)_{n_{i}}}\prod_{j=1}^{s_{i}}\frac{\theta(b_{ij}z_{i},b_{ij}/z_{i};p)}{\theta(b_{ij}a_{i},b_{ij}/a_{i};p)}\right)$
$\displaystyle{}\times\prod_{1\leq i<j\leq
m}\left(a_{i}^{w_{ij}}z_{i}^{-w_{ij}}\theta(z_{i}z_{j},z_{i}/z_{j};p)^{w_{ij}}\prod_{l_{ij}=1}^{r_{ij}}\frac{\theta(\alpha_{l_{ij}}z_{i}z_{j},\alpha_{l_{ij}}z_{i}/z_{j},\alpha_{l_{ij}}z_{j}/z_{i},\alpha_{l_{ij}}/z_{i}z_{j};p)}{\theta(\alpha_{l_{ij}}a_{i}a_{j},\alpha_{l_{ij}}a_{i}/a_{j},\alpha_{l_{ij}}a_{j}/a_{i},\alpha_{l_{ij}}/a_{i}a_{j};p)}\right)$
$\displaystyle=\sum_{k_{1},\dots,k_{m}=0}^{n_{1},\dots,n_{m}}\prod_{i=1}^{m}\\!\left(q^{k_{i}}\frac{\theta(a_{i}^{2}q^{2k_{i}};p)}{\theta(a_{i}^{2};p)}\frac{(q^{-n_{i}},a_{i}^{2},a_{i}z_{i},a_{i}/z_{i};q,p)_{k_{i}}}{(q,a_{i}^{2}q^{n_{i}+1},a_{i}qz_{i},a_{i}q/z_{i};q,p)_{k_{i}}}\prod_{j=1}^{s_{i}}\\!\frac{\theta(a_{i}b_{ij}q^{k_{i}},a_{i}q^{k_{i}}/b_{ij};p)}{\theta(a_{i}b_{ij},a_{i}/b_{ij};p)}\right)$
$\displaystyle{}\times\prod_{1\leq i<j\leq
m}\prod_{l_{ij}=1}^{r_{ij}}q^{-2k_{i}}\frac{\theta\big{(}\alpha_{l_{ij}}a_{i}a_{j}q^{k_{i}+k_{j}},q^{k_{i}+k_{j}}a_{i}a_{j}/\alpha_{l_{ij}},\alpha_{l_{ij}}a_{i}q^{k_{i}-k_{j}}/a_{j},q^{k_{i}-k_{j}}a_{i}/a_{j}\alpha_{l_{ij}};p\big{)}}{\theta(\alpha_{l_{ij}}a_{i}a_{j},a_{i}a_{j}/\alpha_{l_{ij}},\alpha_{l_{ij}}a_{i}/a_{j},a_{i}/a_{j}\alpha_{l_{ij}};p)}$
$\displaystyle{}\times\prod_{1\leq i<j\leq
m}q^{-w_{ij}k_{i}}\theta\big{(}a_{i}a_{j}q^{k_{i}+k_{j}},a_{i}q^{k_{i}-k_{j}}/a_{j};p\big{)}^{w_{ij}},$

where

$\displaystyle
n_{i}=s_{i}+\sum_{j=1}^{i-1}w_{ji}+\sum_{j=i+1}^{m}w_{ij}+2\sum_{j=1}^{i-1}r_{ji}+2\sum_{j=i+1}^{m}r_{ij},$

for $i=1,\dots,m$.

###  2.1 A quadratic elliptic Taylor expansion theorem

In [13] Ismail and Stanton also considered the basis $\\{\phi_{k}(z),\,0\leq
k\leq n\\}$ where $\phi_{k}(z)=\big{(}q^{1/4}z,q^{1/4}/z;q^{1/2}\big{)}_{k}$.
The set

$\displaystyle\left\\{\frac{\big{(}q^{1/4}z,q^{1/4}/z;q^{1/2},p\big{)}_{k}}{(cz,c/z;q,p)_{k}},\,0\leq
k\leq n\right\\}$

apparently forms a basis for $W_{c}^{n}$. We now provide a Taylor expansion
theorem with respect to this basis.

