Abstract
F. H. Jackson defined a generalization of the factorial function by
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)=[(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.
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References
Artin, E.,The gamma function. Holt, Rinehart and Winston, New York, 1964.
Askey, R.,The q-gamma and q-beta functions. Applicable Anal. to appear.
Bohr, H. andMollerup, J.,Laerebog Matematisk Analyse, Vol. III, Copenhagen, 1922.
Jackson, F. H.,On q-definite integrals. Quart. J. Pure Appl. Math.41 (1910), 193–203.
John, F.,Special solutions of certain difference equations. Acta Math.71 (1939), 175–189.
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Moak, D.S. Theq-gamma function forq>1. Aeq. Math. 20, 278–285 (1980). https://doi.org/10.1007/BF02190519
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DOI: https://doi.org/10.1007/BF02190519