NAME: Steve Milne

ADDRESS: Department of Mathematics
         The Ohio State University
         231 W. 18th Ave.
         Columbus, OH 43210-1174

EMAIL ADDRESS: milne@math.ohio-state.edu

AUTHOR: Stephen C. Milne (joint with Verne E. Leininger)

TITLE OF TALK: 

Some new infinite families of eta function identities

ABSTRACT:

Ever since Euler proved his expansion for 
$\prod_{i=1}^\infty (1-q^i)\equiv (q;q)_\infty$
mathematicians have been looking for other identities of 
this form.  In 1829, Jacobi utilized his triple product
identity to derive an elegant expansion for 
$(q;q)_\infty^3$.  Since then, expansions have been found
for $(q;q)_\infty^c$ for many values of $c$.
These included several infinite families of expansions
and a few exceptional cases.  
In 1892 F. Klein and R. Fricke gave a result for
$c=8$.  This was rediscovered by S. Ramanujan in 1916.  
L. Winquist in 1969 proved a result for
$c=10$ but stated that this was first found by J.
Rushforth, then independently discovered by A. Atkin.  
L. Winquist also noted that A. Atkin had formulae for
$c=14$ and $c=26$.  The existence of these identities had
been suggested in 1955 by M. Newman. In
1972 F. Dyson gave his famous formula for
$c=24$ and stated that formulae corresponding to  
$c = 3,8,10,14,15,21,24,26,28,35,36,\dots$ had been
found, but noted that these ad hoc results had been
unified by I. Macdonald.  In his landmark 1972 paper,
Macdonald related most of these expansions for
$(q;q)_\infty^c$ to affine root systems.  
(This connection with Lie Algebras has been the
main focus of work since I. Macdonald.)  A few
notable exceptions remained: $c=2$ found by Hecke and
Rogers, $c=4$ by Ramanujan, and $c=26$ by Atkin.  
\medskip

In this talk we discuss our derivation of new, more
symmetrical expansions for $(q;q)_\infty^{n^2+2n}$ by
means of our multivariable generalization of Andrews'
variation of the standard proof of Jacobi's
$(q;q)_\infty^3$ result.  We also present examples of
our general expansion for $(q;q)_\infty^c$ 
where $c=3,8,15,24$.  Our proof relies upon Milne's new
$U(n)$ multivariable extension of the Jacobi triple
product identity.  This result is deduced from a
$U(n)$ multiple basic hypergeometric series
generalization of Watson's very--well--poised 
$\_8\phi_7$ transformation. The derivation of our
$(q;q)_\infty^{n^2+2n}$ expansion also utilizes partial
derivatives and dihedral group symmetries to write the
sum over regions in $n$-space.  We note that
our expansions for $(q;q)_\infty^{n^2+2n}$ are equivalent
to Macdonald's $A_n$ family of eta-function identities.
In addition, we utilize various summation and
transformation formulas for $U(n+1)$, equivalently $A_n$,
multiple basic hypergeometric series to derive new
infinite families of expansions for
$(q;q)_\infty^{n^2+2}$ and $(q;q)_\infty^{n^2}$, similar
products of these, and the corresponding powers of the 
$\eta$-function.  (Recall that the eta function is defined
by $\eta(q):= q^{1/24}(q;q)_\infty$.)  These additional
infinite families of expansions extend the list in
Appendix I of Macdonald's 1972 paper.  All of this work
is motivated by Milne's $U(n+1)$ multiple basic
hypergeometric series treatment of the Macdonald
identities for $A_{\ell}^{(1)}$.