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\begin{document}

\setbox9\vbox{
\centerline{Institut f\"ur Mathematik der Universit\"at Wien,}
\centerline{Strudlhofgasse 4, A-1090 Wien, Austria.}
\centerline{E-mail: {\tt\footnotesize kratt@@pap.univie.ac.at,
    schlosse@@pap.univie.ac.at}}
\centerline{WWW: {\tt\footnotesize http://radon.mat.univie.ac.at/People/kratt}}
\centerline{\tt\footnotesize 
http://radon.mat.univie.ac.at/People/mschloss}
\vskip18pt
\centerline{\em Dedicated to H. W. Gould}
}


\title[A New Multidimensional Matrix Inverse]
  {A New Multidimensional Matrix Inverse with applications to
   multiple $q$-series}
\author[Christian Krattenthaler and Michael Schlosser]
   {Christian Krattenthaler$^\dagger$ and Michael
  Schlosser$^\dagger$\\[18pt]\box9}
\address{Institut f\"ur Mathematik der Universit\"at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.}
\email{KRATT@@Pap.Univie.Ac.At, SCHLOSSE@@Pap.Univie.Ac.At,\newline
WWW: {\tt http://radon.mat.univie.ac.at/People/kratt},\newline
{\tt http://radon.mat.univie.ac.at/People/mschloss}}
\thanks{$^\dagger$ The authors were supported by the Austrian
Science Foundation FWF, grant P12094-MAT\\
1991 {\em Mathematics Subject Classification}: Primary 33D70; 
Secondary 05A19 05A30 11B65 15A09 33C70 33D20.\\ 
{\em Keywords}: matrix inversion, inverse relations, basic hypergeometric
  series, $q$-Abel identities, $q$-Rothe identities,
  MacMahon's Master Theorem}

\maketitle

\begin{abstract}
We compute the inverse of a specific infinite $r$-dimensional matrix,
extending a matrix inverse of Krattenthaler.
Our inversion is different from the $r$-dimensional matrix inversion
recently found by Schlosser but generalizes a multidimensional matrix
inversion previously found by Chu. As applications of our
matrix inversion we derive some multidimensional $q$-series identities.
Among these are $q$-analogues of Carlitz' multidimensional
Abel-type expansion formulas. Furthermore, we derive a $q$-analogue
of MacMahon's Master Theorem.
\end{abstract}

%\maketitle


\section{Introduction}

Matrix inversions are very important tools in combinatorics and
special functions theory. In particular, it is a widely spread and
often used method to derive and prove identities for (basic)
hypergeometric series with the help of so-called ``inverse
relations" (see Section~\ref{secappl}), 
which are immediate consequences of matrix inversions. (An inverse
relation is in fact {\em equivalent} to its corresponding matrix inversion.) 
In order to be able to apply this method, {\em explicit\/} matrix
inversions must be at hand.

At this point it seems appropriate to elaborate a little on the
history of (explicit) matrix inversions and inverse relations, in
particular, since H.~W.~Gould's name is inevitably tied with it.
Over time, people came across an increasing number of such explicit
matrix inversions. In the 1960s, in his book \cite{riord}, Riordan
provided lists of known matrix inversions and, in fact, dedicated two
complete chapters of his book to inverse relations and their
applications. (Riordan's inverse relations were classified and given a
unified method of proof by Egorychev \cite{EgorAA}.) 
A prominent part of these inverse relations were due to Gould, who
studied them in a series of papers
\cite{GOulAE}, \cite{GOulAG}, \cite{GOulAI}, \cite{GOulAJ}. 
This study culminated in the
important discovery, jointly with Hsu, of a very general matrix
inversion \cite{GOHsAA}, which contained a lot of inverse relations
of, what is now called, Gould-type and Abel-type as special cases. The
problem, posed by Gould and Hsu, of finding a $q$-analogue of their
formula was immediately solved thereafter by Carlitz
\cite{carlitz}. He did not give any applications, however. The
significance of Carlitz' matrix inversion showed up first when
Andrews \cite{andrews} discovered that the Bailey transform
\cite{BailAB}, \cite{BailAC}, which is one of the corner stones in the
development of the theory of (basic) hypergeometric series, is
equivalent to a certain matrix inversion that is just a very special
case of Carlitz'. Some time later, while further developing on Andrews' idea,
Gessel and Stanton \cite{gesstant}, \cite{gesstant2} used another special case
of Carlitz' matrix inversion (a bibasic extension of the inversion
Andrews considered) to derive a number of basic
hypergeometric summations and transformations, and identities of
Rogers--Ramanujan type. Finite forms of identities of
Rogers--Ramanujan type were considered by Bressoud \cite{BresAA}.
The transform which he used to prove them is equivalent to a matrix
inversion \cite{bress}, which has some overlap with Carlitz' matrix
inversion (namely in the one Andrews considered), but in general 
is not covered by Carlitz' result. A few years later, Gasper and
Rahman proved a bibasic matrix inversion \cite{gas}, \cite{RahmAC}
which unifies 
the matrix inversions of Gessel and Stanton, and Bressoud.
It enabled them to derive numerous beautiful new quadratic, cubic, and 
quartic summation formulas for basic hypergeometric series. 
(They also extended this method to obtain bibasic,
cubic, and quartic transformation formulas 
\cite{gasrahm}, \cite{RahmAA}, \cite[Sec.~3.6]{grhyp}.)
The end of this line of development came 
with the attempt of the first author to combine all
these recent matrix inversions into one formula. Indeed, in 1989, he
discovered a matrix inversion, published in \cite{kratt-mi}, which
subsumes most of Riordan's inverse relations and all the other
aforementioned matrix inversions, as it contains them all as special
cases. 

This matrix inversion is the following:
The matrices
$(f_{nk})_{n,k\in\Bbb Z}$ and $(g_{kl})_{k,l\in\Bbb Z}$
($\Bbb Z$ denotes the set of integers), are inverses of each other, where
\begin{equation}\label{krattf}
f_{n k}=\frac {\prodl _{j=k} ^{n-1} (a_j+b_jc_k)}
{\prodl _{j=k+1} ^{n} (c_j-c_k)},
\end{equation}
and
\begin{equation}\label{krattg}
g_{k l}=\frac {(a_l+b_lc_l)} {(a_k+b_kc_k)}
\frac {\prodl _{j=l+1} ^{k} (a_j+b_jc_k)}
{\prodl _{j=l} ^{k-1} (c_j-c_k)}.
\end{equation}

Starting in the late 1970s, Milne and co-authors,
in a long series of papers (cf\@. \cite{lilmil}, \cite{MilnAM},
\cite{MilnAC}, \cite{MilnAK}, \cite{MiLiAB}, \cite{MiNeAA}, 
and the references cited therein), developed a theory of
multiple (basic) hypergeometric series associated to root systems.
In order to have an equivalent of the (one-dimensional) Bailey
transform at hand, to conveniently extend the development of the theory
of (one-dimensional) basic hypergeometric series to an analogous
theory for multiple series, matrix inversions in
this multidimensional setting needed to be found. 
In \cite{lilmil} and \cite{MiLiAB},
Lilly and Milne provided multidimensional extensions of the earlier
mentioned matrix inversion that Andrews considered. Subsequently,
Bhatnagar and Milne \cite{bhatmil} found a matrix
inversion that extended Gasper and Rahman's (one-dimensional) matrix
inversion to the multidimensional setting. 
(According to our terminology ``multidimensional'' matrix inversions 
are matrix inversions that arise 
in the theory of multiple series, whereas ``one-dimensional'' matrix
inversions are matrix inversions which arise in the theory of
one-dimensional series; see
Section~\ref{secom} for a precise explanation.) 
At the end of this line of
development stand the second author's matrix inversions
\cite{schlossmmi}. 
Theorems~3.1 and 4.1 of \cite{schlossmmi} do indeed cover all previously
mentioned matrix inversions in that area
as special cases.
In particular, these matrix inversions also contain the inversion 
(\ref{krattf})/(\ref{krattg}) as one-dimensional special case.

The main result of this paper is another multidimensional extension
of the matrix inverse (\ref{krattf})/(\ref{krattg}) 
(see Theorem~\ref{mainres}). This matrix inversion finds its
applications in the theory of ``ordinary" multiple series. It does
not ``belong", as far as we can tell, to the theory of multiple
series associated to root systems. Also here, special cases of this 
matrix inversion appeared earlier in the literature. Aside from
reducing to (\ref{krattf})/(\ref{krattg}), the matrix inversion from 
\cite{kratt-mi}, in the one-dimensional case, it also
contains Chu's multidimensional matrix inversion \cite{chu} and a
two-dimensional matrix inversion \cite{kratt-2var} by the first
author. We demonstrate the usefulness of our new multidimensional
matrix inverse by deriving several multidimensional $q$-series
identities, among them $q$-analogues of Carlitz' multidimensional
Abel-type expansion formulas, and a $q$-analogue
of MacMahon's Master Theorem.

Our paper is organized as follows. In order to prove our matrix
inversion, we need some preparations, which we provide in
Section~\ref{secom}. There we review the first author's operator 
method~\cite{kratt-opm}. We adapt a main theorem of
\cite{kratt-opm} and add an appropriate multidimensional corollary
(see Corollary~\ref{corol}). Then, in Section~\ref{secmain} we state
and prove our multidimensional matrix inversion. We also add a
companion inversion (Theorem~\ref{mainres2}) 
which we use later in the applications. In
Section~\ref{secappl} the notion and use of inverse relations is
explained, together with the standard basic hypergeometric notation.
The following sections contain applications of our matrix
inversion. In Section~\ref{secdouble} we derive some new basic
hypergeometric double summations. A multidimensional extension of
a very-well-poised $_{10}\phi_9$-summation is the contents of
Section~\ref{sec10phi9}. In Sections~\ref{seccarlitz} and
\ref{secmore} we present $q$-analogues of Carlitz' multidimensional
Abel-type expansion formulas \cite{carlitz1}, \cite{carlitz2}, \cite{CarlAQ}. 
These $q$-analogues are new even in the one-dimensional case. 
Related multiple $q$-Abel and $q$-Rothe summations are presented
in Section~\ref{secrothe}. Finally, in Section~\ref{secmaster}, 
we find, for the first time, a (noncommutative) $q$-analogue
of MacMahon's Master Theorem. 

\section{An operator method for proving matrix inversions}\label{secom}
%\setcounter{equation}{0}

Let $F=(f_{\bold{n}\bold{k}})_{\bold{n},\bold{k}\in{\Bbb Z}^r}$ (as before,
$\Bbb Z$ denotes the set of integers) be an infinite lower-triangular
$r$-dimensional matrix; i.e., $f_{\bold{n}\bold{k}}=0$ unless
$\bold{n}\ge\bold{k}$, by which we mean $n_i\ge k_i$ for all $i=1,\dots,r$.
The matrix $G=(g_{\bold{k}\bold{l}})_{\bold{k},\bold{l}\in{\Bbb Z}^r}$ is said
to be the {\em inverse matrix} of $F$ if and only if
$$
\sum_{\bold{n}\ge\bold{k}\ge\bold{l}} f_{\bold{n}\bold{k}}
g_{\bold{k}\bold{l}}=\delta_{\bold{nl}}
$$
for all $\bold{n},\bold{l}\in{\Bbb Z}^r$, where $\delta_{\bold{nl}}$ is the
usual Kronecker delta.

In \cite{kratt-opm}, the first author gave a method for solving Lagrange
inversion problems, which are closely connected with the problem of inverting
lower-triangular matrices. We will use his operator method for proving our
new theorems. 

First we need to introduce some notation and terminology.
By a {\em formal Laurent series} we mean a series of the form
$\sum _{\bold{n}\ge\bold{k}} a_{\bold{n}} \bold{z}^{\bold{n}}$, for some
$\bold{k}\in{\Bbb Z}^r$, where $\bold{z}^{\bold{n}}=
z _1 ^{n_1} z _2 ^{n_2}\cdots z _r ^{n_r}$.
Given the formal Laurent series $a(\bold{z})$ and $b(\bold{z})$ we introduce
the bilinear form $\langle\, ,\, \rangle$ by
$$
\langle a(\bold{z}),b(\bold{z})\rangle = \langle\bold{z}^{\bold{0}}\rangle
(a(\bold{z})\cdot b(\bold{z})),
$$
where $\langle\bold{z}^{\bold{0}}\rangle c(\bold{z})$ denotes the coefficient
of $\bold{z}^{\bold{0}}$ in $c(\bold{z})$. Given any linear operator
$L$ acting on formal Laurent series, $L^*$ denotes the adjoint of $L$ with
respect to $\langle\,,\,\rangle$; i.e.,
$\langle L a(\bold{z}),b(\bold{z})\rangle
=\langle a(\bold{z}),L^* b(\bold{z})\rangle$ for all formal Laurent series
$a(\bold{z})$ and $b(\bold{z})$. We need the following special case of
\cite[Theorem~ 1]{kratt-opm}.

\begin{Lemma}\label{lemma}
Let $F=(f_{\bold{n k}})_{\bold{n,k}\in{\Bbb Z}^r}$ be an infinite
lower-triangular $r$-dimensional matrix with $f_{\bold{k k}}\ne 0$ for all
$\bold{k}\in{\Bbb Z}^r$. For $\bold{k}\in{\Bbb Z}^r$, define the formal Laurent
series $f_{\bold{k}}(\bold{z})$ and $g_{\bold{k}}(\bold{z})$ by
$f_{\bold{k}}(\bold{z})=\sum_{\bold{n\ge k}} f_{\bold{n k}}
\bold{z}^{\bold{n}}$ and $g_{\bold{k}}(\bold{z})=\sum_{\bold{l\le k}}
g_{\bold{k l}}\bold{z}^{-\bold{l}}$, where $(g_{\bold{k l}})_{\bold{k,l}\in
{\Bbb Z}^r}$ is the uniquely determined inverse matrix of $F$. Suppose that
for $\bold{k}\in{\Bbb Z}^r$ a system of equations of the form
$$
U_j f_{\bold{k}}(\bold{z})=c_j(\bold{k}) V f_{\bold{k}}(\bold{z}), 
\qquad j=1,\dots,r,
$$
holds, where $U_j, V$ are linear operators acting on formal Laurent series,
$V$ being bijective, and where $(c_j(\bold{k}))_{\bold{k}\in{\Bbb Z}^r}$ are
arbitrary sequences of constants. Moreover, we suppose that
\begin{equation}\label{cjk}
\text {for all $\bold{m,n}\in{\Bbb Z}^r$, $\bold{m\neq n}$, there exists a $j$
with $1\le j\le r$ and $c_j(\bold{m})\neq c_j(\bold{n})$.}
\end{equation}
Then, if $h_{\bold{k}}(\bold{z})$ is a solution of the dual system
$$
U_j^* h_{\bold{k}}(\bold{z})=c_j(\bold{k}) V^* h_{\bold{k}}(\bold{z}),
\qquad j=1,\dots,r,
$$
with $h_{\bold{k}}(\bold{z})\not\equiv 0$ for all $\bold{k}\in{\Bbb Z}^r$,
the series $g_{\bold{k}}(\bold{z})$ are given by
$$
g_{\bold{k}}(\bold{z})=\frac{1}{\langle f_{\bold{k}}(\bold{z}),
V^*h_{\bold{k}}(\bold{z})\rangle} V^* h_{\bold{k}}(\bold{z}).
$$
\end{Lemma}