######  Theorem 2.15.

If $f$ is in $W_{c}^{n}$, then

$\displaystyle
f(z)=\sum_{k=0}^{n}f_{k}\frac{\big{(}q^{1/4}z,q^{1/4}/z;q^{1/2},p\big{)}_{k}}{(cz,c/z;q,p)_{k}},$
(2.5)

where

$\displaystyle
f_{k}=\frac{(-1)^{k}q^{-k/4}\theta(q;p)^{k}}{2^{k}(q;q,p)_{k}\big{(}cq^{\frac{k}{2}-\frac{3}{4}};q^{1/2},p\big{)}_{2k}}\big{[}\mathcal{D}_{c,q,p}^{(k)}f(z)\big{]}_{z=q^{1/4}}.$

###### Proof.

Note that

$\displaystyle\mathcal{D}_{c,q,p}^{(k)}\left(\frac{\big{(}q^{1/4}z,q^{1/4}/z;q^{1/2},p\big{)}_{n}}{(cz,c/z;q,p)_{n}}\right)$
$\displaystyle\qquad{}=\frac{(-2)^{k}q^{k/4}\big{(}cq^{\frac{n}{2}-\frac{3}{4}};q^{1/2},p\big{)}_{2k}(q;q,p)_{n}}{(q;q,p)_{n-k}\theta(q;p)^{k}}\frac{\big{(}q^{1/4}z,q^{1/4}/z;q^{1/2},p\big{)}_{n-k}}{\big{(}cq^{\frac{3}{2}k}z,cq^{\frac{3}{2}k}/z;q,p\big{)}_{n-k}},$

which can be proved by induction. The theorem then follows by applying
$\mathcal{D}_{c,q,p}^{(j)}$ to both sides of (2.5) and then setting
$z=q^{1/4}$. ∎

In the following, we recover an elliptic quadratic summation by Warnaar [24,
Corollary 4.4; $b=a$], which was originally proved by using inverse relations.
Its $p=0$ case has been given earlier by Gessel and Stanton [10, equation
(1.4)].

######  Corollary 2.16.

We have the following summation

$\displaystyle\frac{(az,a/z;q,p)_{n}}{(cz,c/z;q,p)_{n}}\frac{\big{(}cq^{-1/4};q^{1/2},p\big{)}_{2n}}{\big{(}aq^{-1/4};q^{1/2},p\big{)}_{2n}}$
$\displaystyle\qquad{}=\sum_{k=0}^{n}q^{\frac{k}{2}}\frac{\theta\big{(}cq^{\frac{3}{2}k-\frac{3}{4}};p\big{)}}{\theta\big{(}cq^{-\frac{3}{4}};p\big{)}}\frac{\big{(}c/a,acq^{n-1},q^{-n};q,p\big{)}_{k}}{(cz,c/z,q;q,p)_{k}}\frac{\big{(}cq^{-\frac{3}{4}},q^{\frac{1}{4}}z,q^{\frac{1}{4}}/z;q^{1/2},p\big{)}_{k}}{\big{(}aq^{-\frac{1}{4}},cq^{n-\frac{1}{4}},q^{\frac{3}{4}-n}/a;q^{1/2},p\big{)}_{k}}.$

###### Proof.

We apply Theorem 2.15 to

$\displaystyle f(z)=\frac{(az,a/z;q,p)_{n}}{(cz,c/z;q,p)_{n}}.$ ∎

######  Remark 2.17.

If we expand

$\displaystyle\frac{\big{(}q^{\frac{1}{4}}z,q^{\frac{1}{4}}/z;q^{\frac{1}{2}},p\big{)}_{n}}{(cz,c/z;q,p)_{n}}$

in terms of

$\displaystyle\frac{(az,a/z;q,p)_{n}}{(cz,c/z;q,p)_{n}}$

using Theorem 2.2, we obtain

$\displaystyle\frac{\big{(}q^{\frac{1}{4}}z,q^{\frac{1}{4}}/z;q^{\frac{1}{2}},p\big{)}_{n}}{\big{(}aq^{\frac{1}{4}},q^{\frac{1}{4}}/a;q^{\frac{1}{2}},p\big{)}_{n}}\frac{(ac,c/a;q,p)_{n}}{(cz,c/z;q,p)_{n}}$
$\displaystyle\qquad{}=\sum_{k=0}^{n}q^{k}\frac{\theta\big{(}acq^{2k-1};p\big{)}}{\theta\big{(}acq^{-1};p\big{)}}\frac{\big{(}q^{-n},acq^{-1},az,a/z,cq^{\frac{n}{2}-\frac{3}{4}},cq^{\frac{n}{2}-\frac{1}{4}};q,p\big{)}_{n}}{\big{(}q,acq^{n},cz,c/z,aq^{\frac{1}{4}-\frac{n}{2}},aq^{\frac{3}{4}-\frac{n}{2}};q,p\big{)}_{n}}.$
(2.6)

