aequationes mathematicae

, Volume 20, Issue 1, pp 278–285

# Theq-gamma function forq>1

• Daniel S. Moak
Research Papers

## 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<q<1, and forq>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)=[(qx−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.

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### References

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Artin, E.,The gamma function. Holt, Rinehart and Winston, New York, 1964.Google Scholar
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Jackson, F. H.,On q-definite integrals. Quart. J. Pure Appl. Math.41 (1910), 193–203.Google Scholar
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John, F.,Special solutions of certain difference equations. Acta Math.71 (1939), 175–189.Google Scholar