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\def\GeKrAA{2}
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\topmatter
\title Alternative proof of a proposition on cylindric partitions 
by Alexei Borodin 
\endtitle
\author C.~Krattenthaler$^\dagger$
\endauthor
\affil 
Fakult\"at f\"ur Mathematik der Universit\"at Wien,\\
Nordbergstra{\ss}e 15, A-1090 Wien, Austria.\\
WWW: \tt http://www.mat.univie.ac.at/\~{}kratt
\endaffil
\address Fakult\"at f\"ur Mathematik der Universit\"at Wien,
Nordbergstrasze 15, A-1090 Wien, Austria.
\endaddress
%\dedicatory \enddedicatory
%\date \enddate
\thanks{$^\dagger$Research partially supported by the Austrian
Science Foundation FWF, grants Z130-N13 and S9607-N13,
the latter in the framework of the National Research Network
``Analytic Combinatorics and Probabilistic Number Theory"}\endthanks
\subjclass Primary 05A15;
 Secondary 05A17, 05A30, 11P81, 33D67
\endsubjclass
\keywords cylindric partitions, 
$\widetilde A_r$ basic hypergeometric series,
$SU(r)$ basic hypergeometric series\endkeywords

\abstract
It is shown how a generating function formula for a certain class
of cylindric partitions originally due to Alexei Borodin can be
derived by applying Gustafson's 
$SU(r)$-extension of Bailey's $_6\psi_6$-summation to a
specialisation of an earlier generating function result on cylindric
partition due to Gessel and the author.
\endabstract
\endtopmatter
\document

%\leftheadtext{C. Krattenthaler}

%\rightheadtext{}

Let $\La$ and $\Mu$ be $r$-tuples of
integers, and let $d$ be an integer. According to \cite{\GeKrAA},
a {\it cylindric
partition of shape\/} $\La/\Mu/d$ is a planar array of non-negative integers of the form 
$$\matrix 
&&&\pi_{1,\mu_1+1}&\innerhdotsfor3\after\quad &\pi_{1,\la_1}\\
&&\pi_{2,\mu_2+1}\quad \dots&\pi_{2,\mu_1+1}&\innerhdotsfor2\after\quad 
&\pi_{2,\la_2}\\
&\iddots&&\vdots&&\iddots\\
\pi_{r,\mu_r+1}&\innerhdotsfor3\after\quad &\pi_{r,\la_r}
\endmatrix$$
such that
$$\align \pi_{i,j}&\ge\pi_{i,j+1}\\
\pi_{i,j}&\ge \pi_{i+1,j}\endalign$$
and
$$\pi_{r,j}\ge \pi_{1,j+d}.
$$
The sum of all the $\pi_{i,j}$'s is denoted by $\vert\pi\vert$.

In \cite{\BoroAC}, Borodin 
proves the following theorem (in a different but equivalent form).

\proclaim{Theorem \smc(\cite{\BoroAC, Proposition~5.1})}
Let $r,d$ be given positive integers,
$\boldsymbol\mu=(\mu_1,\mu_2,\dots,\mu_r)$, $d>-r$ and 
$\mu_1\ge\mu_2\ge\dots\ge\mu_r\ge\mu_1-d$.
The generating function $
\sum _{\pi} ^{}q^{\vert\pi\vert}$, where the sum is over all cylindric partitions
of shape $(\infty^r)/ \boldsymbol\mu/d$, is equal to
$$\multline 
\frac {(q^{r+d};q^{r+d})_\infty^{r-1}} {(q;q)_\infty^r}\\
\times
\prod _{1\le i<j\le r}
^{}(1-q^{\mu_i-i-\mu_j+j})\,(q^{r+d+\mu_i-i-\mu_j+j};q^{r+d})_\infty\,
(q^{r+d+\mu_j-j-\mu_i+i};q^{r+d})_\infty. 
\endmultline\tag1$$
\endproclaim

This theorem is proved in \cite{\BoroAC} by making use of 
a formula for the partition function of the periodic Schur
process, which is the main subject of \cite{\BoroAC}.
The purpose of this note is to show that this theorem follows
straightforwardly when a result on cylindric partitions
due to Gessel and the author is combined with Gustafson's 
$SU(r)$-extension of Bailey's $_6\psi_6$-summation.


