• Speaker: Prof Richard Brent, CARMA, The University of Newcastle
  • Title: Some Identities involving Products of Gamma Functions: a Case Study in Experimental Mathematics
  • Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
  • Time and Date: 2:00 pm, Tue, 27th Oct 2015
  • Abstract:

    We consider identities satisfied by discrete analogues of Mehta-like integrals. The integrals are related to Selberg’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.

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