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Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions. (English) Zbl 0809.33008
The $q$-gamma function was introduced by Thomae. It is a slight variation of an infinite product studied by Euler, but the change allows a nicer behavior as $q$-changes. The present paper contains an asymptotic formula for the logarithm of the $q$-gamma function which reduces to Stirling’s series when $q\to 1$. Earlier, Moak had found a different expansion for the log of the $q$-gamma function which reduces to Stirling’s series when $q$ is 1. The author uses the Abel-Plana formula, while Moak used the Euler-Maclaurin formula. The Abel-Plana formula is also used to find asymptotic formulas for two $q$-extensions of the exponential function. The result here is similar to one of Littlewood, although the proofs are different. The $q$-Bessel functions treated is the ${}\sb 1\psi\sb 1$ one studied by Hahn and Exton. Here the $q$-difference equation is used to obtain an asymptotic expansion.

33D05$q$-gamma functions, $q$-beta functions and integrals
33D15Basic hypergeometric functions of one variable, ${}_r\phi_s$
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