At first glance this appears to be a true quadratic summation formula.
However, the right-hand side of (2.6) is

$\displaystyle{}_{10}V_{9}\big{(}acq^{-1};az,a/z,cq^{\frac{n}{2}-\frac{3}{4}},cq^{\frac{n}{2}-\frac{1}{4}},q^{-n};q,p\big{)},$

which, by Frenkel and Turaev’s summation formula (1.2), can be reduced to

$\displaystyle\frac{\big{(}ac,c/a,q^{\frac{3}{4}-\frac{n}{2}}z,q^{\frac{1}{4}-\frac{n}{2}}/z;q,p\big{)}_{n}}{\big{(}cz,c/z,aq^{\frac{3}{4}-\frac{n}{2}},q^{\frac{3}{4}-\frac{n}{2}}/a;q,p\big{)}_{n}}.$

Elementary manipulations can now be applied to transform this expression to
the left-hand side of (2.6).

##  3 Expansions involving cubic theta functions

The cubic theta function $\gamma(z,a;p)$ with two independent variables $z$
and $a$ in addition to the nome $p$ was considered by S. Bhargava [3]. (For a
thorough treatment of the theory of cubic theta functions in analogy to the
theory of the classical Jacobi theta functions, see [22].) It is defined by

$\displaystyle\gamma(z,a;p)=\sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty}p^{k^{2}+kl+l^{2}}a^{k+l}z^{k-l}.$
(3.1)

This function, up to a normalization factor
$\big{(}p^{2};p^{2}\big{)}_{\infty}^{2}$ (independent from $a$ and $z$), is
almost equal to the following product of two modified Jacobi theta functions

$\displaystyle\big{(}p^{2};p^{2}\big{)}_{\infty}^{2}\theta\big{(}{-}paz;p^{2}\big{)}\theta\big{(}{-}pa/z;p^{2}\big{)}=\sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty}p^{k^{2}+l^{2}}a^{k+l}z^{k-l},$

which differs by the factor $p^{kl}$ to the summand of the double series in
(3.1). Because of this additional factor $p^{kl}$, the cubic theta function
does not factorize into a product of two modified Jacobi theta functions of
such a simple form. In principle though, the cubic theta function could be
factorized into two modified Jacobi theta functions, but their arguments would
have nontrivial expansions in $a$, $z$, and $p$.

From (3.1), by replacing $(k,l)$ by $(l,k)$, or $(k,l)$ by $(-l,-k)$,
respectively, we immediately deduce the symmetries [3]

$\displaystyle\gamma(1/z,a;p)=\gamma(z,a;p),$ (3.2a) and
$\displaystyle\gamma(z,1/a;p)=\gamma(z,a;p).$ (3.2b)

Further, from (3.1), by replacing $(k,l)$ by $(k+\lambda+\mu,l+\lambda)$, it
is easy to verify that for all integers $\lambda$ and $\mu$ the following
functional equation holds [3]:

$\displaystyle\gamma(z,a;p)=p^{3\lambda^{2}+3\lambda\mu+\mu^{2}}a^{2\lambda+\mu}z^{\mu}\gamma\big{(}p^{\mu/2}z,p^{3(2\lambda+\mu)/2}a;p\big{)}.$

In particular, we have the quasi periodicities

$\displaystyle\gamma(pz,a;p)=\frac{1}{pz^{2}}\gamma(z,a;p),$ (3.3a) and
$\displaystyle\gamma(z,p^{3}a;p)=\frac{1}{p^{3}a^{2}}\gamma(z,a;p).$ (3.3b)

Further, by separating the terms in the expansion of $p$ according to whether
the exponents of $p$ are divisible by $3$ or not, one can show [3]

$\displaystyle\gamma(z,a;p)=\gamma\big{(}\sqrt{az^{3}},\sqrt{a^{3}/z^{3}};p^{3}\big{)}+paz^{-1}\gamma\big{(}\sqrt{az^{3}},p^{3}\sqrt{a^{3}/z^{3}};p^{3}\big{)},$

while separating the terms in the expansion of $z$ according to whether the
exponents of $z$ are even or odd, one has [4]

$\displaystyle\gamma(z,a;p)=\big{(}p^{6};p^{6}\big{)}_{\infty}\big{(}p^{2};p^{2}\big{)}_{\infty}\left[\theta\big{(}{-}p^{3}a;p^{6}\big{)}\theta\big{(}{-}pz^{2};p^{2}\big{)}+paz\theta\big{(}{-}p^{6}a^{2};p^{6}\big{)}\theta\big{(}{-}p^{2}z^{2};p^{2}\big{)}\right].$

Cooper and Toh [6] proved the following addition formulae which will be useful
in our computations.

