Theq-gamma function forx<0

Abstract

F. H. Jackson defined a generalization of the factorial function by $$1(1 + q)(1 + q + q^2 ) \cdot \cdot \cdot (1 + q + q^2 + \cdot \cdot \cdot + q^{n - 1} ) = (n!)_q $$ forq>0. He also generalized the gamma function, both for 0 1. Askey then obtained analogues of many of the classical facts about theq-gamma function for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q x −1)/(q−1)]f(x) is theq-gamma function. He also considered the behavior of theq-gamma function asq changes, and showed that asq→1−, theq-gamma function becomes the ordinary gamma function. In this paper we will state two analogues of the Bohr-Mollerup theorem forq>1. It turns out that the log convexity off together with the initial condition and the functional equation no longer forcesf to be theq-gamma function. A stronger condition is needed than the log convexity, and two sufficient conditions are given in this paper. Also we will consider the behavior of theq-gamma function asq-changes forq>1.