aequationes mathematicae

, Volume 20, Issue 1, pp 278–285

Theq-gamma function forq>1

  • Daniel S. Moak
Research Papers


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.

AMS (1970) subject classification

Primary 33A15 


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  1. [1]
    Artin, E.,The gamma function. Holt, Rinehart and Winston, New York, 1964.Google Scholar
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    Askey, R.,The q-gamma and q-beta functions. Applicable Anal. to appear.Google Scholar
  3. [3]
    Bohr, H. andMollerup, J.,Laerebog Matematisk Analyse, Vol. III, Copenhagen, 1922.Google Scholar
  4. [4]
    Jackson, F. H.,On q-definite integrals. Quart. J. Pure Appl. Math.41 (1910), 193–203.Google Scholar
  5. [5]
    John, F.,Special solutions of certain difference equations. Acta Math.71 (1939), 175–189.Google Scholar

Copyright information

© Birkhäuser Verlag 1980

Authors and Affiliations

  • Daniel S. Moak
    • 1
  1. 1.Mathematics DepartmentUniversity of WisconsinMadisonU.S.A.

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