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