We will use the following corollary of Lemma~\ref{lemma}:

\begin{Corollary}\label{corol}
Let $W_i$, $V_{ij}$ be linear operators acting on formal Laurent series,
$c_j(\bold{k})$ arbitrary constants for $\bold{k}\in{\Bbb Z}^r$ and
$i,j=1,\dots,r$. Suppose the operators $W_i,V_{ij}$, $i,j=1,\dots,r$,
satisfy the commutation relations
\begin{alignat}3
V_{i_1j}W_{i_2}&=W_{i_2}V_{i_1j},&\qquad i_1&\neq i_2;\quad
&1&\le i_1,i_2,j\le r,\label{wi1}\\
V_{i_1j_1}V_{i_2j_2}&=V_{i_2j_2}V_{i_1j_1},
&\qquad i_1&\neq i_2;\quad &1&\le i_1,i_2,j_1,j_2\le r.\label{wi2}
\end{alignat}
Moreover, the $c_j(\bold{k})$ are assumed to satisfy $($\ref{cjk}$)$,
and $\Detl_{1\le i,j\le r}(V_{ij})$ is assumed to be invertible.
With the notation of Lemma~\ref{lemma}, if
\begin{equation}\label{co1}
\suml _{j=1} ^{r} c_j(\bold{k}) V_{ij} f_{\bold{k}}(\bold{z})=
W_i f_{\bold{k}}(\bold{z}),\qquad i=1,\dots,r,
\end{equation}
then
\begin{equation}\label{cog}
g_{\bold{k}}(\bold{z})=\frac{1}{\langle f_{\bold{k}}(\bold{z}),
\Detl (V_{ij}^*)h_{\bold{k}}(\bold{z})\rangle}\Detl
(V_{ij}^*)h_{\bold{k}}(\bold{z}),
\end{equation}
where $h_{\bold{k}}(\bold{z})$ is a solution of
\begin{equation}\label{co2}
\suml _{j=1} ^{r} c_j(\bold{k}) V_{ij}^* h_{\bold{k}}(\bold{z})=
W_i^* h_{\bold{k}}(\bold{z}),\qquad i=1,\dots,r,
\end{equation}
with $h_{\bold{k}}(\bold{z})\not\equiv 0$ for all $\bold{k}\in{\Bbb Z}^r$.
\end{Corollary}

\begin{pf}
Due to (\ref{wi2}), we can apply Cramer's rule to (\ref{co1}) to obtain
$$
c_j(\bold{k})\Detl_{1\le i,l\le r} (V_{il}) f_{\bold{k}}(\bold{z})
=\suml_{i=1}^{r}(-1)^{i+j}V^{(i,j)}W_if_{\bold{k}}(\bold{z}),
$$
for $j=1,\dots,r$, $V^{(i,j)}$ being the minor of $(V_{st})_{1\le
s,t\le r}$ with the $i$-th row
and $j$-th column being omitted. The dual system (in the sense of
Lemma~\ref{lemma}) reads
\begin{align}\label{dualwi}
c_j(\bold{k})\Detl_{1\le i,l\le r} (V_{il}^*) h_{\bold{k}}(\bold{z})
&=\suml_{i=1}^{r}(-1)^{i+j}W_i^*{V^*}^{(i,j)}
h_{\bold{k}}(\bold{z})\\
&=\suml_{i=1}^{r}(-1)^{i+j}{V^*}^{(i,j)}W_i^*
h_{\bold{k}}(\bold{z}),\notag
\end{align}
for $j=1,\dots,r$, and is easily seen to be equivalent to 
(\ref{co2}). Notice that condition (\ref{wi2})
justifies to write the dual of $\det(V_{il})$ as $\det(V^*_{il})$
(and similarly for $V^{(i,j)}$), and that, because of (\ref{wi1}), 
we may commute $W_i^*$ and ${V^*}^{(i,j)}$ in (\ref{dualwi}). Now apply
Lemma~\ref{lemma} with $V=\Detl(V_{ij})$ and
$U_j=\suml_{i=1}^{r}(-1)^{i+j}V^{(i,j)}W_i$.
\end{pf}

\begin{Remark}
A slightly more general corollary is given in
\cite[Corollary~2.14]{schlossmmi}
which was needed to prove another multidimensional matrix inversion
which lead to the derivation of several interesting identities for
multidimensional basic hypergeometric series associated to root
systems.
\end{Remark}

\section{A multidimensional matrix inversion}\label{secmain}
%\setcounter{equation}{0}

\begin{Theorem}\label{mainres}
Let $(a_i(t))_{t\in\Bbb Z}$, $(b_{ij}(t))_{t\in\Bbb Z}$,
and $(c_i(t))_{t\in\Bbb Z}$, $i,j=1,\dots,r$ be arbitrary sequences
such that $c_i(s)\neq c_i(t)$ for $s\neq t$. Then
$(f_{\bold{nk}})_{\bold{n,k}\in{\Bbb Z}^r}$ and
$(g_{\bold{kl}})_{\bold{k,l}\in{\Bbb Z}^r}$ are inverses of each other, where
\begin{equation}\label{alf}
f_{\bold{n k}}=\prodl _{i=1} ^{r}
\frac {\prodl _{t_i=k_i} ^{n_i-1}
(a_i(t_i)+\sum _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=k_i+1}^{n_i}
(c_i(t_i)-c_i(k_i))}
\end{equation}
and
\begin{multline}\label{alinv}
g_{\bold{k l}}=
\frac {\Detl_{1\le i,j\le r}\big(
(a_i(l_i)+\sum _{s=1} ^{r}b_{is}(l_i)c_s(k_s))\delta_{ij}+
b_{ij}(l_i)(c_i(l_i)-c_i(k_i))\big)}
{\prodl _{i=1} ^{r} (a_i(k_i)+\sum _{j=1} ^{r}b_{ij}(k_i)c_j(k_j))}\\
\times\prodl _{i=1} ^{r}
\frac {\prodl _{t_i=l_i+1} ^{k_i}
(a_i(t_i)+\sum _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=l_i}^{k_i-1}
(c_i(t_i)-c_i(k_i))}.
\end{multline}
\end{Theorem}

\begin{Remark}
For $r=1$, Theorem~\ref{mainres} reduces to the first author's matrix inverse
(\ref{krattf})/(\ref{krattg}).
The special case $c_i(t)=t$, $i=1,\dots,r$, is equivalent to
Chu's~\cite[Eqs.~(2.3)/(2.4)]{chu} matrix inversion result.
Setting $r=2$, $c_i(t)=q^t$, $a_i(t)=0$, $i=1,2$, 
$$\big(b_{ij}(t_i)\big)_{1\le i,j\le 2}=\begin{pmatrix} -1&q^{\mu+t_1}\\
q^{\lambda+t_2}&-1\end{pmatrix},$$
and simplifying a
bit, we recover the first author's two-dimensional inversion
\cite[(4.15)/(4.16)]{kratt-2var}, which was used there to derive many
two-dimensional expansion formulas.
\end{Remark}

\begin{pf*}{Proof of Theorem~\ref{mainres}}
We will use the operator method of Section~\ref{secom}.
From (\ref{alf}) we deduce for $\bold{n\ge k}$ the recursion
\begin{equation}\label{mrec}
\textstyle (c_i(n_i)-c_i(k_i))f_{\bold{n k}}
=(a_i(n_i-1)+\sum\nolimits _{j=1} ^{r}b_{ij}(n_i-1)c_j(k_j))
f_{\bold{n}-\bold{e}_i,\bold{k}},\quad i=1,\dots,r, 
\end{equation}
where $\bold{e}_i$ denotes
the vector of ${\Bbb Z}^r$ where all components are zero
except the $i$-th, which is 1. We write
$$
f_{\bold{k}}(\bold{z})=\suml_{\bold{n}\ge\bold{k}}f_{\bold n\bold
k}\bold z^{\bold n}
=\suml_{\bold{n}\ge\bold{k}}
\prodl _{i=1} ^{r}
\frac {\prodl _{t_i=k_i} ^{n_i-1}
(a_i(t_i)+\sum\nolimits _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=k_i+1}^{n_i}
(c_i(t_i)-c_i(k_i))} \bold{z}^{\bold{n}}.
$$
Moreover, we define linear operators
${\cal B}_{ij},{\cal A}_i,{\cal C}_i$ by
${\cal B}_{ij}\bold{z}^{\bold{n}}=b_{ij}(n_i)\bold{z}^{\bold{n}}$,
${\cal A}_i\bold{z}^{\bold{n}}=a_i(n_i)\bold{z}^{\bold{n}}$, and
${\cal C}_i\bold{z}^{\bold{n}}=c_i(n_i)\bold{z}^{\bold{n}}$, for all
$i,j=1,\dots,r$. Then we may write (\ref{mrec}) in the form
\begin{equation}\label{moprec}
\textstyle ({\cal C}_i-c_i(k_i))f_{\bold{k}}(\bold{z})
=(z_i{\cal A}_i+z_i\sum\nolimits _{j=1} ^{r}{\cal B}_{ij}c_j(k_j))
f_{\bold{k}}(\bold{z}),\quad i=1,\dots,r, 
\end{equation}
valid for all $\bold{k}\in{\Bbb Z}^r$. We rewrite our system of
equations in a way such that Corollary~\ref{corol} is applicable:
\begin{equation}\label{moprecel}
\textstyle (c_i(k_i)+z_i\sum\nolimits _{j=1} ^{r}c_j(k_j){\cal B}_{ij})
f_{\bold{k}}(\bold{z})
=({\cal C}_i-z_i{\cal A}_i)f_{\bold{k}}(\bold{z}),\quad i=1,\dots,r.
\end{equation}
Now (\ref{moprecel}) is a system of type (\ref{co1}) with
$V_{ij}=\delta_{ij}+z_i{\cal B}_{ij}$,
$W_i={\cal C}_i-z_i{\cal A}_i$, and
$c_j(\bold{k})=c_j(k_j)$. The conditions (\ref{cjk}), (\ref{wi1}), and
(\ref{wi2}) are satisfied. Hence we may apply
Corollary~\ref{corol}. The dual system (\ref{co2}) for the auxiliary
formal Laurent series $h_{\bold{k}}(\bold{z})$ in this case reads
\begin{equation*}
\textstyle (c_i(k_i)+\sum\nolimits _{j=1} ^{r}c_j(k_j){\cal B}_{ij}^*z_i)
h_{\bold{k}}(\bold{z})
=({\cal C}_i^*-{\cal A}_i^*z_i)h_{\bold{k}}(\bold{z}),\quad i=1,\dots,r.
\end{equation*}
Equivalently, we have
\begin{equation}\label{moprecdu}
\textstyle ({\cal C}_i^*-c_i(k_i))h_{\bold{k}}(\bold{z})
=({\cal A}_i^*z_i+\sum\nolimits _{j=1} ^{r}{\cal B}_{ij}^*c_j(k_j)z_i)
h_{\bold{k}}(\bold{z}),\quad i=1,\dots,r,
\end{equation}
for all $\bold{k}\in{\Bbb Z}^r$.
As is easily seen, we have
${\cal B}_{ij}^*\bold{z}^{-\bold{l}}=b_{ij}(l_i)\bold{z}^{-\bold{l}}$,
${\cal A}_i^*\bold{z}^{-\bold{l}}=a_i(l_i)\bold{z}^{-\bold{l}}$, and
${\cal C}_i^*\bold{z}^{-\bold{l}}=c_i(l_i)\bold{z}^{-\bold{l}}$,
for $i,j=1,\dots,r$. Thus, with $h_{\bold{k}}(\bold{z})=
\sum_{\bold{l\le k}}h_{\bold{kl}}\bold{z}^{-\bold{l}}$,
by comparing coefficients of $\bold{z}^{-\bold{l}}$ in
(\ref{moprecdu}) we obtain
\begin{equation*}
\textstyle (c_i(l_i)-c_i(k_i))h_{\bold{k l}}
=(a_i(l_i)+\sum\nolimits _{j=1} ^{r}b_{ij}(l_i)c_j(k_j))
h_{\bold{k},\bold{l}+\bold{e}_i},\quad i=1,\dots,r.
\end{equation*}
If we set $h_{\bold{k k}}=1$, we get
$$
h_{\bold{k l}}=\prodl _{i=1} ^{r}
\frac {\prodl _{t_i=l_i} ^{k_i-1}
(a_i(t_i)+\sum _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=l_i}^{k_i-1} (c_i(t_i)-c_i(k_i))}.
$$
Taking into account (\ref{cog}), we have to compute the action of
\begin{equation}\label{detc1}
\Detl_{1\le i,j\le r} (V_{ij}^*)=\Detl_{1\le i,j\le r}
\left(\delta_{ij}+{\cal B}_{ij}^*z_i\right)
\end{equation}
when applied to
$$
h_{\bold{k}}(\bold{z})=\suml_{\bold{l}\le\bold{k}}
\prodl _{i=1} ^{r} \frac {\prodl _{t_i=l_i} ^{k_i-1}
(a_i(t_i)+\sum _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=l_i}^{k_i-1} (c_i(t_i)-c_i(k_i))} \bold{z}^{-\bold{l}}.
$$
Since
$$
z_ih_{\bold{k}}(\bold{z})=\suml _{\bold{l\le k}}
\frac {(c_i(l_i)-c_i(k_i))}
{(a_i(l_i)+\sum _{j=1} ^{r}b_{ij}(l_i)c_j(k_j))}
h_{\bold{kl}}\bold{z}^{-\bold{l}},
$$
we conclude that
\begin{equation}\label{vhk}
\Detl_{1\le i,j\le r}(V_{ij}^*)h_{\bold{k}}(\bold{z})=
\suml _{\bold{l\le k}}\det_{1\le i,j\le r}\left(
\delta_{ij}
+\frac {b_{ij}(l_i)(c_i(l_i)-c_i(k_i))}
{(a_i(l_i)+\sum _{s=1} ^{r}b_{is}(l_i)c_s(k_s))}\right)
h_{\bold{kl}}\bold{z}^{-\bold{l}}.
\end{equation}
Note that since $f_{\bold{kk}}=1$,
the pairing $\langle f_{\bold{k}}(\bold{z}),
\Detl (V_{ij}^*)h_{\bold{k}}(\bold{z})\rangle$ is simply the coefficient of
$\bold{z}^{\bold{-k}}$ in (\ref{vhk}) which is easily seen to be one.
By taking the denominators out of the rows of the determinant,
equation (\ref{cog}) is turned into
\begin{multline}\label{gkz}
g_{\bold{k}}(\bold{z})=\Detl_{1\le i,j\le r}(V_{ij}^*)
h_{\bold{k}}(\bold{z})\\
=\suml _{\bold{l\le k}}\Bigg(
\frac {\Detl_{1\le i,j\le r}\big(
(a_i(l_i)+\sum _{s=1} ^{r}b_{is}(l_i)c_s(k_s))\delta_{ij}+
b_{ij}(l_i)(c_i(l_i)-c_i(k_i))\big)}
{\prodl _{i=1} ^{r}
(a_i(k_i)+\sum _{j=1} ^{r}b_{ij}(k_i)c_j(k_j))}\\
\times\prodl _{i=1} ^{r}
\frac {\prodl _{t_i=l_i+1} ^{k_i}
(a_i(t_i)+\sum _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=l_i}^{k_i-1}
(c_i(t_i)-c_i(k_i))}\bold{z}^{-\bold{l}}\Bigg),
\end{multline}
where $g_{\bold{k}}(\bold{z})=\sum_{\bold{l\le k}}
g_{\bold{k l}}\bold{z}^{-\bold{l}}$. So, by extracting the coefficient of
$\bold{z}^{-\bold{l}}$ in (\ref{gkz}) we  obtain exactly (\ref{alinv}). 
\end{pf*}