######  Lemma 3.1 ([6, Corollary 4.5]).

The following identities connecting modified Jacobi theta functions and cubic
theta functions hold:

$\displaystyle\gamma(z_{1},\alpha;p)\theta(z_{3}/z_{2},z_{2}z_{3};p)-\gamma(z_{2},\alpha;p)\theta(z_{3}/z_{1},z_{1}z_{3};p)$
$\displaystyle\qquad{}=\frac{z_{3}}{z_{1}}\gamma(z_{3},\alpha;p)\theta(z_{1}/z_{2},z_{1}z_{2};p),$
(3.4a) and
$\displaystyle\gamma\big{(}z,\alpha_{1};p^{\frac{1}{3}}\big{)}\theta(\alpha_{3}/\alpha_{2},\alpha_{2}\alpha_{3};p)-\gamma\big{(}z,\alpha_{2};p^{\frac{1}{3}}\big{)}\theta(\alpha_{3}/\alpha_{1},\alpha_{1}\alpha_{3};p)$
$\displaystyle\qquad{}=\frac{\alpha_{3}}{\alpha_{1}}\gamma\big{(}z,\alpha_{3};p^{\frac{1}{3}}\big{)}\theta(\alpha_{1}/\alpha_{2},\alpha_{1}\alpha_{2};p).$
(3.4b)

These two identities were proved in [6] by specializing a $(3\times 3)$
determinant evaluation involving cubic theta functions. They can also be
proved directly, expanding the cubic theta functions and modified Jacobi theta
functions as infinite series, together with clever series rearrangement.

Now we introduce the first cubic theta analogue of the $q$-shifted factorial
by

$\displaystyle\langle
az,a/z;q,p\rangle_{n}:=\prod_{j=0}^{n-1}\gamma\big{(}zq^{\frac{1-n}{2}+j},aq^{\frac{n-1}{2}};p\big{)}.$

From (3.3a) it is easy to see that the cubic shifted factorial satisfies

$\displaystyle\langle apz,a/pz;q,p\rangle_{n}=\frac{1}{p^{n}z^{2n}}\langle
az,a/z;q,p\rangle_{n}.$

Together with (3.2a), this implies that the quotient

$\displaystyle\frac{\langle az,a/z;q,p\rangle_{n}}{(cz,c/z;q,p)_{n}}$

is in the space $W_{c}^{n}$. Hence we can apply Theorem 2.2 to it, by which we
obtain the first cubic theta extension of Jackson’s ${}_{8}\phi_{7}$ summation
(1.3).

######  Corollary 3.2.

We have the following summation

$\displaystyle(bc,c/b;q,p)_{n}\frac{\langle
az,a/z;q,p\rangle_{n}}{(cz,c/z;q,p)_{n}}=\sum_{k=0}^{n}q^{nk}\left(\frac{c}{b}\right)^{k}\frac{\theta(bcq^{2k-1};p)}{\theta(bcq^{-1};p)}\frac{(q^{-n},bcq^{-1},bz,b/z;q,p)_{k}}{(q,bcq^{n},cz,c/z;q,p)_{k}}$
$\displaystyle\qquad{}\times\big{\langle}acq^{n-1},aq^{1-k}/c;q,p\big{\rangle}_{k}\frac{\langle
abq^{k},aq^{-k}/b;q,p\rangle_{n}}{\langle abq^{n},aq^{-k}/b;q,p\rangle_{k}}.$
(3.5)

