Abstract:
We consider identities satisfied by discrete analogues of Mehta-like integrals.
The integrals are related to Selbergâ€™s integral and the Macdonald conjectures.
Our discrete analogues have the form
$$S_{\alpha,\beta,\delta} (r,n) :=
\sum_{k_1,...,k_r\in\mathbb{Z}}
\prod_{1\leq i < j\leq r}
|k_i^\alpha - k_j^\alpha|^\beta
\prod_{j=1}^r |k_j|^\delta
\binom{2n}{n+k_j},$$
where $\alpha,\beta,\delta,r,n$ are non-negative integers subject to certain restrictions.
In the cases that we consider, it is possible to express $S_{\alpha,\beta,\delta} (r,n)$ as a
product of Gamma functions and simple functions such as powers of two.
For example, if $1 \leq r \leq n$, then
$$S_{2,2,3} (r,n) =
\prod_{j=1}^r
\frac{(2n)!j!^2}{(n-j)!^2}.$$
The emphasis of the talk will be on how such identities can be obtained,
with a high degree of certainty, using numerical computation. In other cases
the existence of such identities can be ruled out, again with a high degree of
certainty. We shall not give any proofs in detail, but will outline the ideas
behind some of our proofs. These involve $q$-series identities and arguments
based on non-intersecting lattice paths.
This is joint work with Christian Krattenthaler and Ole Warnaar.