By a slightly modified application of the operator method
of Section~\ref{secom} 
one can show that the determinant in (\ref{alinv}) can be
``transferred" from $g_{\bold{k l}}$ to $f_{\bold{n k}}$.
The corresponding Theorem reads as follows.

\begin{Theorem}\label{mainres2}
Assume the conditions of Theorem~\ref{mainres}. Then
$(f_{\bold{nk}})_{\bold{n,k}\in{\Bbb Z}^r}$ and
$(g_{\bold{kl}})_{\bold{k,l}\in{\Bbb Z}^r}$ are inverses of each other, where
\begin{multline}\label{alf2}
f_{\bold{n k}}=
\frac {\Detl_{1\le i,j\le r}\big(
(a_i(n_i)+\sum _{s=1} ^{r}b_{is}(n_i)c_s(k_s))\delta_{ij}+
b_{ij}(n_i)(c_i(n_i)-c_i(k_i))\big)}
{\prodl _{i=1} ^{r} (a_i(k_i)+\sum _{j=1} ^{r}b_{ij}(k_i)c_j(k_j))}\\
\times\prodl _{i=1} ^{r}
\frac {\prodl _{t_i=k_i} ^{n_i-1}
(a_i(t_i)+\sum _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=k_i+1}^{n_i}
(c_i(t_i)-c_i(k_i))}
\end{multline}
and
\begin{equation}\label{alinv2}
g_{\bold{k l}}=
\prodl _{i=1} ^{r}
\frac {\prodl _{t_i=l_i+1} ^{k_i}
(a_i(t_i)+\sum _{j=1} ^{r}b_{ij}(t_i)c_j(k_j))}
{\prodl_{t_i=l_i}^{k_i-1}
(c_i(t_i)-c_i(k_i))}.
\end{equation}
\end{Theorem}

\begin{Remark} The special case $c_i(t)=t$, $i=1,\dots,r$, is equivalent to
Chu's~\cite[Eq.~(2.9)/(2.10)]{chu} companion matrix inversion result.
\end{Remark}

\section{Preliminaries on inverse relations and basic hypergeometric
notation}\label{secappl}
%\setcounter{equation}{0}

Here we introduce the basic concept of ``inverse relations"
and introduce some standard $q$-series notation.

There is a standard technique for deriving new summation formulas
from known ones by using inverse matrices (cf.~\cite{andrews},
\cite{bhatmil}, \cite{ChuWAM},
\cite{gas}, \cite{gasrahm}, \cite[Sec.~3.8]{grhyp},
\cite{gesstant}, \cite{gesstant2}, \cite{kratt-mi}, 
\cite{lilmil}, \cite{MilnAM}, \cite{MiLiAB}, \cite{RahmAA},
\cite{RahmAC}, \cite{riord}, \cite{schlossmmi}).
If $(f_{\bold{n k}})_{\bold{n,k}\in{\Bbb Z}^r}$ and
$(g_{\bold{k l}})_{\bold{k,l}\in{\Bbb Z}^r}$ are lower triangular matrices
being inverses of each other, then of course the following is true:
\begin{align}\label{invrelf}
\suml _{\bold{0\le k\le n}}f_{\bold{n k}}a_{\bold{k}}=b_{\bold{n}}
\end{align}
{\em if and only if}
\begin{align}\label{invrelg}
\suml _{\bold{0\le l\le k}}g_{\bold{k l}}b_{\bold{l}}=a_{\bold{k}}.
\end{align}
If either \eqref{invrelf} or \eqref{invrelg} is known, then the other
produces another summation formula.
The less used dual version, the so-called ``rotated inversion'', can be
used to derive nonterminating summations. It reads
\begin{align}\label{rotinvf}
\suml _{\bold{k\le n\le{\boldsymbol \infty}}}
f_{\bold{n k}}a_{\bold{n}}=b_{\bold{k}}
\end{align}
{\em if and only if}
\begin{align}\label{rotinvg}
\suml _{\bold{l\le k\le{\boldsymbol \infty}}}
g_{\bold{k l}}b_{\bold{k}}=a_{\bold{l}},
\end{align}
subject to suitable convergence conditions. Again, if
one of \eqref{rotinvf} or \eqref{rotinvg} is known, the other
produces a possibly new identity.

In the subsequent sections
we use special cases of our Theorems~\ref{mainres} and \ref{mainres2}
to derive a couple of higher dimensional summations for $q$-series.

Before we start to develop the applications of our Theorems, we need
to recall the standard basic hypergeometric notation 
(cf.~\cite{grhyp}). Let $q$ be a complex number such that $|q|<1$. Define
\begin{align}
(a;q)_{\infty}:=\prodl_{j\ge 0}(1-aq^j),
\end{align}
and,
\begin{align}\label{pochinf}
(a;q)_k:={}&\frac{(a;q)_{\infty}}{(aq^k;q)_{\infty}}\\ \label{pochf}
={}&\prodl_{j=0}^{k-1}(1-aq^j),
\end{align}
where the equality \eqref{pochf} holds
when $k$ is a non-negative integer. We also make use of the standard
notation for basic hypergeometric series,
\begin{equation}\label{defhyp}
{_s\phi_t}\!\left[\begin{matrix}a_1,\dots,a_s\\
b_1,\dots,b_t\end{matrix};q,z\right]:=
\suml _{k=0} ^{\infty}\frac {(a_1;q)_k\,(a_2;q)_k\,\cdots\,(a_s;q)_k}
{(q;q)_k\,(b_1;q)_k\,\cdots\,(b_t;q)_k}\,
\left((-1)^k q^{k\choose 2}\right)^{1+t-s}z^k.
\end{equation}

Finally, for multidimensional series, we also employ the notation
$|{\bold k}|$ for $(k_1+\dots+k_r)$ where ${\bold k}=(k_1,\dots,k_r)$.
Concerning the nonterminating multiple series given in this paper,
we have stated their regions of
convergence explicitly. The convergence of these series
can be checked by application of the multiple power series ratio
test~\cite{horn}, \cite{sri}. In cases where the summand of
the multiple series contains a determinant we would first have to
expand the determinant appearing in the summand according to
its definition as a sum over the symmetric group, then
interchange summations and apply the multiple power series ratio
test to each of its resulting $r!$ multiple sums.
In our proofs, however, we have not carried out
such calculations explicitly.
For explicit examples of how to use the multiple power series ratio
test, see \cite[Sec.~5]{MilnAM}.

\section{Some identities for double series}\label{secdouble}

In our first application we use the two-dimensional special case (i.e.,
the $r=2$ case) of our matrix inversion (\ref{mainres}) to derive a few 
basic hypergeometric double summation theorems. These developments
are very much in the spirit of \cite{kratt-2var}, although 
the particular case of (\ref{mainres}) that we consider here is a different 
one than in \cite{kratt-2var}. Namely,
the particular choice of the parameters in (\ref{mainres}) that 
we make is $a_1(t)=a_2(t)= 0$, $c_1(t)=c_2(t)=q^t$, and
$$\big(b_{ij}(t_i)\big)_{1\le i,j\le 2}
=\begin{pmatrix} Cq^{t_1}&-1\\-1&Dq^{t_2}\end{pmatrix}.$$
Thus, after little simplification, we obtain that {\em the matrices 
$(f_{\bold n\bold k})_{{\bold n},{\bold k}\in {\Bbb Z}^2}$ and
$(g_{\bold k\bold l})_{{\bold k},{\bold l}\in {\Bbb Z}^2}$
are inverses of each other, where
\begin{equation} \label{DInv1}
f_{\bold n\bold k}=q^{(k_1-k_2)(n_2-k_2-n_1+k_1)}
\frac {(Cq^{2k_1-k_2};q)_{n_1-k_1}\,(Dq^{2k_2-k_1};q)_{n_2-k_2}} 
{(q;q)_{n_1-k_1}\,(q;q)_{n_2-k_2}}
\end{equation}
and
\begin{multline} \label{DInv2}
g_{\bold k\bold l}=q^{(k_1-k_2)(k_2-l_2-k_1+l_1)}
\frac {\big((C q^{2l_1}-q^{k_2})(D q^{2l_2}-q^{k_1})-(q^{l_1}-q^{k_1})(q^{l_2}-q^{k_2})\big)} 
{(C q^{2k_1}-q^{k_2})(D q^{2k_2}-q^{k_1})}\\
\times
\frac {(C q^{2k_1-k_2};q^{-1})_{k_1-l_1} \,
        (D q^{2k_2-k_1};q^{-1})_{k_2-l_2}} 
{(q^{-1};q^{-1})_{k_1-l_1}\,(q^{-1};q^{-1})_{k_2-l_2}}        .
\end{multline}
}

Now, in (\ref{invrelf}) we take
\begin{equation} \label{DA1}
a_{\bold k}=C^{k_2}D^{k_1}q^{-k_1^2+3k_1k_2-k_2^2}
\frac {(C;q)_{2k_1-k_2}\,(D;q)_{2k_2-k_1}} {(q;q)_{k_1}\,(q;q)_{k_2}}.
\end{equation}
By two-fold use of $q$-Chu--Vandermonde summation (cf\@. \cite[Eq.~(1.5.3); Appendix
(II.6)]{grhyp}),
\begin{equation}\label{qChuVandermonde}
{}_2\phi _1\!\left [ \begin{matrix} \let\over/ a,{q^{-n}}\\ \let\over/  c\end{matrix} ;q,q\right ] 
=  {{{a^n} {(\let\over/ {c\over a};q)}_{n}}\over {{(\let\over/ c;q)}_{n}}}
\end{equation}
we have by (\ref{invrelf}) and a bit of manipulation
\begin{align} \notag
b_{\bold n}&=\sum_{{k_1} = 0}^{n_1}\sum _{k_2=0} ^{n_2}
       {q^{{k_1} {k_2} - {k_1} {n_1} + {k_2} {n_1} + 
           {k_1} {n_2} - {k_2} {n_2}}}
{C^{{k_2}}} {D^{{k_1}}} 
{{ 
       ({\let \over / C}; q) _{n_1+{k_1} - {k_2} }  \,
       ({\let \over / D}; q) _{n_2+k_2-{k_1} } }\over 
     {({\let \over / q}; q) _{{k_1}} \, ({\let \over / q}; q) _{{k_2}}  \,
       ({\let \over / q}; q) _{{n_1}-{k_1}}  \,
       ({\let \over / q}; q) _{n_2-{k_2} } }}\\
&\notag
=\sum _{k_2=0} ^{n_2}{q^{{k_2} {n_1} - {k_2} {n_2}}}{C^{{k_2}}} 
{{ 
     ({\let \over / C}; q) _{n_1-{k_2}}  \,
     ({\let \over / D}; q) _{{k_2} + {n_2}} }\over 
   {({\let \over / q}; q) _{{k_2}} \, ({\let \over / q}; q) _{{n_1}}  \,
     ({\let \over / q}; q) _{n_2-{k_2}} }}
  {} _{2} \phi _{1} \! \left [ \matrix \let \over / C {q^{n_1-{k_2} }}, 
   {q^{-{n_1}}}\\ \let \over / {{{q^{1 - {k_2} - {
      n_2}}}}\over D}\endmatrix ;q, {\displaystyle q} \right ]\\
&\notag
=\sum_{{k_2} = 0}^{{n_2}}{q^{{{{n_1}}^2} - {k_2} {n_2}}}{C^{{k_2} + {n_1}}} 
{{ 
       ({\let \over / C}; q) _{n_1-{k_2}}  \,
       ({\let \over / D}; q) _{{k_2} + {n_2}}  \,
       ({\let \over / {{{q^{1 - {n_1} - {n_2}}}}\over {C D}}}; q)
        _{{n_1}} }\over 
     {({\let \over / q}; q) _{{k_2}} \, ({\let \over / q}; q) _{{n_1}}  \,
       ({\let \over / q}; q) _{n_2-{k_2}}  \,
       ({\let \over / {{{q^{1 - {k_2} - {n_2}}}}\over D}}; q) _{{
        n_1}} }}\\
&\notag
={q^{{{{n_1}}^2}}}{C^{{n_1}}} {{ 
     ({\let \over / C}; q) _{{n_1}} \, ({\let \over / D}; q) _{{n_2}}  \,
     ({\let \over / {{{q^{1 - {n_1} - {n_2}}}}\over {C D}}}; q) _{{
      n_1}} }\over 
   {({\let \over / q}; q) _{{n_1}} \, ({\let \over / q}; q) _{{n_2}}  \,
     ({\let \over / {{{q^{1 - {n_2}}}}\over D}}; q) _{{n_1}} }}
     {} _{2} \phi _{1} \! \left [ \matrix \let \over / D {q^{-{n_1} + {
      n_2}}}, {q^{-{n_2}}}\\ \let \over / {{{q^{1 - {n_1}}}}\over
      C}\endmatrix ;q, {\displaystyle q} \right ]\\
&\label{DB1}
=     {q^{{{{n_1}}^2} - {n_1} {n_2} + {{{n_2}}^2}}}
{C^{{n_1}}} {D^{{n_2}}} 
{{ 
     ({\let \over / C}; q) _{{n_1}}  \, ({\let \over / D}; q) _{{n_2}}  \,
     ({\let \over / {{{q^{1 - {n_1} - {n_2}}}}\over {C D}}}; q) _{{
      n_1} + {n_2}} }\over 
   {({\let \over / q}; q) _{{n_1}}  \, ({\let \over / q}; q) _{{n_2}}  \,
     ({\let \over / {{{q^{1 - {n_1}}}}\over C}}; q) _{{n_2}}  \,
     ({\let \over / {{{q^{1 - {n_2}}}}\over D}}; q) _{{n_1}} }}.
\end{align}
Substituting this into the inverse relation (\ref{invrelg}) gives, after 
some simplification,
\begin{multline} \label{DoppelSumme1}
\sum _{l_1=0} ^{k_1}\sum _{l_2=0} ^{k_2}
 q^{(k_1-k_2)(l_1-l_2)} 
\frac {\big(( C q^{2 l_1}-q^{k_2} ) ( D q^{2 l_2}-q^{k_1} )
    -( q^{l_1}-q^{k_1} ) ( q^{l_2}-q^{k_2} ) \big)} 
{(1-C)(1-D)} \\
\cdot
\frac {(C; q)_{ l_1 - l_2} \,
    (D; q)_{ l_2-l_1 }\, (C D; q)_{ l_1 + l_2}\, (q^{-k_1}; q)_{ l_1} \,
    (q^{-k_2}; q)_{ l_2}} {  (Cq^{1+k_1-k_2}; q)_{ l_1} \,
    (Dq^{1+k_2-k_1}; q)_{ l_2}\,(q; q)_{ l_1} \,(q; q)_{ l_2}}\\
=(-1)^{k_1 + k_2}C^{k_2}D^{k_1}q^{k_1k_2-\binom {k_1}2 -\binom {k_2}2}
(Cq;q)_{k_1-k_2}\, (Dq;q)_{k_2-k_1}.
\end{multline}
Setting $k_2=0$ in this identity, we obtain 
\begin{equation} \label{EinfachSumme1}
{{     {} _{5} \phi _{5} \! \left [ \matrix \let \over / C, {\sqrt{C}} q,
      - {\sqrt{C}} q  , C D, {q^{-{k_1}}}\\ \let \over /
      {\sqrt{C}}, -{\sqrt{C}}, {q\over D}, C {q^{1 + {k_1}}}, 0\endmatrix
      ;q, {\displaystyle {{{q^{1 + {k_1}}}}\over D}} \right ]}}
=\frac {(qC;q)_{k_1}} {(q/D;q)_{k_1}}.
\end{equation}
which is a terminating limiting case 
of the very-well-poised $_6\phi_5$-summation (cf\@. \cite[Eq.~(2.7.1); 
Appendix (II.20)]{grhyp}).
Hence, our double sum identity (\ref{DoppelSumme1}) is a 
two-dimensional extension of the $_5\phi_5$-summation (\ref{EinfachSumme1}).