###### Proof.

By using (3.4a) in Lemma 3.1, we can prove by induction that

$\displaystyle\mathcal{D}_{c,q,p}^{(k)}\left(\frac{\langle
az,a/z;q,p\rangle_{n}}{(cz,c/z;q,p)_{n}}\right)$
$\displaystyle\qquad{}=(2c)^{k}q^{\frac{3}{4}k(k-1)}\frac{(q^{n};q^{-1},p)_{k}}{\theta(q;p)^{k}}\prod_{j=0}^{k-1}\gamma(cq^{\frac{n-1}{2}+j},aq^{\frac{n-1}{2}};p)\frac{\langle
aq^{\frac{k}{2}}z,aq^{\frac{k}{2}}/z;q,p\rangle_{n-k}}{(cq^{\frac{3}{2}k}z,cq^{\frac{3}{2}k}/z;q,p)_{n-k}}$
$\displaystyle\qquad{}=(2c)^{k}q^{\frac{3}{4}k(k-1)}\frac{(q^{n};q^{-1},p)_{k}}{\theta(q;p)^{k}}\langle
acq^{n-1},aq^{1-k}/c;q,p\rangle_{k}\frac{\langle
aq^{\frac{k}{2}}z,aq^{\frac{k}{2}}/z;q,p\rangle_{n-k}}{(cq^{\frac{3}{2}k}z,cq^{\frac{3}{2}k}/z;q,p)_{n-k}}.$

Then the corollary follows from Theorem 2.2 while expanding in the basis

$\displaystyle f(z)=\frac{(bz,b/z;q,p)_{n}}{(cz,c/z;q,p)_{n}}.$ ∎

To recover Jackson’s ${}_{8}\phi_{7}$ summation from Corollary 3.2, substitute

$\displaystyle a\mapsto\frac{-a}{p(1+a^{2}q^{n-1})}$

in (3.5), multiply both sides of the identity by $(1+a^{2}q^{n-1})^{n}$ and
let $p\to 0$. When $p\to 0$, the usual theta shifted factorials clearly reduce
to the $q$-shifted factorials. That is, the quotient on the left-hand side
reduces to

$\displaystyle\lim_{p\to
0}\frac{(bc,c/b;q,p)_{n}}{(cz,c/z;q,p)_{n}}=\frac{(bc,c/b;q)_{n}}{(cz,c/z;q)_{n}}.$

What happens with the cubic theta shifted factorial? We have

$\displaystyle\lim_{p\to
0}\big{(}1+a^{2}q^{n-1}\big{)}^{n}\Big{\langle}\frac{-az}{p(1+a^{2}q^{n-1})},\frac{-a}{p(1+a^{2}q^{n-1})z};p\Big{\rangle}_{n}$
$\displaystyle=\big{(}1+a^{2}q^{n-1}\big{)}^{n}\lim_{p\to
0}\prod_{j=0}^{n-1}\gamma\left(zq^{\frac{1-n}{2}+j},\frac{-aq^{\frac{n-1}{2}}}{p(1+a^{2}q^{n-1})};p\right)$
$\displaystyle=\big{(}1+a^{2}q^{n-1}\big{)}^{n}\prod_{j=0}^{n-1}\lim_{p\to
0}\sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty}(-1)^{k+l}p^{k^{2}+kl+l^{2}-k-l}\left(\frac{aq^{\frac{n-1}{2}}}{1+a^{2}q^{n-1}}\right)^{k+l}\big{(}zq^{\frac{1-n}{2}+j}\big{)}^{k-l}.$

Now it is easy to see that for $p\to 0$ only three terms in the various double
infinite series survive. These three terms correspond to the cases
$(k,l)=(0,0),(1,0),(0,1)$. The last expression thus reduces to

$\displaystyle\big{(}1+a^{2}q^{n-1}\big{)}^{n}\prod_{j=0}^{n-1}\left(1-\frac{aq^{\frac{n-1}{2}}}{1+a^{2}q^{n-1}}\big{(}zq^{\frac{1-n}{2}+j}+z^{-1}q^{\frac{n-1}{2}-j}\big{)}\right)$
$\displaystyle\qquad{}=\prod_{j=0}^{n-1}\left(1+a^{2}q^{n-1}-aq^{\frac{n-1}{2}}\big{(}zq^{\frac{1-n}{2}+j}+z^{-1}q^{\frac{n-1}{2}-j}\big{)}\right)$
$\displaystyle\qquad{}=\prod_{j=0}^{n-1}(1-azq^{j})\big{(}1-aq^{n-1-j}/z\big{)}=(az,a/z;q)_{n}.$

We take similar limits on the right-hand side of (3.5).

Our next result involves elliptic interpolation of cubic theta shifted
factorials.

