4 February 2016, 16:00 - 17:00, in HG00.310
Erdal Emsiz (Pontificia Universidad Catolica de Chile)
Difference equation for the Heckman-Opdam hypergeometric function and its confluent Whittaker limit
We will discuss explicit difference equations for the Heckman-Opdam
hypergeometric function associated with root systems (a generalization of
the Gauss hypergeometric function to various variables). Our method
exploits the fact that for discrete spectral values on a (translated) cone
of dominant weights the Heckman-Opdam hypergeometric function truncates in
terms of Heckman-Opdam Jacobi polynomials. This permits us to derive/prove
the desired difference equations in two steps: first for the discrete
spectral values by performing a $q\to 1$ degeneration of a recently found
Pieri formula for the celebrated Macdonald polynomials, and then for
arbitrary spectral values upon invoking an analytic continuation argument
borrowed from Rösler (based on known growth estimates for the Heckman-Opdam
hypergeometric function that enable one to apply Carlson's theorem).
If time permits we will also mention analogous difference equation for the class-one Whittaker function diagonalizing the open quantum Toda chain associated with reduced root systems.
Based on joint work with Jan Felipe van Diejen (Universidad de Talca).