For our second application of the matrix inverse (\ref{DInv1})/(\ref{DInv2}),
in (\ref{invrelf}) we choose 
\begin{equation} \label{DA2}
a_{\bold k}=\delta_{k_1k_2}A^{k_1}q^{k_1(k_1-1)}
\frac {(C;q)_{k_1}\,(D;q)_{k_1}} {(q;q)_{k_1}\,(A;q)_{k_1}},
\end{equation}
where $\delta_{i,j}$ denotes the Kronecker delta, $\delta_{i,j}=1$ if $i=j$
and $\delta_{i,j}=0$ otherwise. Again, by using $q$-Chu--Vandermonde summation
(\ref{qChuVandermonde}), we obtain from (\ref{invrelf}),
\begin{equation} \label{DB2}
b_{\bold n}=\frac {(C;q)_{n_1}\,(D;q)_{n_2}\,(A;q)_{n_1+n_2}} 
{(q;q)_{n_1}\,(q;q)_{n_2}\,(A;q)_{n_1}\,(A;q)_{n_2}}.
\end{equation}
With these values of $a_{\bold k}$ and $b_{\bold n}$, the inverse relation
(\ref{invrelg})
then becomes
\begin{multline} \label{DoppelSumme2}
\sum _{l_1=0} ^{k_1} \sum _{l_2=0} ^{k_2}
q^{(k_1-k_2)(l_1-l_2)} 
\frac {\big(( C q^{2 l_1}-q^{k_2} ) ( D q^{2 l_2}-q^{k_1} )
   -( q^{l_1}-q^{k_1} ) ( q^{l_2}-q^{k_2} ) \big)} 
{(1-C)(1-D)} \\
\cdot
\frac {(A; q)_{ l_1 + l_2} \,
    (C; q)_{ l_1}\, (D; q)_{ l_2}\, (q^{-k_1}; q)_{ l_1} \,
    (q^{-k_2}; q)_{ l_2}} 
{ (A; q)_{ l_1}\, (A; q)_{ l_2} \,
    (C q^{1 + k_1 - k_2}; q)_{ l_1}\, (D q^{1 + k_2 - k_1 }; q)_{l_2}\,
   (q; q)_{ l_1}\, (q; q)_{ l_2} } \\
=\delta_{k_1k_2}A^{k_1}\frac {(q;q)_{k_1}} {(A)_{k_1}}.
\end{multline}
This is a two-dimensional extension of the terminating very-well-poised 
$_4\phi_3$-summation (cf\@. \cite[Eq.~(2.3.4)]{grhyp}), which in Chapter~2 of
\cite{grhyp} is used as one of the corner stones of building up the
summation theory for very-well-poised basic hypergeometric series.

\section{A ``twisted" multidimensional extension of a very-well-poised
$_{10}\phi_9$-summation}\label{sec10phi9}

In this section we bring an application of the companion inversion
(\ref{alf2})/(\ref{alinv2}). We start by making the replacements $c_i(t)\to 
{1}/ {c_i(t)}+A_ic_i(t)$,
$a_i(t)\to (A_{i+1}+a_i(t)^2)/c_{i}(t+1)$, and
\begin{equation}\label{detdiff}
\big(b_{ij}(t_i)\big)_{1\le i,j\le r}=
\begin{pmatrix} 0&-\frac {a_1(t_1)} {c_1(t_1+1)}&0&\dots&\dots&0\\
&\ddots&-\frac {a_2(t_2)} {c_2(t_2+1)}&\ddots&&\vdots\\
\vdots&&\ddots&\ddots&\ddots&\vdots\\
&&&\ddots&\ddots&0\\
0&&&&\ddots&-\frac {a_{r-1}(t_{r-1})} {c_{r-1}(t_{r-1}+1)}\\
-\frac {a_r(t_r)} {c_r(t_r+1)}&0&&\dots&&0\end{pmatrix}
\end{equation}
in (\ref{mainres2}). Upon little simplification we obtain that {\em the
matrices $(f_{\bold{nk}})_{\bold{n,k}\in{\Bbb Z}^r}$ and
$(g_{\bold{kl}})_{\bold{k,l}\in{\Bbb Z}^r}$ are inverses of each other, where
\begin{multline} \label{bressf}
f_{\bold n\bold k}=
{\textstyle \Big(\prodl _{i=1}
^{r}c_i(n_i)A_{i+1}(1-a_i(n_i)c_{i+1}(k_{i+1}))
(1-a_i(n_i)/A_{i+1}c_{i+1}(k_{i+1}))-}\\
\frac {\kern6cm
\prodl _{i=1} ^{r}a_i(n_i)(1-A_ic_i(n_i)c_i(k_i)) (1-c_i(n_i)/c_i(k_i))\Big)} 
{\prodl _{i=1} ^{r}(1-a_i(k_i)c_{i+1}(k_{i+1}))
(A_{i+1}c_{i+1}(k_{i+1})-a_i(k_i))}\\
\times
\prodl _{i=1} ^{r}\frac {\prodl _{t_i=k_i} ^{n_i-1}
(1-a_i(t_i)c_{i+1}(k_{i+1}))(1-a_i(t_i)/A_{i+1}c_{i+1}(k_{i+1}))} 
{{\prodl _{t_i=k_i+1} ^{n_i}
(1-A_ic_i(t_i)c_{i}(k_{i}))(1-c_i(t_i)/c_{i}(k_{i}))}}
\end{multline}
and
\begin{equation} \label{bressg}
g_{\bold k\bold l}=\prodl _{i=1} ^{r}\frac {\prodl _{t_i=l_i+1} ^{k_i}
(1-a_i(t_i)c_{i+1}(k_{i+1}))(1-a_i(t_i)/A_{i+1}c_{i+1}(k_{i+1}))} 
{{\prodl _{t_i=l_i} ^{k_i-1}
(1-A_ic_i(t_i)c_{i}(k_{i}))(1-c_i(t_i)/c_{i}(k_{i}))}}.
\end{equation}
}
Here and in the following we make the convention that indices have to
be taken modulo $r$, i.e., by $k_{r+1}$ we mean $k_1$, etc.
The above matrix inversion is a ``twisted" extension to several dimensions
of the ``Bressoud-type" writing~\cite[(1.5)]{kratt-mi} of the
one-dimensional matrix inversion (\ref{krattf})/(\ref{krattg}), 
to which it reduces for $r=1$.

Now, in (\ref{bressf})/(\ref{bressg}) we specialize $c_i(t)=q^t$ and
$a_i(t)=a_iq^t$. Thus, we obtain the inverse pair of matrices 
$(f_{\bold{nk}})_{\bold{n,k}\in{\Bbb Z}^r}$ and
$(g_{\bold{kl}})_{\bold{k,l}\in{\Bbb Z}^r}$, where
\begin{multline} \label{bressf1}
f_{\bold n\bold k}=q^{\vert\bold n\vert}\frac {\Big(\prodl _{i=1}
^{r}A_i(1-a_iq^{n_i+k_{i+1}})(1-a_iq^{n_i-k_{i+1}}/A_{i+1})-
\prodl _{i=1} ^{r}a_i(1-A_iq^{n_i+k_i})(1-q^{n_i-k_i})\Big)} 
{\prodl _{i=1}
^{r}(1-a_iq^{k_i+k_{i+1}})(A_{i+1}q^{k_{i+1}}-a_iq^{k_i})}\\
\times
\prod _{i=1} ^{r}\frac {(a_iq^{k_i+k_{i+1}};q)_{n_i-k_i}\,
(a_iq^{k_i-k_{i+1}}/A_{i+1};q)_{n_i-k_i}} {(A_iq^{2k_i+1};q)_{n_i-k_i}\,
(q;q)_{n_i-k_i}}
\end{multline}
and
\begin{equation} \label{bressg1}
g_{\bold k\bold l}=\prod _{i=1} ^{r}\frac {(a_iq^{k_i+k_{i+1}};q^{-1})_{k_i-l_i}\,
(a_iq^{k_i-k_{i+1}}/A_{i+1};q^{-1})_{k_i-l_i}} {(A_iq^{2k_i-1};q^{-1})_{k_i-l_i}\,
(q^{-1};q^{-1})_{k_i-l_i}}.
\end{equation}
This matrix inversion is a ``twisted" extension to several dimensions
of Bressoud's matrix inversion \cite{bress},
to which it reduces for $r=1$.

For our application of (\ref{bressf1})/(\ref{bressg1}), in
(\ref{invrelg}) we choose 
\begin{equation} \label{bressb}
b_{\bold l}=q^{\vert\bold l\vert}\prod _{i=1} ^{r}\frac
{(qa_i^2/A_iA_{i+1};q)_{l_i}} {(q;q)_{l_i}}.
\end{equation}
Then the left-hand side of (\ref{invrelg}) can be written as
\begin{equation} \notag
\prod _{i=1} ^{r}\left(\frac {(q^{k_{i+1}+1}a_i;q)_{k_i}\,
(a_iq^{1-k_{i+1}}/A_{i+1};q)_{k_i}} {(q^{-k_i};q)_{k_i}\,
(A_iq^{k_i};q)_{k_i}}
{}_3\phi_2\!\left[\begin{matrix} q^{k_i}A_i,qa_i^2/A_iA_{i+1},q^{-k_i}\\
q^{1-k_{i+1}}a_i/A_{i+1},q^{k_{i+1}+1}a_i\end{matrix}; q,
q\right]\right).
\end{equation}
The $_3\phi_2$-series appearing in this expression can be evaluated
by means of the $q$-Pfaff--Saalsch\"utz summation (cf\@.
\cite[(1.7.2); Appendix (II.12)]{grhyp}),
$$
{}_3\phi _2\!\left [ \begin{matrix} \let\over/ a,b,{q^{-n}}\\ \let\over/  c,{{a b {q^{1 - n}}}\over
   c}\end{matrix} ;q,q\right ] = {{{(\let\over/ {c\over a};q)}_{n} {(\let\over/ {c\over b};q)}_{n}}\over 
    {{(\let\over/ c;q)}_{n} {(\let\over/ {c\over {a b}};q)}_{n}}},
$$
where $n$ is a nonnegative integer. Thus we obtain
\begin{equation} \label{bressa}
a_{\bold k}=\prod _{i=1} ^{r}q^{k_i(k_i+k_{i+1}+1)}a_i^{k_i}
\frac {(A_iq^{-k_{i+1}}/a_i;q)_{k_i}\,
(q^{1-k_i-k_{i+1}}/A_iA_{i+1};q)_{k_i}} {(q^{k_i}A_i;q)_{k_i}\,
(q;q)_{k_i}}.
\end{equation}
Substitution of (\ref{bressb}) and (\ref{bressa}) in (\ref{invrelf}), and some
simplification, leads to the following summation theorem.
\begin{Theorem} \label{T10phi9}Let $a_i$ and $A_i$ be indeterminates,
$i=1,2,\dots,r$. Then
\begin{multline} \label{eq10phi9}
\sum _{\bold 0\le \bold k\le \bold n} ^{}
\frac {\Big(\prodl _{i=1}
^{r}A_i(1-a_iq^{n_i+k_{i+1}})(1-a_iq^{n_i-k_{i+1}}/A_{i+1})-
\prodl _{i=1} ^{r}a_i(1-A_iq^{n_i+k_i})(1-q^{n_i-k_i})\Big)} 
{\prodl _{i=1}
^{r}(1-a_iq^{k_i+k_{i+1}})(A_{i+1}q^{k_{i+1}}-a_iq^{k_i})}\\
\cdot
\prod _{i=1} ^{r}q^{2k_i}\left(\frac {a_i^2q^{n_i}}
{A_iA_{i+1}}\right)^{k_i-k_{i+1}}\frac {(1-q^{2k_i}A_i)} {(1-A_i)}
\frac {(A_i;q)_{k_i}\, (q^{-n_i};q)_{k_i}} 
{(q;q)_{k_{i}}\, (q^{1+n_i}A_i;q)_{k_{i}}}\kern2.5cm\\
\cdot
\frac {(q^{n_i}a_i;q)_{k_{i+1}}\, 
(qa_i/A_i;q)_{k_{i+1}}\, (A_iA_{i+1}/a_i;q)_{k_i+k_{i+1}}\, 
(A_i/a_i;q)_{k_i-k_{i+1}}} 
{(q^{1-n_i}A_{i+1}/a_i;q)_{k_{i+1}}\, (A_iA_{i+1}/a_i;q)_{k_{i+1}}\, 
(a_i;q)_{k_i+k_{i+1}}\, (a_i/A_{i+1};q)_{k_i-k_{i+1}}}\\
=\prod _{i=1} ^{r}\frac {(qa_i^2/A_iA_{i+1};q)_{n_i}\, (qA_i;q)_{n_i}} 
{(a_i;q)_{n_i}\, (a_i/A_{i+1};q)_{n_i}},
\end{multline}
where, by convention, $A_{r+1}=A_1$ and $k_{r+1}=k_1$.
\end{Theorem}