######  Corollary 3.3.

We have the following Karlsson–Minton type identity involving cubic theta
functions

$\displaystyle\frac{\big{(}a^{2}q,q;q,p\big{)}_{n}}{(aqz,aq/z;q,p)_{n}}\langle
bz,b/z;q,p\rangle_{n}$
$\displaystyle\qquad{}=\sum_{k=0}^{n}q^{k(n+1)}\frac{\theta\big{(}a^{2}q^{2k};p\big{)}}{\theta\big{(}a^{2};p\big{)}}\frac{\big{(}q^{-n},a^{2},az,a/z;q,p\big{)}_{k}}{\big{(}q,a^{2}q^{n+1},aqz,aq/z;q,p\big{)}_{k}}\big{\langle}abq^{k},bq^{-k}/a;q,p\big{\rangle}_{n}.$

###### Proof.

We apply Theorem 2.6 to

$\displaystyle f(z)=\frac{\langle bz,b/z;q,p\rangle_{n}}{(cz,c/z;q,p)_{n}}.$ ∎

More generally, we have the following Karlsson–Minton type identity involving
cubic theta functions.

######  Corollary 3.4.

We have

$\displaystyle\frac{\big{(}a^{2}q,q;q,p\big{)}_{n}}{(aqz,aq/z;q,p)_{n}}\prod_{i=1}^{s}\theta(b_{i}z,b_{i}/z;p)\prod_{j=1}^{n-s}\gamma(z,d_{j};p)=\sum_{k=0}^{n}q^{k(n+1)}\frac{\theta(a^{2}q^{2k};p)}{\theta(a^{2};p)}$
$\displaystyle\qquad{}\times\frac{\big{(}q^{-n},a^{2},az,a/z;q,p\big{)}_{k}}{\big{(}q,a^{2}q^{n+1},aqz,aq/z;q,p\big{)}_{k}}\prod_{i=1}^{s}\theta\big{(}ab_{i}q^{k},b_{i}q^{-k}/a;p\big{)}\prod_{j=1}^{n-s}\gamma\big{(}aq^{k},d_{j};p\big{)}.$

###### Proof.

We apply Theorem 2.6 to

$\displaystyle
f(z)=\frac{\prod\limits_{i=1}^{s}\theta(b_{i}z,b_{i}/z;p)\prod\limits_{j=1}^{n-s}\gamma(z,d_{j};p)}{(cz,c/z;q,p)_{n}}.$
∎

Our next result concerns a cubic theta extension of Gessel and Stanton’s
quadratic summation [10, equation (1.4)].

######  Corollary 3.5.

We have the following summation

$\displaystyle\frac{\langle
az,a/z;q,p\rangle_{n}}{(cz,c/z;q,p)_{n}}\big{(}cq^{-\frac{1}{4}},cq^{\frac{1}{4}};q,p\big{)}_{n}$
$\displaystyle\qquad{}=\sum_{k=0}^{n}c^{k}q^{\frac{k}{4}(k-2)+nk}\frac{\theta\big{(}cq^{\frac{3}{2}k-\frac{3}{4}};p\big{)}}{\theta\big{(}cq^{\frac{k}{2}-\frac{3}{4}};p\big{)}}\frac{\big{(}q^{-n};q,p\big{)}_{k}}{(q;q,p)_{k}}\frac{\big{(}cq^{-\frac{1}{4}};q^{\frac{1}{2}},p\big{)}_{k}}{\big{(}cq^{n-\frac{1}{4}};q^{\frac{1}{2}},p\big{)}_{k}}\frac{\big{(}q^{\frac{1}{4}}z,q^{\frac{1}{4}}/z;q^{\frac{1}{2}},p\big{)}_{k}}{(cz,c/z;q,p)_{k}}$
$\displaystyle\qquad\quad{}\times\big{\langle}acq^{n-1},aq^{1-k}/c;q,p\big{\rangle}_{k}\big{\langle}aq^{\frac{k}{2}+\frac{1}{4}},aq^{\frac{k}{2}-\frac{1}{4}};q,p\big{\rangle}_{n-k}.$
(3.6)

###### Proof.

We apply Theorem 2.15 to

$\displaystyle f(z)=\frac{\langle az,a/z;q,p\rangle_{n}}{(cz,c/z;q,p)_{n}}.$ ∎

Similarly to the way we recovered Jackson’s ${}_{8}\phi_{7}$ summation from
Corollary 3.2, Gessel and Stanton’s quadratic summation can be readily
obtained by substituting $a\mapsto-p^{-1}a/\big{(}1+a^{2}q^{n-1}\big{)}$ in
(3.6), multiplying both sides by $(1+a^{2}q^{n-1})^{n}$ and taking the limit
$p\to 0$.

