Zeta Functions, Heat Kernel Expansions, and Asymptotics for q-Bessel Functions

Abstract

Analytic structure of the zeta functions ζ ν( z; q) = Σ ∞ n=1 [ j ν n ( q)] − z of the zeros j ν n ( q) of the q-Bessel functions J ν( x; q) and J (2) ν( x; q) is studied. All poles and corresponding residues of ζ ν are found. Explicit formulas for ζ ν(2 n; q) at n = ±1, ±2, ... are obtained. Asymptotics of the sum Z ν( t; q) = Σ n exp[− tj 2 ν n ( q)] as t ↓ 0 ("heat kernel expansion") is derived. Asymptotics of the q-Bessel functions at large arguments are found.