\begin{Remark} The special case $r=1$ of (\ref{eq10phi9}) can be
rewritten as (when writing $a$ for $a_1$, $A$ for $A_1$, and $n$ for
$n_1$)
\begin{multline}
{} _{10} \phi _{9} \! \left [ \matrix \let \over / A, {\sqrt{A}} q, -
  {\sqrt{A}} q  , {A\over {{\sqrt{a}}}}, -{A\over {{\sqrt{a}}}},
  {{A {\sqrt{q}}}\over {{\sqrt{a}}}}, -{{A {\sqrt{q}}}\over {{\sqrt{a}}}},
  {{a q}\over A}, a {q^n}, {q^{-n}}\\ \let \over / {\sqrt{A}}, -{\sqrt{A}},
  {\sqrt{a}} q, - {\sqrt{a}} q  , {\sqrt{a q}},
  -{\sqrt{a q}}, {{{A^2}}\over a}, {{A {q^{1 - n}}}\over a}, A {q^{1 +
  n}}\endmatrix ;q, {\displaystyle q} \right ]\\
=\frac {(1-a)} {(1-aq^{2n})}\frac {(qa^2/A^2;q)_n\, (qA;q)_{n}}
{(a;q)_n\, (a/A;q)_n}.
\end{multline}
This $_{10}\phi_9$-summation can be obtained by specializing Bailey's
transformation formula \cite{bailey} (see \cite[(2.8.5); 
Appendix~(III.27)]{grhyp}; the
specializations that have to be performed there are $d=1$, $b\to
a^2q/cA$, $a\to A^2/aq$, in that order).
\end{Remark}




\section{$q$-Analogues of Carlitz' Abel-type expansion formulas}\label{seccarlitz}

In \cite{carlitz2}, Carlitz gave multidimensional extensions of
Euler's formulas
\begin{align} \label{euler}
e^{AZ}=\suml _{k=0} ^{\infty}\frac {A(A+Bk)^{k-1}} {k!}
Z^ke^{-BZk},
\end{align}
where $\left|BZe^{1-BZ}\right|<1$ \cite[p.~354]{euler}
(cf.~\cite[Sec.~4.5]{riord}), and 
\begin{align} \label{euler2}
(1+Z)^A=\suml _{k=0} ^{\infty}\frac {A}{A+Bk} {A+Bk\choose k}
Z^k(1+Z)^{-Bk},
\end{align}
where $\left|\frac {(B-1)Z}{(1+Z)^B}\right|<1$ \cite[p.~350]{euler} 
(cf.~\cite[Sec.~4.5]{riord}).
The purpose of this section is to present $q$-analogues of
(\ref{euler}) and (\ref{euler2}) and to derive $q$-analogues of
Carlitz' multidimensional extensions thereof.

First we derive a simple multidimensional extension of the
expansion formula
\begin{align} \label{qeuler}
1=\suml _{k=0} ^{\infty}\frac {(a+b) (a+bq^k)^{k-1}} {(q;q)_k}
(z(a+bq^k);q)_{\infty}\,z^k,
\end{align}
valid for $|az|<1$, which is a $q$-analogue of Euler's formula
(\ref{euler}).

To see that \eqref{qeuler} is a $q$-analogue of \eqref{euler},
do the replacements $a\to 1-q^A+\frac {1-q^B}{1-q}$,
$b\to-\frac {1-q^B}{1-q}$, $z\to Z$ and then let $q\to 1$.
In this case, $\lim_{q\to1}\frac{a+bq^k}{1-q}=A+Bk$.
Also, recall that $\lim_{q\to 1}((1-q)Z;q)_{\infty}=e^{-Z}$.

The second formula that we extend to several dimensions is
\begin{align} \label{qeuler2}
(z;q)_{\infty}=\suml _{k=0} ^{\infty}
\frac {1-(a+b)}{1-(aq^{-k}+b)}\frac {(aq^{-k}+b;q)_{k}} {(q;q)_k}
(-1)^kq^{k\choose 2}(z(a+bq^k);q)_{\infty}\,z^k,
\end{align}
valid for $|az|<1$, which in turn is a $q$-analogue of
Euler's formula (\ref{euler2}).

To see that \eqref{qeuler2} is a $q$-analogue of \eqref{euler2},
do the replacements $a\to q^A-\frac{1-q^B}{1-q}$,
$b\to\frac {1-q^B}{1-q}$, $z\to-Z$ and then let $q\to 1$.
In this case, $\lim_{q\to1}\frac{1-(aq^{-k}+b)q^j}{1-q}=A+Bk+j-k$.
Furthermore, we use $\lim_{q\to 1}\frac {(z(a+bq^k);q)_{\infty}}
{(z;q)_{\infty}}=(1+Z)^{-A-Bk}$.
Similar limiting processes apply for others of the identities
given in the following sections, especially for our multidimensional formulas.

Now we state our multiple extension of \eqref{qeuler}:

\begin{Theorem}\label{meuler}
Let $a_i,b_{ij},z_i$, $i,j=1,\dots,r$, be indeterminate.
Then there holds
\begin{multline}\label{meulergl}
1=\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\Detl_{1\le i,j\le r}\Big(
\left(a_i+{\textstyle\sum\nolimits
_{s=1} ^{r}b_{is}q_s^{k_s}}\right)\delta_{ij}+
b_{ij}(1-q_i^{k_i})\Big)\\
\times\prodl_{i=1}^{r}\frac
{\left(a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j}\right)^{k_i-1}}{(q_i;q_i)_{k_i}}
\left(z_i(a_i+{\textstyle\sum\nolimits
_{j=1} ^{r}b_{ij}q_j^{k_j}});q_i\right)_{\infty}\,
z_i^{k_i}\Bigg),
\end{multline}
provided $|a_iz_i|<1$ for $i=1,\dots,r$.
\end{Theorem}

\begin{Remark} The expansion \eqref{meulergl} is a $q$-analogue
of Carlitz' formula~\cite[Eq.~(3.5)]{carlitz2} which he derived
via MacMahon's Master Theorem. To obtain his result we would have
to do the replacements
$a_i\to 1-q_i^{A_i}+\sum\nolimits_{j=1}^r\frac {1-q_i^{B_{ij}}}{1-q_i}$,
$b_{ij}\to -\frac {1-q_i^{B_{ij}}}{1-q_i}$,
and then let $q_i\to 1$ for $i=1,\dots,r$ (compare with our observation
concerning equation~\eqref{qeuler}).
\end{Remark}

\begin{pf*}{Proof of Theorem~\ref{meuler}}
Setting $c_i(t_i)\to q_i^{t_i}$, $a_i(t_i)\to a_i$,
$b_{ij}(t_i)\to b_{ij}$, $i,j=1,\dots,r$ in Theorem~\ref{mainres} we see
that the following pair of matrices are inverses of each other: 
\begin{equation*}
f_{\bold{n k}}=
(-1)^{|\bold n|-|\bold k|}
\prodl_{i=1}^{r}\frac{\left(a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j}\right)
^{n_i-k_i}}{(q_i;q_i)_{n_i-k_i}}q_i^{n_i-k_i \choose 2}
\end{equation*}
and
\begin{equation*}
g_{\bold{k l}}=
\frac {\Detl_{1\le i,j\le r}\Big(
\left(a_i+\sum _{s=1} ^{r}b_{is}q_s^{k_s}\right)\delta_{ij}+
b_{ij}(q_i^{l_i}-q_i^{k_i})\Big)}
{\prodl_{i=1}^{r}\left(a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j}\right)}
\prodl_{i=1}^{r}\frac
{\left(a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j}\right)
^{k_i-l_i}}{(q_i;q_i)_{k_i-l_i}}.
\end{equation*}
Now \eqref{rotinvf} holds for
$$a_{\bold{n}}=\prod\nolimits_{i=1}^rz_i^{n_i}\qquad\qquad
\text{and}\qquad\qquad
b_{\bold{k}}=\prod\nolimits_{i=1}^rz_i^{k_i}
\left(z_i(a_i+{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j})};q_i\right)_{\infty},$$
by $r$-fold application of the $_0\phi_0$-summation
(which is a $q$-analogue of the exponential function
\cite[Eq.~(1.3.16); Appendix~(II.2)]{grhyp}). This implies the inverse relation
\eqref{rotinvg}, with the above values of $a_{\bold n}$ and $b_{\bold
k}$. After shifting the indices $k_i\to k_i+l_i$,
$i=1,\dots,r$, and substituting the variables
$b_{ij}\to b_{ij}q_j^{-l_j}$, $i,j=1,\dots,r$, we get rid of
the $l_i$ and eventually obtain \eqref{meulergl}.
\end{pf*}

Next we give a multidimensional extension of \eqref{qeuler2}:

\begin{Theorem}\label{meuler2}
Let $a_i,b_{ij},z_i$, $i,j=1,\dots,r$, be indeterminate.
Then there holds
\begin{multline}\label{meuler2gl}
\prodl_{i=1}^r(z_i;q_i)_{\infty}=
\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\frac{\Detl_{1\le i,j\le r}\Big(
\left(a_i-1+\sum _{s=1} ^{r}b_{is}q_s^{k_s}\right)\delta_{ij}+
b_{ij}(1-q_i^{k_i})\Big)}
{\prodl_{i=1}^{r}\left(a_i-q_i^{k_i}+\sum _{j=1}
^{r}b_{ij}q_j^{k_j}\right)}\\
\times\,\prodl_{i=1}^{r}\frac
{\left((a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j})q_i^{-k_i};q_i\right)
_{k_i}}{(q_i;q_i)_{k_i}}\,q_i^{k_i+1\choose 2}\cdot (-1)^{|\bold k|}\\
\times\,\prodl_{i=1}^{r}\left(z_i(a_i+
{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j}});q_i\right)_{\infty}\,
z_i^{k_i}\Bigg),
\end{multline}
provided $|a_iz_i|<1$ for $i=1,\dots,r$.
\end{Theorem}

\begin{Remark}\label{remeuler2}
The expansion~\eqref{meuler2gl} is a $q$-analogue of Carlitz'
formula~\cite[Eq.~(6.5)]{carlitz2} which he also derived
via MacMahon's Master Theorem. To obtain his result we would
have to do the replacements
$a_i\to q_i^{A_i}-\sum\nolimits_{j=1}^r\frac {1-q_i^{B_{ij}}}{1-q_i}$,
$b_{ij}\to\frac {1-q_i^{B_{ij}}}{1-q_i}$,
$z_i\to -Z_i$, and then let $q_i\to 1$ for $i=1,\dots,r$
(compare with our observation concerning equation~\eqref{qeuler2}).
\end{Remark}

\begin{pf*}{Proof of Theorem~\ref{meuler2}}
Setting $c_i(t_i)\to q_i^{t_i}$, $a_i(t_i)\to a_i-q_i^{t_i}$,
$b_{ij}(t_i)\to b_{ij}$, $i,j=1,\dots,r$ in Theorem~\ref{mainres}
we see that the following pair of matrices are inverses of each other: 
\begin{equation*}
f_{\bold{n k}}=
(-1)^{|\bold n|-|\bold k|}
\prodl_{i=1}^{r}\frac{\left(q_i^{k_i}/
(a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j});q_i\right)_{n_i-k_i}}
{(q_i;q_i)_{n_i-k_i}}
\left(a_i+{\textstyle\sum\nolimits
_{j=1} ^{r}b_{ij}q_j^{k_j}}\right)^{n_i-k_i}
q_i^{n_i-k_i \choose 2}
\end{equation*}
and
\begin{multline*}
g_{\bold{k l}}=
\frac {\Detl_{1\le i,j\le r}\Big(
\left(a_i-q_i^{l_i}+\sum _{s=1} ^{r}b_{is}q_s^{k_s}\right)\delta_{ij}+
b_{ij}(q_i^{l_i}-q_i^{k_i})\Big)}
{\prodl_{i=1}^{r}\left(a_i-q_i^{k_i}+
\sum _{j=1} ^{r}b_{ij}q_j^{k_j}\right)}\\
\times(-1)^{|{\bold k}|-|{\bold l}|}
\prodl_{i=1}^{r}\frac
{\left((a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j})q_i^{-k_i};q_i\right)
_{k_i-l_i}}{(q_i;q_i)_{k_i-l_i}}\,
q_i^{{k_i+1\choose 2}-{l_i+1\choose 2}}.
\end{multline*}
Now \eqref{rotinvf} holds for
$$
a_{\bold{n}}=
\prod_{i=1}^r\frac{z_i^{n_i}}{(z_i;q_i)_{n_i}}\qquad\quad
\text{and}\qquad\quad
b_{\bold{k}}=
\prod_{i=1}^r\frac{\left(z_i(a_i+
{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j}});q_i\right)_{\infty}}
{(z_i;q_i)_{\infty}}
z_i^{k_i},$$
by $r$-fold application of the ${}_1\phi_1$-summation
(see~\cite[Appendix~(II.5)]{grhyp}). This implies the inverse relation
\eqref{rotinvg}, with the above values of $a_{\bold n}$ and $b_{\bold
k}$. After shifting the indices $k_i\to k_i+l_i$,
$i=1,\dots,r$, and substituting the variables
$z_i\to z_iq_i^{-l_i}$,
$a_i\to a_iq_i^{l_i}$, $b_{ij}\to b_{ij}q_i^{l_i}q_j^{-l_j}$,
$i,j=1,\dots,r$, we get
rid of the $l_i$ and eventually obtain \eqref{meuler2gl}.
\end{pf*}

We consider Theorems~\ref{meuler} and~\ref{meuler2} as being
of ``simple type", as we have started with
products of classical (one-dimensional) summations, even with
independent bases ($q_1,\dots,q_r$), and then applied
inversion. Our next theorem, a multiple extension of \eqref{qeuler2}, is not as
``simple" in this respect, as we start with a genuine $r$-dimensional
summation which needs to have only one base $q$. For the derivation
of our multiple extension of \eqref{qeuler2}, in
Theorem~\ref{meuler2a} we apply rotated inversion to the
following multidimensional $_1\phi_1$-sum:

\begin{Lemma}[An $\bar$ $\boneph$-sum]\label{phi11}
Let $x_1,\dots,x_r$, $a$ and $c$ be indeterminate.
There holds the following summation: 
\begin{multline}\label{phi11gl}
\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\prodl_{1\le i<j\le r}\left(\frac{q^{-k_i}/x_i-q^{-k_j}/x_j}
{1/x_i-1/x_j}\right)\,
\prodl_{i=1}^{r}\frac {(ax_i;q)_{k_i}}
{(q;q)_{k_i}(cx_i;q)_{k_i}}\,\cdot
(-1)^{|{\bold k}|}q^{\sum_{i=1} ^{r}{k_i\choose 2}}
\left(\frac {c} {a}\right)^{|{\bold k}|}\Bigg)\\
=\prodl_{i=1}^{r}\frac{(cq^{1-i}/a;q)_{\infty}}
{(cx_i;q)_{\infty}}.
\end{multline}
\end{Lemma}

Lemma~\ref{phi11} (which extends the classical $_1\phi_1$-summation
formula~\cite[Appendix~(II.5)]{grhyp}) is a special case of a
more general multidimensional $q$-Gau\ss{} summation formula which
originally came up in~\cite[(4.3.12)]{krattmaj}.
Such type of identities occurred
there in combinatorial studies of generating functions for specific
plane partitions. In the sequel the special type
of series has been studied by Gustafson and
the first author~\cite{guskranew}, \cite{guskradet},
and more recently by the second author~\cite{schlsumdet}.

