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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 q1. 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 1 ψ 1 one studied by Hahn and Exton. Here the q-difference equation is used to obtain an asymptotic expansion.

MSC:
33D05q-gamma functions, q-beta functions and integrals
33D15Basic hypergeometric functions of one variable, r φ s