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q-gamma function

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1: 5.18 q -Gamma and q -Beta Functions
§5.18 q -Gamma and q -Beta Functions
§5.18(ii) q -Gamma Function
5.18.5 Γ q ( 1 ) = Γ q ( 2 ) = 1 ,
Also, ln Γ q ( x ) is convex for x > 0 , and the analog of the Bohr-Mollerup theorem (§5.5(iv)) holds. …
2: 17.13 Integrals
17.13.2 - c d ( - q x / c ; q ) ( q x / d ; q ) ( - x q α / c ; q ) ( x q β / d ; q ) d q x = Γ q ( α ) Γ q ( β ) Γ q ( α + β ) c d c + d ( - c / d ; q ) ( - d / c ; q ) ( - q β c / d ; q ) ( - q α d / c ; q ) .
17.13.3 0 t α - 1 ( - t q α + β ; q ) ( - t ; q ) d t = Γ ( α ) Γ ( 1 - α ) Γ q ( β ) Γ q ( 1 - α ) Γ q ( α + β ) ,
17.13.4 0 t α - 1 ( - c t q α + β ; q ) ( - c t ; q ) d q t = Γ q ( α ) Γ q ( β ) ( - c q α ; q ) ( - q 1 - α / c ; q ) Γ q ( α + β ) ( - c ; q ) ( - q / c ; q ) .
3: 5.1 Special Notation
The main functions treated in this chapter are the gamma function Γ ( z ) , the psi function (or digamma function) ψ ( z ) , the beta function B ( a , b ) , and the q -gamma function Γ q ( z ) . …
4: 5.21 Methods of Computation
For the computation of the q -gamma and q -beta functions see Gabutti and Allasia (2008).
5: Bibliography G
  • B. Gabutti and G. Allasia (2008) Evaluation of q -gamma function and q -analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.
  • 6: 17.6 ϕ 1 2 Function
    7: Bibliography O
  • A. B. Olde Daalhuis (1994) Asymptotic expansions for q -gamma, q -exponential, and q -Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.