\demo{Proof of the Theorem} 
We let $c\to\infty$ in \cite{\GeKrAA, Theorem~5}.
This yields that the generating function that one is interested in
is given by
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{r(r+d)\sum _{i=1}
^{r}k_i^2/2 +d\sum _{i=1} ^{r}ik_i+r\sum _{i=1} ^{r}\mu_ik_i}\\
\cdot
\frac {\prod _{1\le i<j\le r}
^{}(1-q^{\mu_i+(r+d)k_i-i-\mu_j-(r+d)k_j+j})} 
{(q;q)_\infty^r}.
\endmultline\tag2$$
The sum over $k_1+\dots+k_r=0$ can be evaluated by means of the
$SU(r)$-extension of the $_6\psi_6$-summation of Bailey, due to
Gustafson \cite{\GustAF, Theorem~1.15; \GustAA, Theorem~4.1}
(see also \cite{\GeKrAA, (7.9)}):
$$\multline \sum _{k_1+\dots+k_r=0} ^{}q^{\sum _{i=1} ^{r}ik_i}\prod
_{1\le i<j\le r} ^{}\frac {1-q^{k_i-k_j}x_i/x_j} {1-x_i/x_j} \prod
_{i,j=1} ^{r}\frac {(a_ix_j;q)_{k_j}} {(b_ix_j;q)_{k_j}}\\
=\frac {(q;q)_\infty^{r-1}\,\prod _{i,j=1} ^{r}(b_i/a_j;q)_\infty
\prod _{1\le i<j\le r}
^{}(qx_i/x_j;q)_\infty\,(qx_j/x_i;q)_\infty} {(q^{1-r}\prod
_{i=1} ^{r} (b_i/a_i);q)_\infty\prod _{i,j=1}
^{r}(q/a_ix_j;q)_\infty\,(b_ix_j;q)_\infty }\\
\tsize\cdot \big(q/\prod _{i=1} ^{r}a_ix_i;q\big)_\infty\,\big(q^{1-r}
\prod _{i=1} ^{r} b_ix_i;q\big)_\infty.
\endmultline$$
To be precise, one replaces $q$ by $q^{r+d}$, one sets
$x_i=q^{\mu_i-i}$, $b_i=0$, and one performs the limit $a_i\to\infty$,
$i=1,2,\dots,r$. In this way, one arrives at
$$\multline 
\kern-2pt
\sum _{k_1+\dots+k_r=0} ^{}q^{
r(r+d)\sum _{i=1} ^{r}k_i^2/2+d\sum _{i=1} ^{r}ik_i+
r\sum _{i=1} ^{r}\mu_ik_i}\prod
_{1\le i<j\le r} ^{}\frac {1-q^{(r+d)k_i-(r+d)k_j+\mu_i-i-\mu_j+j}} 
{1-q^{\mu_i-i-\mu_j+j}} \\
= {(q^{r+d};q^{r+d})_\infty^{r-1}\,
\prod _{1\le i<j\le r}
^{}(q^{r+d+\mu_i-i-\mu_j+j};q^{r+d})_\infty\,
(q^{r+d+\mu_j-j-\mu_i+i};q^{r+d})_\infty}. 
\endmultline$$
Using this equation, we see that the sum in (2) simplifies to (1),
as desired.
\quad \quad \qed
\enddemo

\Refs

\ref\no \BoroAC\by A.    Borodin\paper Periodic Schur process 
and cylindric partitions\jour Duke Math\. J. \vol 140\yr 2007\pages 391--468\endref

\ref\no \GeKrAA\by I. M. Gessel and C. Krattenthaler \yr 1997 \paper 
Cylindric partitions\jour Trans\. Amer\. Math\. Soc\.\vol 349\pages 
429--479\endref

\ref\no \GustAF\by R. A. Gustafson \yr 1987 \paper Multilateral
summation theorems for ordinary and basic hypergeometric series in
$U(n)$\jour SIAM J. Math\. Anal\.\vol 18\pages 1576--1596\endref

\ref\no \GustAA\by R. A. Gustafson \yr 1989 \paper The Macdonald
identities for affine root systems of classical type and
hypergeometric series very-well-poised on semisimple Lie
algebras\inbook Ramanujan International Symposium on Analysis
(Dec.~26th to 28th, 1987, Pune, India)\ed N.~K.~Thakare \pages
187--224\endref 

\endRefs

\enddocument

