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Date: 10-6-2019
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A q-analog of the gamma function defined by
(1) |
where is a q-Pochhammer symbol (Koepf 1998, p. 26; Koekoek and Swarttouw 1998). The -gamma function satisfies
(2) |
where is the gamma function (Andrews 1986).
The -gamma function is implemented in the Wolfram Language as QGamma[z, q].
The -gamma function satisfies the functional equation
(3) |
with (Koekoek and Swarttouw 1998, p. 10), which simplifies to
(4) |
as . A curious identity for the functional equation
(5) |
where
(6) |
is given by
(7) |
for any .
REFERENCES:
Andrews, G. E. "W. Gosper's Proof that ." Appendix A in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 11 and 109, 1986.
Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.
Koekoek, R. and Swarttouw, R. F. "The q-Gamma Function and the q-Binomial Coefficient." §0.3 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 10-11, 1998.
Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.
Wenchang, C. Problem 10226 and Solution. "A q-Trigonometric Identity." Amer. Math. Monthly 103, 175-177, 1996.
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