Michitomo Nishizawa

On a q-analogue of the multiple gamma functions

A q-analogue of the multiple gamma functions is introduced and is shown to satisfy the generalized Bohr-Morellup theorem. Furthermore, we give some expressions of these functions.

An elliptic analogue of the multiple gamma function

A hierarchy of functions including the elliptic gamma function is introduced. It can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q-analogue of the multiple gamma function. Some properties of the functions are considered.

Evaluation of a certain q-determinant

Abstract A q -analogue of Mehta–Wangs determinant is introduced and evaluated. In the case when q =1, this determinant contains a factor represented by using a special case of the Meixner–Pollaczek polynomial. A recurrence relation for the factor is derived in the case when 0 q |

q-special functions with |q| = 1 and their application to discrete integrable systems

Solutions for certain discrete integrable systems are constructed by using integral solutions for hypergeometric q-difference systems with |q| = 1. We apply a solution for Lauricellas D-type hypergeometric q-difference system with |q| = 1 to construct a solution for the discrete KP-hierarchy. Furthermore, an integral solution of a q-difference analogue of Bessels equation is newly introduced and applied to construct a solution for a q-difference analogue of the cylindrical Toda equation with |q| = 1.