Zeta functions, heat kernel expansions, and asymptotics for \(q\)-Bessel functions
If the heat-kernel expansion of a spectral zeta function is known, then the corresponding Mellin transform gives rise to an analytic continuation in the complex plane. This property is classically applied to situations where the Bessel operator plays a significant role. The purpose of this study is to furnish analogues of this work associated with two \(q\)-analogues of the Bessel function: the Jackson \(q\)-Bessel function and the Hahn-Exton \(q\)-Bessel function [\textit{M. E. H. Ismail}, J. Math. Anal. Appl. 86, 1-16 (1982; Zbl 0483.33004) and \textit{H. T. Koelink} and \textit{R. F. Swarttouw}, ibid. 186, No. 3, 690-710 (1994; Zbl 0811.33013)]. Corresponding asymptotic properties of the \(q\)-zeta functions are investigated, and asymptotic properties of the \(q\)-Bessel functions for large values of their arguments are deduced. It is pointed out that certain of these results do not hold in the limit as the base \(q\) tends to unity from below.
scientific article; zbMATH DE number 861861