Next, we define the second cubic theta shifted factorial, with base $p^{1/3}$:

$\displaystyle\langle\langle
az,a/z;q,p^{\frac{1}{3}}\rangle\rangle_{n}:=\prod_{j=0}^{n-1}\gamma\big{(}aq^{\frac{n-1}{2}},zq^{\frac{1-n}{2}+j};p^{\frac{1}{3}}\big{)}.$

Recalling equations (3.3a) and (3.3b) (which we reformulate after
interchanging $a$ and $z$),

$\displaystyle\gamma(a,z;p)=\gamma(a,1/z;p),\qquad\gamma(a,z;p)=p^{3}z^{2}\gamma\big{(}a,p^{3}z;p\big{)},$

we see that

$\displaystyle\big{\langle}\big{\langle}apz,a/pz;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}=\frac{1}{p^{n}z^{2n}}\big{\langle}\big{\langle}az,a/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}.$

This implies that the quotient

$\displaystyle\frac{\big{\langle}\big{\langle}az,a/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}}{(cz,c/z;q,p)_{n}}$

is also in the space $W_{c}^{n}$. Thus, Theorem 2.2 can be applied to it, by
which we obtain the second cubic theta extension of Jackson’s ${}_{8}\phi_{7}$
summation (1.3).

######  Corollary 3.6.

We have the following summation

$\displaystyle\frac{\big{\langle}\big{\langle}bz,b/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}}{(cz,c/z;q,p)_{n}}(ac,c/a;q,p)_{n}=\sum_{k=0}^{n}q^{nk}\left(\frac{c}{a}\right)^{k}\frac{\theta\big{(}acq^{2k-1};p\big{)}}{\theta\big{(}acq^{-1};p\big{)}}\frac{\big{(}q^{-n},acq^{-1},az,a/z;q,p\big{)}_{k}}{\big{(}q,acq^{n},cz,c/z;q,p\big{)}_{k}}$
$\displaystyle\qquad\quad{}\times\big{\langle}\big{\langle}bcq^{n-1},bq^{1-k}/c;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{k}\big{\langle}\big{\langle}abq^{k},b/a;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n-k}.$
(3.7)

###### Proof.

Note that by using (3.4b) in Lemma 3.1, we can show by induction that

$\displaystyle\mathcal{D}_{c,q,p}^{(k)}\left(\frac{\big{\langle}\big{\langle}bz,b/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}}{(cz,c/z;q,p)_{n}}\right)$
$\displaystyle\qquad{}=(2c)^{k}q^{\frac{3}{4}k(k-1)}\frac{\big{(}q^{n};q^{-1},p\big{)}_{k}}{\theta(q;p)^{k}}\prod_{j=0}^{k-1}\gamma\big{(}bq^{\frac{n-1}{2}},cq^{\frac{n-1}{2}+j};p^{\frac{1}{3}}\big{)}\frac{\big{\langle}\big{\langle}bq^{\frac{k}{2}}z,bq^{\frac{k}{2}}/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n-k}}{\big{(}cq^{\frac{3}{2}k}z,cq^{\frac{3}{2}k}/z;q,p\big{)}_{n-k}}$
$\displaystyle\qquad{}=(2c)^{k}q^{\frac{3}{4}k(k-1)}\frac{\big{(}q^{n};q^{-1},p\big{)}_{k}}{\theta(q;p)^{k}}\big{\langle}\big{\langle}bcq^{n-1},bq^{1-k}/c;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{k}\frac{\big{\langle}\big{\langle}bq^{\frac{k}{2}}z,bq^{\frac{k}{2}}/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n-k}}{\big{(}cq^{\frac{3}{2}k}z,cq^{\frac{3}{2}k}/z;q,p\big{)}_{n-k}}.$

Using this, we apply Theorem 2.2 to

$\displaystyle
f(z)=\frac{\big{\langle}\big{\langle}bz,b/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}}{(cz,c/z;q,p)_{n}}.$
∎

To recover Jackson’s ${}_{8}\phi_{7}$ summation from Corollary 3.6, substitute

$\displaystyle b\mapsto\frac{-b}{p^{\frac{1}{3}}\big{(}1+b^{2}q^{n-1}\big{)}}$

in (3.7), multiply both sides of the identity by
$\big{(}1+b^{2}q^{n-1}\big{)}^{n}$ and let $p\to 0$. When $p\to 0$, the usual
theta shifted factorials reduce to the $q$-shifted factorials and the cubic
theta shifted factorial on the left-hand side of (3.7) becomes

