Asymptotic expansions for \(q\)-gamma, \(q\)-exponential, and \(q\)-Bessel functions (Q1340530)

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 \({}_ 1\psi_ 1\) one studied by Hahn and Exton. Here the \(q\)-difference equation is used to obtain an asymptotic expansion.