Here is our other extension of \eqref{qeuler2}:

\begin{Theorem}\label{meuler2a}
Let $b_i,x_i$, $i=1,\dots,r$, $a$ and $z$ be indeterminate.
Then there holds
\begin{multline}\label{meuler2gla}
\prodl_{i=1}^r(zx_i;q)_{\infty}=
\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\prodl_{1\le i<j\le r}\left(\frac{q^{-k_i}/x_i-q^{-k_j}/x_j}
{1/x_i-1/x_j}\right)\,(-1)^{|\bold k|}\,
q^{\sum_{i=1}^r{k_i\choose 2}}\\
\times\left(1-\suml_{i=1}^r\frac {b_i(1-q^{k_i})}
{(x_i-a-{\textstyle\sum\nolimits
_{j=1}^rb_j q^{k_j}})}\right)\,\prodl_{i=1}^{r}\frac
{\left((a+\sum _{j=1} ^{r}b_j q^{k_j})q^{1-k_i}/x_i;q\right)
_{k_i}}{(q;q)_{k_i}}\\
\times\prodl_{i=1}^{r}\left(zq^{1-i}(a+
{\textstyle\sum\nolimits _{j=1} ^{r}b_j q^{k_j}});q\right)_{\infty}\,
x_i^{k_i}\,\cdot z^{|\bold k|}\Bigg),
\end{multline}
provided $|azq^{1-r}|<1$.
\end{Theorem}

\begin{Remark}
To the authors' knowledge, the expansion~\eqref{meuler2gla}
is not a $q$-analogue of any of the identities which have appeared
in literature yet. In particular, it is of different type than Carlitz'
formulas in \cite{carlitz1}, \cite{carlitz2}.
\end{Remark}

\begin{pf*}{Proof of Theorem~\ref{meuler2a}}
Setting $c_i(t_i)\to q^{t_i}$, $a_i(t_i)\to a-x_iq^{t_i}$,
$b_{ij}(t_i)\to b_j$, $i,j=1,\dots,r$ in Theorem~\ref{mainres} we see
that the following pair of matrices are inverses of each other: 
\begin{equation*}
f_{\bold{n k}}=
(-1)^{|\bold n|-|\bold k|}
\prodl_{i=1}^{r}\frac{\left(x_iq^{k_i}/
(a+\sum _{j=1} ^{r}b_{j}q^{k_j});q\right)_{n_i-k_i}}
{(q;q)_{n_i-k_i}}
\left(a+{\textstyle\sum\nolimits
_{j=1} ^{r}b_{j}q^{k_j}}\right)^{n_i-k_i}
q^{n_i-k_i \choose 2}
\end{equation*}
and
\begin{multline*}
g_{\bold{k l}}=
\frac {\Detl_{1\le i,j\le r}\Big(
\left(a-x_iq^{l_i}+\sum _{s=1} ^{r}b_s q^{k_s}\right)\delta_{ij}+
b_j(q^{l_i}-q^{k_i})\Big)}
{\prodl_{i=1}^{r}\left(a-x_iq^{k_i}+\sum _{j=1} ^{r}b_j q^{k_j}\right)}\\
\times (-1)^{|{\bold k}|-|{\bold l}|}
\prodl_{i=1}^{r}\frac
{\left((a+\sum _{j=1} ^{r}b_j q^{k_j})q^{-k_i}/x_i;q\right)
_{k_i-l_i}}{(q;q)_{k_i-l_i}}\,x_i^{k_i-l_i}
q^{{k_i+1\choose 2}-{l_i+1\choose 2}}.
\end{multline*}
Now \eqref{rotinvf} holds for
$$
a_{\bold{n}}=
\prodl_{1\le i<j\le r}\left(q^{-n_i}/x_i-q^{-n_j}/x_j\right)\,
\frac {z^{|{\bold n}|}}{\prod_{i=1}^r(zx_i;q)_{n_i}}$$
and
$$b_{\bold{k}}=
\prodl_{1\le i<j\le r}\left(q^{-k_i}/x_i-q^{-k_j}/x_j\right)
\frac {z^{|{\bold k}|}}{\prod_{i=1}^r(zx_i;q)_{\infty}}
\prod_{i=1}^r\left(zq^{1-i}(a+
{\textstyle\sum\nolimits _{j=1} ^{r}b_j
q^{k_j}});q\right)_{\infty},$$
by Lemma~\ref{phi11}.
This implies the inverse relation~\eqref{rotinvg}, with 
the above values of $a_{\bold n}$ and $b_{\bold k}$.
After shifting the indices $k_i\to k_i+l_i$,
$i=1,\dots,r$, and substituting the variables
$x_i\to x_iq^{-l_i}$, $b_i\to b_iq^{-l_i}$, $i=1,\dots,r$, we get
rid of the $l_i$.
In addition, we can simplify our determinant due to the rule
\begin{equation}\label{detsimpl}
\Detl_{1\le i,j\le r}\big(A_i\delta_{ij}+B_iC_j\big)=
{\textstyle\big(1+\sum_{i=1}^r\frac
{B_iC_i} {A_i}\big)\prod_{i=1}^rA_i},
\end{equation}
and eventually obtain \eqref{meuler2gla}.
\end{pf*}

Of course, by specializing equation~\eqref{meuler2gla}
we may also obtain an interesting formula for ordinary series.
If we do the replacements 
$a\to q^A-\sum\nolimits_{j=1}^r\frac {1-q^{B_j}}{1-q}$,
$b_i\to\frac{1-q^{B_i}}{1-q}$, $x_i\to q^{A-A_i}$, $i=1,\dots,r$, 
$z\to -Z$, then let $q\to 1$ and rewrite 
(compare with Remark~\ref{remeuler2}), we obtain the following
multidimensional generalization of \eqref{euler2}:

\begin{Theorem}\label{meuler2o}
Let $A_i,B_i$, $i=1,\dots,r$, and $Z$ be indeterminate.
Then there holds
\begin{multline}\label{meuler2glo}
(1+Z)^{{r\choose 2}+\sum_{i=1}^rA_i}
=\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\prodl_{1\le i<j\le r}\left(\frac{A_i-k_i-A_j+k_j}
{A_i-A_j}\right)\cdot\left(1-\suml_{i=1}^r\frac {B_ik_i}
{(A_i+{\textstyle\sum\nolimits
_{j=1}^rB_jk_j})}\right)\\
\times\prodl_{i=1}^{r}{A_i+{\textstyle\sum\nolimits
_{j=1}^rB_j k_j}\choose k_i}\,
Z^{k_i}\,(1+Z)^{-B_ik_i}\Bigg),
\end{multline}
provided\/ $\big|\frac {(B_i-1)Z}{(1+Z)^{B_i}}\big|<1$ for $i=1,\dots,r$.
\end{Theorem}

\section{Multiple $q$-Abel and $q$-Rothe summations}\label{secrothe}
%\setcounter{equation}{0}

We can use the expansions~\eqref{qeuler} and \eqref{qeuler2}
to obtain terminating $q$-Abel and $q$-Rothe summations,
respectively (for $q$-summations of this type, also
see~\cite{bhatmil}, \cite{jacksonabel}, \cite{johnson}).
Later, we will apply the same method to derive
multidimensional generalizations of these formulas.

First, we apply the $q$-binomial theorem in the form
$$(z(a+bq^k);q)_{\infty}=(z;q)_{\infty}
\suml_{j=0}^{\infty} \frac{(a+bq^k;q)_j}{(q;q)_j}z^j
$$
to the right-hand side of \eqref{qeuler} and move $(z;q)_{\infty}$
to the left-hand side of \eqref{qeuler}.
Next, by equating coefficients of $z^n/(q;q)_n$ in the resulting
identity (again making use of the $q$-binomial theorem),
and employing the $q$-binomial coefficient notation
$$\begin{bmatrix}n\\ k\end{bmatrix}_q:=\frac {(q;q)_n} {(q;q)_k(q;q)_{n-k}},
$$
for nonnegative integers $k\le n$ (cf.~\cite[Appendix~(I.39)]{grhyp}),
we arrive at the following terminating summation:
\begin{equation} \label{qabel}
1=\suml _{k=0} ^n\begin{bmatrix}n\\ k\end{bmatrix}_q
(a+b) (a+bq^k)^{k-1}(a+bq^k;q)_{n-k}
\end{equation}
(see \cite{jacksonabel}). This is a $q$-analogue of Abel's theorem 
\begin{equation} \label{abel}
(A+C)^n=\suml _{k=0} ^{n}{n\choose k}
A(A+Bk)^{k-1}(C-Bk)^{n-k}.
\end{equation}
For, \eqref{abel} can easily be
obtained from \eqref{qabel} via the substitutions
$a\to \frac{1-q^{A/(A+C)}}{1-q}+\frac{1-q^{B/(A+C)}}{(1-q)^2}$,
$b\to -\frac{1-q^{B/(A+C)}}{(1-q)^2}$, and then letting $q\to 1$.

On the other hand, if we iterate \eqref{qeuler2} $r-1$ times we get
\begin{multline} \label{qeuler2it}
(z;q)_{\infty}=\suml _{k_1,\dots,k_r=0} ^{\infty}
\Bigg(\prodl_{i=1}^{r}\frac {1-(a_i+b_i)}{1-(a_iq^{-k_i}+b_i)}
\frac {(a_iq^{-k_i}+b_i;q)_{k_i}} {(q;q)_{k_i}}(-1)^{k_i}q^{k_i\choose 2}\\
\times\,\big(z\,{\textstyle\prod\nolimits_{i=1}^r
(a_i+b_iq^{k_i})};q\big)_{\infty}\,z^{|{\bold k}|}\,
\prodl_{i=1}^{r}\big(a_i+b_iq^{k_i}\big)^{\sum_{j=i+1}^rk_j}\Bigg).
\end{multline}
Now, after the following application of the $q$-binomial theorem
$$
\big(z\,{\textstyle\prod\nolimits_{i=1}^r
(a_i+b_iq^{k_i})};q\big)_{\infty}=(z/c;q)_{\infty}
\suml_{j=0}^{\infty} \frac{\big(c\,{\textstyle\prod\nolimits_{i=1}^r
(a_i+b_iq^{k_i})};q\big)_{j}}{(q;q)_j}\left(\frac {z} {c}\right)^j
$$
to the right-hand side of \eqref{qeuler2it} we may put
$(z/c;q)_{\infty}$ to the left-hand side of \eqref{qeuler2it}.
Next, by equating coefficients of $(z/c)^N$
(again making use of the $q$-binomial theorem),
we arrive at the following terminating summation:
\begin{multline} \label{qeuler2t}
\frac{(c;q)_N}{(q;q)_N}=\underset{0\le |{\bold k}|\le N}
{\suml _{k_1,\dots,k_r\ge 0}}
\Bigg(\prodl_{i=1}^{r}\frac {1-(a_i+b_i)}{1-(a_iq^{-k_i}+b_i)}
\frac {(a_iq^{-k_i}+b_i;q)_{k_i}} {(q;q)_{k_i}}(-1)^{k_i}q^{k_i\choose 2}\\
\times\,\frac{\big(c\,{\textstyle\prod\nolimits_{i=1}^r
(a_i+b_iq^{k_i})};q\big)_{N-|{\bold k}|}}{(q;q)_{N-|{\bold k}|}}
\prodl_{i=1}^{r}\big(a_i+b_iq^{k_i}\big)^{\sum_{j=i+1}^rk_j}
c^{|{\bold k}|}\Bigg).
\end{multline}
Identity \eqref{qeuler2t} may be viewed as a Gould-type generalization
of the $q$-multinomial theorem. The case $r=1$,
\begin{equation} \label{qrothe}
\frac{(c;q)_n}{(q;q)_n}=\suml _{k=0} ^{n}
\frac {1-(a+b)}{1-(aq^{-k}+b)}
\frac {(aq^{-k}+b;q)_{k}}{(q;q)_k}\frac{(c\,(a+bq^k);q)_{n-k}}
{(q;q)_{n-k}}\,(-1)^kq^{k\choose 2}\,c^{k}
\end{equation}
(see \cite{johnson}),
is a $q$-analogue of the (Hagen-)Rothe summation formula~\cite{gouldfvc}
\begin{equation} \label{rothe}
{A+C\choose n}=\suml _{k=0} ^{n}
\frac{A}{A+Bk} {A+Bk\choose k} {C-Bk\choose n-k},
\end{equation}
for \eqref{rothe} can be obtained from \eqref{qrothe} via the
substitutions $a\to q^A-\frac{1-q^B}{1-q}$,
$b\to\frac {1-q^B}{1-q}$, $c\to q^{-A-C}$, and then letting $q\to 1$.

Many more similar convolution formulas, in the $q=1$ case,
are listed in~\cite{gouldci} (also see~\cite{strehlrothe}),
whereas more $q$-Abel and $q$-Rothe
summations can be found in~\cite{johnson},
where these are derived by means of umbral calculus. 