$\displaystyle\lim_{p\to
0}\big{(}1+b^{2}q^{n-1}\big{)}^{n}\Big{\langle}\Big{\langle}\frac{-bz}{p^{\frac{1}{3}}\big{(}1+b^{2}q^{n-1}\big{)}},\frac{-b}{p^{\frac{1}{3}}\big{(}1+b^{2}q^{n-1}\big{)}z};p^{\frac{1}{3}}\Big{\rangle}\Big{\rangle}_{n}$
$\displaystyle\qquad{}=\big{(}1+b^{2}q^{n-1}\big{)}^{n}\lim_{p\to
0}\prod_{j=0}^{n-1}\gamma\left(\frac{-bq^{\frac{n-1}{2}}}{p^{\frac{1}{3}}(1+b^{2}q^{n-1})},zq^{\frac{1-n}{2}+j};p^{\frac{1}{3}}\right)$
$\displaystyle\qquad{}=\big{(}1+b^{2}q^{n-1}\big{)}^{n}\prod_{j=0}^{n-1}\lim_{p\to
0}\sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty}(-1)^{k-l}p^{\frac{1}{3}(k^{2}+kl+l^{2}-k+l)}$
$\displaystyle\qquad\quad{}\times\big{(}zq^{\frac{1-n}{2}+j}\big{)}^{k+l}\left(\frac{bq^{\frac{n-1}{2}}}{1+b^{2}q^{n-1}}\right)^{k-l}.$

Now it is easy to see that for $p\to 0$ only three terms in the various double
infinite series survive. These correspond to the cases
$(k,l)=(0,0),(1,0),(0,-1)$. The last expression thus reduces to

$\displaystyle\big{(}1+b^{2}q^{n-1}\big{)}^{n}\prod_{j=0}^{n-1}\left(1-\frac{bq^{\frac{n-1}{2}}}{1+b^{2}q^{n-1}}\big{(}zq^{\frac{1-n}{2}+j}+z^{-1}q^{\frac{n-1}{2}-j}\big{)}\right)$
$\displaystyle\qquad{}=\prod_{j=0}^{n-1}\left(1+b^{2}q^{n-1}-bq^{\frac{n-1}{2}}\big{(}zq^{\frac{1-n}{2}+j}+z^{-1}q^{\frac{n-1}{2}-j}\big{)}\right)$
$\displaystyle\qquad{}=\prod_{j=0}^{n-1}\big{(}1-bzq^{j}\big{)}\big{(}1-bq^{n-1-j}/z\big{)}=(bz,b/z;q)_{n}.$

We take similar limits on the right-hand side of equation (3.7).

Our final result concerns another cubic theta extension of Gessel and
Stanton’s quadratic summation [10, equation (1.4)].

######  Corollary 3.7.

We have the following summation

$\displaystyle\frac{\big{\langle}\big{\langle}az,a/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}}{(cz,c/z;q,p)_{n}}\big{(}cq^{-\frac{1}{4}},cq^{\frac{1}{4}};q,p\big{)}_{n}$
$\displaystyle\qquad{}=\sum_{k=0}^{n}c^{k}q^{\frac{k}{4}(k-2)+nk}\frac{\theta\big{(}cq^{\frac{3}{2}k-\frac{3}{4}};p\big{)}}{\theta\big{(}cq^{\frac{k}{2}-\frac{3}{4}};p\big{)}}\frac{\big{(}q^{-n};q,p\big{)}_{k}}{(q;q,p)_{k}}\frac{\big{(}cq^{-\frac{1}{4}};q^{\frac{1}{2}},p\big{)}_{k}}{\big{(}cq^{n-\frac{1}{4}};q^{\frac{1}{2}},p\big{)}_{k}}\frac{\big{(}q^{\frac{1}{4}}z,q^{\frac{1}{4}}/z;q^{\frac{1}{2}},p\big{)}_{k}}{(cz,c/z;q,p)_{k}}$
$\displaystyle\qquad\quad{}\times\big{\langle}\big{\langle}acq^{n-1},aq^{1-k}/c;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{k}\big{\langle}\big{\langle}aq^{\frac{k}{2}+\frac{1}{4}},aq^{\frac{k}{2}-\frac{1}{4}};q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n-k}.$

###### Proof.

We apply Theorem 2.15 to

$\displaystyle
f(z)=\frac{\big{\langle}\big{\langle}az,a/z;q,p^{\frac{1}{3}}\big{\rangle}\big{\rangle}_{n}}{(cz,c/z;q,p)_{n}}.$
∎

### Acknowledgements

The work in this paper has been supported by FWF Austrian Science Fund grant
F50-08 within the SFB “Algorithmic and enumerative combinatorics”.

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