# The*q*-gamma function for*q*>1

## Authors

Research Papers

- Received:
- Revised:

DOI: 10.1007/BF02190519

- Cite this article as:
- Moak, D.S. Aeq. Math. (1980) 20: 278. doi:10.1007/BF02190519

## Abstract

F. H. Jackson defined a generalization of the factorial function by for

$$1(1 + q)(1 + q + q^2 ) \cdot \cdot \cdot (1 + q + q^2 + \cdot \cdot \cdot + q^{n - 1} ) = (n!)_q $$

*q*>0. He also generalized the gamma function, both for 0<*q*<1, and for*q*>1. Askey then obtained analogues of many of the classical facts about the*q*-gamma function for 0<*q*<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfying*f*(1)=1 and*f*(*x*+1)=[(*q*^{ x }−1)/(*q*−1)]*f*(*x*) is the*q*-gamma function. He also considered the behavior of the*q*-gamma function as*q*changes, and showed that as*q*→1^{−}, the*q*-gamma function becomes the ordinary gamma function.In this paper we will state two analogues of the Bohr-Mollerup theorem for*q*>1. It turns out that the log convexity of*f* together with the initial condition and the functional equation no longer forces*f* to be the*q*-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 the*q*-gamma function as*q*-changes for*q*>1.

### AMS (1970) subject classification

Primary 33A15## Copyright information

© Birkhäuser Verlag 1980