We start our multidimensional exposition with a
multiple $q$-Abel summation:
\begin{Theorem}\label{qabelm}
Let $a_i,b_{ij}$, $i,j=1,\dots,r$, be indeterminate,
and let $n_1,\dots,n_r$ be nonnegative integers. Then there holds
\begin{multline} \label{qabelmgl}
1=\suml _{\bold{0\le k\le n}}\Bigg(
\Detl_{1\le i,j\le r}\Big(
\left(a_i+{\textstyle\sum\nolimits
_{s=1} ^{r}b_{is}q_s^{k_s}}\right)\delta_{ij}+
b_{ij}(1-q_i^{k_i})\Big)\\
\times\prodl_{i=1}^{r}
\begin{bmatrix}n_i\\ k_i\end{bmatrix}_{q_i}
\left(a_i+{\textstyle\sum\nolimits
 _{j=1} ^{r}b_{ij}q_j^{k_j}}\right)^{k_i-1}
\big(a_i+{\textstyle\sum\nolimits
_{j=1} ^{r}b_{ij}q_j^{k_j}};q_i\big)_{n_i-k_i}\Bigg).
\end{multline}
\end{Theorem}
Starting with the identity~\eqref{meulergl},
Theorem~\ref{qabelm} is proved exactly as in the one
variable case~\eqref{qabel},
by an $r$-fold application of the $q$-binomial theorem
and a comparison of coefficients.
Our multidimensional $q$-Abel summation theorem is a $q$-analogue
of Carlitz' formula~\cite[Eq.~(3.8)]{carlitz2} which basically
can be obtained from~\eqref{qabelmgl} via the substitutions
$a_i\to \frac{1-q^{A_i/(A_i+C_i)}}{1-q_i}+
\sum_{j=1}^r\frac{1-q^{B_{ij}/(A_i+C_i)}}{(1-q_i)^2}$,
$b_{ij}\to -\frac{1-q^{B_{ij}/(A_i+C_i)}}{(1-q_i)^2}$, 
and then letting $q_i\to 1$, for $i=1,\dots,r$.

A multidimensional $q$-Abel summation of a different type is given
in Bhatnagar and Milne~\cite{bhatmil}.

Concerning the next two theorems, we could also have given
multidimensional generalizations of the Gould-type
$q$-multinomial convolution~\eqref{qeuler2t},
but have decided to restrict ourselves to stating
the special cases which are multiple $q$-Rothe summations:
\begin{Theorem}\label{qrothem}
Let $a_i,b_{ij},c_i$, $i,j=1,\dots,r$, be indeterminate,
and let $n_1,\dots,n_r$ be nonnegative integers. Then there holds
\begin{multline}\label{qrothemgl}
\prodl_{i=1}^r\frac{(c_i;q_i)_{n_i}}{(q_i;q_i)_{n_i}}=
\suml _{\bold{0\le k\le n}}\Bigg(
\frac{\Detl_{1\le i,j\le r}\Big(
\left(a_i-1+\sum _{s=1} ^{r}b_{is}q_s^{k_s}\right)\delta_{ij}+
b_{ij}(1-q_i^{k_i})\Big)}
{\prodl_{i=1}^{r}\left(a_i-q_i^{k_i}+\sum _{j=1}
^{r}b_{ij}q_j^{k_j}\right)}\\
\times\,\prodl_{i=1}^{r}\frac
{\left((a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j})q_i^{-k_i};q_i\right)
_{k_i}}{(q_i;q_i)_{k_i}}\,q_i^{k_i+1\choose 2}\cdot (-1)^{|\bold k|}\\
\times\,\prodl_{i=1}^{r}\frac{\left(c_i(a_i+
{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j}});q_i\right)_{n_i-k_i}}{(q_i;q_i)_{n_i-k_i}}\,
c_i^{k_i}\Bigg).
\end{multline}
\end{Theorem}
Starting with the identity~\eqref{meuler2gl},
Theorem~\ref{qrothem} is proved exactly as in the one
variable case~\eqref{qrothe},
by an $r$-fold application of the $q$-binomial theorem
and a comparison of coefficients.
Our multidimensional $q$-Rothe summation theorem is a $q$-analogue
of Carlitz' formula~\cite[Eq.~(6.10)]{carlitz2} which basically
can be obtained from~\eqref{qrothemgl} via the substitutions
$a_i\to q^{A_i}-\sum_{j=1}^r\frac{1-q^{B_{ij}}}{1-q_i}$,
$b_{ij}\to\frac{1-q^{B_{ij}}}{1-q_i}$, $c_i\to q_i^{-A_i-C_i}$,
and then letting $q_i\to 1$, for $i=1,\dots,r$.

\begin{Theorem}\label{qrothema}
Let $b_i,c_i,x_i$, $i=1,\dots,r$, and $a$ be indeterminate,
and let $n_1,\dots,n_r$ be nonnegative integers. Then there holds
\begin{multline}\label{qrothemagl}
\prodl_{i=1}^r\frac{(c_i;q)_{n_i}}{(q;q)_{n_i}}=
\suml _{\bold{0\le k\le n}}\Bigg(
\prodl_{1\le i<j\le r}\left(\frac{q^{-k_i}/x_i-q^{-k_j}/x_j}
{1/x_i-1/x_j}\right)\,(-1)^{|\bold k|}\,
q^{\sum_{i=1}^r{k_i\choose 2}}\\
\times\left(1-\suml_{i=1}^r\frac {b_i(1-q^{k_i})}
{(x_i-a-{\textstyle\sum\nolimits
_{j=1}^rb_j q^{k_j}})}\right)\,\prodl_{i=1}^{r}\frac
{\left((a+\sum _{j=1} ^{r}b_j q^{k_j})q^{1-k_i}/x_i;q\right)
_{k_i}}{(q;q)_{k_i}}\\
\times\prodl_{i=1}^{r}\frac{\left((a+
{\textstyle\sum\nolimits _{j=1} ^{r}b_j q^{k_j}})
c_iq^{1-i}/x_i;q\right)_{n_i-k_i}}
{(q;q)_{n_i-k_i}}\,c_i^{k_i}\Bigg).
\end{multline}
\end{Theorem}

Starting with the identity~\eqref{meuler2gla},
Theorem~\ref{qrothema} is proved exactly as in the one
variable case~\eqref{qrothe},
by an $r$-fold application of the $q$-binomial theorem
and a comparison of coefficients.
In identity~\eqref{qrothemagl}, if we make the substitutions
$a\to q^A-\sum_{j=1}^r\frac{1-q^{B_j}}{1-q}$,
$b_i\to\frac{1-q^{B_i}}{1-q}$, $c_i\to q^{-A_i-C_i}$,
$x_i\to q^{A-A_i}$, $i=1,\dots,r$, and then let $q\to 1$,
we obtain the following nice multidimensional
Rothe summation:
\begin{Theorem}\label{mrotheo}
Let $A_i,B_i,C_i$, $i=1,\dots,r$, be indeterminate,
and let $n_1,\dots,n_r$ be nonnegative integers. Then there holds
\begin{multline}\label{mrotheogl}
\prod_{i=1}^r{A_i+C_i\choose n_i}=
\suml _{\bold{0\le k\le n}}\Bigg(
\prodl_{1\le i<j\le r}\left(\frac{A_i-k_i-A_j+k_j}
{A_i-A_j}\right)\cdot\left(1-\suml_{i=1}^r\frac {B_ik_i}
{(A_i+{\textstyle\sum\nolimits
_{j=1}^rB_jk_j})}\right)\\
\times\prodl_{i=1}^{r}{A_i+{\textstyle\sum\nolimits
_{j=1}^rB_j k_j}\choose k_i}
{C_i+i-1-{\textstyle\sum\nolimits
_{j=1}^rB_j k_j}\choose n_i-k_i}\Bigg).
\end{multline}
\end{Theorem}

\begin{Remark}
In this section, we derived (multiple) $q$-Abel and $q$-Rothe
summations by manipulating the series expansions we had
obtained by rotated inversion in Section~\ref{seccarlitz} and
then extracting coefficients from them.
However, we also could have derived these terminating summations
directly by applying the inverse relations
\eqref{invrelf}/\eqref{invrelg}
combined with terminating $q$-binomial and $q$-Chu--Vandermonde
summations. In this case we would have utilized
the companion matrix inversion in Theorem~\ref{mainres2}. 
\end{Remark}

\section{A $q$-analogue of MacMahon's Master Theorem}\label{secmaster}
%\setcounter{equation}{0}

Here we derive a $q$-extension of MacMahon's Master
Theorem~\cite{macmahon}. Chu~\cite[Sec.~5]{chu} observed that
inverse relations imply MacMahon's Master Theorem. Basically, he
recovered Carlitz' multidimensional extension
of~\eqref{euler}~\cite[Eq.~(3.5)]{carlitz2} (or rather
the related formula in \cite[Eq.~(4.3)]{carlitz1})
by inverse relations, which by some further manipulations he
showed to be equivalent to MacMahon's celebrated theorem.

Letting $\langle{\bold z}^{\bold n}\rangle
f(\bold z)$ denote 
the coefficient of ${\bold z}^{\bold n}$ in $f(\bold z)$, the classical
version of MacMahon's Master Theorem can be stated as follows:
\begin{Theorem} \label{master}
Let $z_i,b_{ij}$, $i,j=1,\dots,r$, be indeterminate,
and let $n_1,\dots,n_r$ be arbitrary nonnegative integers. Then there holds
$$
\langle{\bold z}^{\bold n}\rangle\;
\prod_{i=1}^{r}\big(\sum\nolimits_{j=1}^rb_{ij}z_j\big)^{n_i}\;=\;
\langle{\bold z}^{\bold n}\rangle\,\left(\Detl_{1\le i,j\le r}
(\delta_{ij}-z_ib_{ij})\right)^{-1}.$$ 
\end{Theorem}

In deriving our $q$-analogue, Theorem~\ref{qmaster},
we basically ``$q$-extend'' Chu's analysis, but if
we would perform the whole matter with Theorem~\ref{meuler},
a $q$-extension of Carlitz' identity mentioned above, we would
just end up with the classical version of the Master Theorem.
Instead, in our derivation we utilize Theorem~\ref{meuler2},
a multiple extension of the $q$-expansion~\eqref{qeuler2}
(i.e., a $q$-analogue of~\cite[Eq.~(6.5)]{carlitz2}),
and are able to extend the whole analysis with additional bases
$q_1,q_2,\dots,q_r$. In our case certain $q$-operators come
into the game. 

Defining the shift operators $\cal E_{(b)}^{q_i}$ by
$\cal E_{(b)}^{q_i}b=q_ib$, our derivation
is based on rewriting the identity~\eqref{meuler2gl}
of Theorem~\ref{meuler2} in the form
\begin{multline*}
\prodl_{i=1}^r(z_i;q_i)_{\infty}=
\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\Detl_{1\le i,j\le r}\left(\delta_{ij}+z_ib_{ij}
\cal E_{(z_i)}^{q_i}\cal E_{(a_i)}^{1/q_i}
{\textstyle\prod\nolimits_{s\neq i}\cal E_{(b_{is})}^{q_s}
\cal E_{(b_{is})}^{1/q_i}}\right)\\
\times\prodl_{i=1}^{r}\frac
{\left((a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j})q_i^{1-k_i};q_i\right)
_{k_i}}{(q_i;q_i)_{k_i}}\,q_i^{k_i\choose 2}
\cdot (-1)^{|\bold k|}\\
\times\prodl_{i=1}^{r}\left(z_i(a_i+
{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j}});q_i\right)_{\infty}\,z_i^{k_i}\Bigg).
\end{multline*}
This is achieved by moving all the terms of the summand in (\ref{meuler2gl})
inside the determinant using linearity in the rows,
by termwise rewriting of the expressions in
the determinant, thereby introducing the shift operators, and then 
moving terms again outside of the determinant by linearity in
the rows.
For our purpose it is particularly pleasant
that now the determinant does not depend on the summation indices. 
If we transfer $\prod_{i=1}^r(z_i;q_i)_{\infty}$
to the right-hand side, we obtain
\begin{multline}\label{detunabh}
1=\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\Detl_{1\le i,j\le r}\left((1-z_i/q_i)\delta_{ij}+z_ib_{ij}
\cal E_{(z_i)}^{q_i}\cal E_{(a_i)}^{1/q_i}
{\textstyle\prod\nolimits_{s\neq i}\cal E_{(b_{is})}^{q_s}
\cal E_{(b_{is})}^{1/q_i}}\right)\\
\times\prodl_{i=1}^{r}\frac
{\left((a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j})q_i^{1-k_i};q_i\right)
_{k_i}}{(q_i;q_i)_{k_i}}\,q_i^{k_i\choose 2}
\cdot (-1)^{|\bold k|}\\
\times\prodl_{i=1}^{r}\frac {\left(z_i(a_i+
{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j}});q_i\right)_{\infty}}
{(z_i/q_i;q_i)_{\infty}}\,z_i^{k_i}\Bigg).
\end{multline}
Since the determinant does not depend on the summation indices
we can multiply both sides of~\eqref{detunabh} with the operator
inverse of the determinant. Then we obtain, after having
replaced $z_i$ by $z_iq_i$, $a_i$ by $a_i/q_i$, and $b_{ij}$ by
$b_{ij}/q_i$, for $i,j=1,\dots,r$, respectively,
\begin{multline}\label{detlgl}
\left(\Detl_{1\le i,j\le r}\left((1-z_i)\delta_{ij}+z_ib_{ij}
\cal E_{(z_i)}^{q_i}\right)\right)^{-1}\,{\bold 1}=
\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\prodl_{i=1}^{r}\frac
{\left((a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j})q_i^{-k_i};q_i\right)
_{k_i}}{(q_i;q_i)_{k_i}}\,q_i^{k_i+1\choose 2}\\
\times (-1)^{|\bold k|}\,
\prodl_{i=1}^{r}\frac {\left(z_i(a_i+
{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j}});q_i\right)_{\infty}}
{(z_i;q_i)_{\infty}}\,z_i^{k_i}\Bigg),
\end{multline}
where we have readily cancelled the operators in the determinant
which produce no powers of $q$ in the left-hand side expansion,
since the operator is applied to the constant polynomial 1
which we have denoted by ${\bold 1}$. 
For our $q$-Master Theorem we set $a_i\to 0$, $i=1,\dots,r$,
in \eqref{detlgl}. Then, we expand the right-hand side further
by means of the $q$-binomial theorem in order to extract
the coefficient of ${\bold z}^{\bold n}$:
\begin{multline}\label{detlgl2}
\left(\Detl_{1\le i,j\le r}\left((1-z_i)\delta_{ij}+z_ib_{ij}
\cal E_{(z_i)}^{q_i}\right)\right)^{-1}\,{\bold 1}\\
=\suml _{k_1,\dots,k_r=0}^{\infty}\prodl_{i=1}^{r}\Bigg(
\frac{(\sum _{j=1} ^{r}b_{ij}q_j^{k_j}q_i^{-k_i};q_i)
_{k_i}}{(q_i;q_i)_{k_i}}(-1)^{k_i}q_i^{k_i+1\choose 2}z_i^{k_i}\,
\suml _{l_i=0}^{\infty}
\frac {(\sum_{j=1} ^{r}
b_{ij}q_j^{k_j};q_i)_{l_i}}
{(q_i;q_i)_{l_i}}\,z_i^{l_i}\Bigg)\\
=\suml _{n_1,\dots,n_r=0}^{\infty}\prodl_{i=1}^{r}\left(
\frac{z_i^{n_i}}{(q_i;q_i)_{n_i}}
\suml _{k_i=0}^{n_i}\begin{bmatrix}n_i\\ k_i\end{bmatrix}_{q_i}
(-1)^{k_i}q_i^{k_i+1\choose 2}
({\textstyle\sum _{j=1} ^{r}b_{ij}q_j^{k_j}q_i^{-k_i}};q_i)
_{n_i}\right).
\end{multline}

Observe that
\begin{multline}\label{zeq1gl}
\prodl_{i=1}^r\frac {1}
{(q_i;q_i)_{n_i}}
\suml_{\bold{0\le k\le n}}\prodl_{i=1}^r\left(
\begin{bmatrix}n_i\\ k_i\end{bmatrix}_{q_i}
(-1)^{k_i}q_i^{{k_i+1\choose 2}-n_ik_i}\right)
p(z_1q_1^{k_1},\dots,z_rq_r^{k_r})
\Bigg|_{z_1=\dots z_r=1}\\
=\;\langle{\bold{z^n}}\rangle\,p(z_1,\dots,z_r),
\end{multline}
for polynomials $p(z_1,\dots,z_r)$ of degree $\le |{\bold n}|$, 
by iterated application of the
$q$-binomial theorem and linearity.
Besides, note that by defining the partial difference operators
$\cal D_i$ by $\cal D_i=((q_i-1)z_i)^{-1}(\cal E_{(z_i)}^{q_i}-I)$,
where $I$ denotes the identity operator (cf.~\cite{cigler}),
and using Sch\"utzenberger's~\cite{schuetzen} observation that
if $yx=qxy$, then
\begin{equation}\label{noncommgl}
(x+y)^n=\suml _{k=0} ^{n}\begin{bmatrix}n\\k\end{bmatrix}_qx^ky^{n-k},
\end{equation}
equation~\eqref{zeq1gl} may also be expressed more compactly as
$$\prodl_{i=1}^r\frac {(1-q_i)^{n_i}} {(q_i;q_i)_{n_i}}{\cal D}_i^{n_i}\,
p({\bold z})\bigg|_{z_1=\dots z_r=1}=
\;\langle{\bold{z^n}}\rangle\,p({\bold z}).
$$

Back to \eqref{detlgl2}, we recognize that we can
apply~\eqref{zeq1gl} to the inner sum on the right-hand side
of~\eqref{detlgl2}, with the instance
$$p(z_1,\dots,z_r)=\prod\nolimits_{i=1}^rz_i^{n_i}
({\textstyle\sum\nolimits_{j=1}^rb_{ij}z_j/z_i};q_i)_{n_i}.
$$

After this observation we have arrived at:
\begin{Theorem}[A $\boldsymbol q$-analogue of
MacMahon's Master Theorem]\label{qmaster}
Let $z_i$, $b_{ij}$, for $i,j=1,\dots,r$, be indeterminate,
and let $n_1,\dots,n_r$ be arbitrary nonnegative integers. Then there holds
\begin{equation}\label{qmastergl}
\langle{\bold z}^{\bold 0}\rangle\,\prod_{i=1}^{r}
\big(\sum\nolimits_{j=1}^rb_{ij}z_j/z_i;q_i\big)_{n_i}\;=\;
\langle{\bold z}^{\bold n}\rangle\,
\left(\Detl_{1\le i,j\le r}
\big((1-z_i)\delta_{ij}+z_ib_{ij}{\cal E}_i\big)\right)^{-1}\,{\bold 1},
\end{equation}
where ${\cal E}_i$ denotes the $q$-shift operator defined
by ${\cal E}_iz_i=q_iz_i$, and ${\bold 1}$ denotes the constant
polynomial 1.
\end{Theorem}

Theorem~\ref{qmaster} indeed includes
MacMahon's Master Theorem as a special case. Namely,
if we write~\eqref{qmastergl} in the way
$$
\langle{\bold z}^{\bold n}\rangle\,\prod_{i=1}^{r}\prod_{s=1}^{n_i}
\big(z_i-\sum\nolimits_{j=1}^rb_{ij}z_j q_i^{s-1}\big)\;=\;
\langle{\bold z}^{\bold n}\rangle\,
\left(\Detl_{1\le i,j\le r}
\big((1-z_i)\delta_{ij}+z_ib_{ij}{\cal E}_i\big)\right)^{-1}\,{\bold 1},
$$
we conveniently can see that the substitutions $q_i\to 1$,
$z_i\to az_i$, $b_{ij}\to -b_{ij}/a$, $i,j=1,\dots,r$, then $a\to 0$,
specialize to the classical case, Theorem~\ref{master}.

For illustration, we quickly verify the statement of Theorem~\ref{qmaster}
for one dimension ($r=1$). Writing $b_{11}=b$, $z_1=z$,
and ${\cal E}_1={\cal E}$, we want to check
$$
(b;q)_n\;=\;
\langle z^n\rangle\,(1-z+zb{\cal E})^{-1}\,{\bold 1}.
$$
We have
$$
(1-z+zb{\cal E})^{-1}{\bold 1}=
\suml_{n=0}^{\infty}(z-zb{\cal E})^n{\bold 1}=
\suml_{n=0}^{\infty}\suml_{k=0}^n
\begin{bmatrix}n\\k\end{bmatrix}_qz^k(-zb{\cal E})^{n-k}{\bold 1}
=\suml_{n=0}^{\infty}z^n\suml_{k=0}^{n}\begin{bmatrix}n\\k\end{bmatrix}_q
(-1)^kq^{k\choose 2}b^k,
$$
the second equality due to~\eqref{noncommgl}.
The last inner sum evaluates to the desired quantity by
the $q$-binomial theorem.

We plan to give a more detailed discussion of our $q$-analogue of
MacMahon's Master Theorem including several applications
in a forthcoming paper~\cite{krattschlqmt}.

\section{Additional expansion formulas}\label{secmore}
%\setcounter{equation}{0}

We want to mention some formulas which are closely related
to those we used and derived in Section~\ref{seccarlitz}, and which may also
be used to derive additional identities.

The expansion formulas (see \cite[Sec.~4.5]{riord})
\begin{align} \label{eulerc}
\frac{e^{AZ}}{1-BZ}=\suml _{k=0} ^{\infty}\frac {(A+Bk)^k} {k!}
Z^ke^{-BZk},
\end{align}
where $\left|BZe^{1-BZ}\right|<1$,
and
\begin{align} \label{euler2c}
\frac{(1+Z)^A}{1-\frac{BZ}{1+Z}}=\suml _{k=0} ^{\infty}{A+Bk\choose k}
Z^k(1+Z)^{-Bk},
\end{align}
where $\left|\frac {(B-1)Z} {(1+Z)^B}\right|<1$,
are companion identities of \eqref{euler} and \eqref{euler2},
respectively. $q$-Analogues of these identities are
\begin{align} \label{qeulerc}
\suml_{k=0}^{\infty}(-1)^kq^{k\choose 2}b^kz^k=
\suml _{k=0} ^{\infty}\frac {(a+bq^k)^k} {(q;q)_k}
(z(a+bq^k);q)_{\infty}\,z^k,
\end{align}
and
\begin{align} \label{qeuler2c}
(z;q)_{\infty}\suml _{k=0} ^{\infty}(b;q)_kz^k
=\suml _{k=0} ^{\infty}
\frac {(aq^{-k}+b;q)_{k}} {(q;q)_k}
(-1)^kq^{k+1\choose 2}(z(a+bq^k);q)_{\infty}\,z^k,
\end{align}
respectively, both being valid for $|az|<1$.
To see that these formulas are $q$-analogues
of the above we can make similar substitutions that were needed
ealier in the respective cases where we showed that
\eqref{qeuler} and \eqref{qeuler2} are $q$-analoques of
\eqref{euler} and \eqref{euler2}.

We have already given a multidimensional version
of~\eqref{qeuler2c}, see equation~\eqref{detlgl},
involving $q$-shift operators,
which appeared in our derivation of the $q$-Master Theorem.

The multidimensional version of \eqref{qeulerc},
\begin{multline}\label{detlgl1}
\left(\Detl_{1\le i,j\le r}\left(\delta_{ij}+z_ib_{ij}
{\cal E}_{(z_i)}^{q_i}\right)\right)^{-1}\,{\bold 1}\\
=\suml _{k_1,\dots,k_r=0}^{\infty}\Bigg(
\prodl_{i=1}^{r}\frac
{\left(a_i+\sum _{j=1} ^{r}b_{ij}q_j^{k_j};q_i\right)
^{k_i}}{(q_i;q_i)_{k_i}}
\left(z_i(a_i+{\textstyle\sum\nolimits _{j=1} ^{r}
b_{ij}q_j^{k_j}});q_i\right)_{\infty}\,z_i^{k_i}\Bigg),
\end{multline}
can easily be deduced from \eqref{meulergl} in the same manner
as \eqref{detlgl} was deduced from \eqref{meuler2gl}.

Identities \eqref{detlgl1} and \eqref{detlgl} themselves can
be used to derive additional higher-dimensional
(terminating) convolutions with the method we
demonstrated in Section~\ref{secrothe}.
These would include $q$-extensions of Carlitz' other multidimensional
convolution formulas~\cite[Eqs.~(3.9) and (6.9)]{carlitz2}.

It is also interesting to look for nontrivial cases where
the determinant in the multiple identities simplify.
These cases have higher chances to occur
naturally in combinatorial enumeration problems.
The specific type of multiple series we consider
occur, e.g., in~\cite{strehlhabil}.
We already had a case of nearly total factorization of the determinant
in Theorem~\ref{meuler2a}, where we could simplify
the determinant
due to \eqref{detsimpl}. Another case concerns the determinant
of a matrix having entries $\neq 0$ only in the principal
diagonal and the diagonal above (indices modulo $r$).
It is easy to see that in this case the determinant
can be reduced to the difference of two products (see for
example Section~\ref{sec10phi9}).

The following theorem provides such an example,
where we use the $q$-binomial coefficient notation
$$\begin{bmatrix}\alpha\\ k \end{bmatrix}_q:=
\frac {(q^{1+\alpha-k};q)_k} {(q;q)_k}
=\frac {(q^{-\alpha};q)_k} {(q;q)_k}(-q^{\alpha})^kq^{-{k\choose 2}},$$
for nonnegative integer $k$ and arbitrary $\alpha$
(cf.~\cite[Appendix (I.42),(I.43)]{grhyp}).

\begin{Theorem}\label{sturt}
Let $\alpha_i,z_i$, $i=1,\dots,r$, be indeterminate. Then there holds
\begin{equation}\label{sturtgl}
\suml_{m=0}^{\infty}q^{r{m\choose 2}+m\sum_{i=1}^r\alpha_i}
\prod_{i=1}^r\frac{z_i^m}{(-z_i;q)_m}
=\suml _{k_1,\dots,k_r=0}^{\infty}
\prod_{i=1}^rq^{k_i\choose 2}
\begin{bmatrix}\alpha_i+k_{i+1}\\ k_i \end{bmatrix}_q
\frac{z_i^{k_i}}{(-z_i;q)_{\alpha_i+k_{i+1}}},
\end{equation}
where the indices are written modulo $r$.
\end{Theorem}

\begin{pf}
We consider the special case of formula~\eqref{detlgl},
where we have $q_1,\dots,q_r=q$, $z_i\to -z_i/q$,
$a_i=0$, $b_{i,i+1}=q^{1+\alpha_i}$, for $i=1,\dots,r$ (mod $r$),
and where $b_{ij}=0$ if $j\neq i+1$ (mod $r$).
We also write ${\cal E}_i$ instead of ${\cal E}_{(z_i)}^q$,
for short. In this special case, the left-hand side of
\eqref{detlgl} is
\begin{multline*}
\left(\prod\nolimits_{i=1}^r(1+z_i/q)-
\prod\nolimits_{i=1}^rq^{\alpha_i}z_i{\cal E}_i\right)^{-1}\,{\bold 1}
=\left(1-\prod\nolimits_{i=1}^r(1+z_i/q)^{-1}
q^{\alpha_i}z_i{\cal E}_i\right)^{-1}
\prod\nolimits_{i=1}^r(1+z_i/q)^{-1}\,{\bold 1}\\
=\suml_{m=0}^{\infty}\prodl_{i=1}^r\left((1+z_i/q)^{-1}
q^{\alpha_i}z_i{\cal E}_i\right)^m (1+z_i/q)^{-1}\,{\bold 1}
=\suml_{m=0}^{\infty}q^{r{m\choose 2}+m\sum_{i=1}^r\alpha_i}
\prod_{i=1}^r\frac{z_i^m}{(-z_i/q;q)_{m+1}}.
\end{multline*}
It is even more straightforward to compute the
right-hand side of \eqref{detlgl} for our particular
choice of parameters. Finally, we multiply both sides
of the resulting identity by $\prod\nolimits_{i=1}^r(1+z_i/q)$
to obtain \eqref{sturtgl}.
\end{pf}

\begin{Remark}
Identity~\eqref{sturtgl} is a $q$-analogue of a special case of
Carlitz' formula~\cite[$\lambda =1$ in Eq.~(2.6)]{CarlAQ},
\begin{equation}\label{sturt1gl}
\frac{\prod_{i=1}^r(1+z_i)^{\alpha_i+1}}
{\prod_{i=1}^r(1+z_i)-\prod_{i=1}^rz_i}
=\suml _{k_1,\dots,k_r=0}^{\infty}
\prod_{i=1}^r\begin{pmatrix}\alpha_i+k_{i+1}\\ k_i \end{pmatrix}
\left(\frac{z_i}{1+z_{i-1}}\right)^{k_i}
\end{equation}
(again, indices are written modulo $r$),
which simply follows from \eqref{sturtgl} by the limit $q\to 1$.
Also Carlitz derived his formula
by specializing a more general expansion. It is worth noting
that \eqref{sturt1gl} (or rather an identity equivalent to
\eqref{sturt1gl} via substitutions)
was given combinatorial proofs~\cite{ConsAA}, \cite{GStuAA}, 
\cite{strehlhabil}.
It would also be interesting to find a combinatorial proof
of the multiple $q$-series identity~\eqref{sturtgl}.
\end{Remark}


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\end{document}










