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

Analytic structure of the zeta functions ζν(z; q) = Σ∞n=1[jνn(q)]−z of the zeros jνn(q) of the q-Bessel functions Jν(x; q) and J(2)ν(x; q) is studied. All poles and corresponding residues of ζν are found. Explicit formulas for ζν(2n; q) at n = ±1, ±2, ... are obtained. Asymptotics of the sum Zν(t; q) = Σn exp[−tj2νn(q)] as t ↓ 0 ("heat kernel expansion") is derived. Asymptotics of the q-Bessel functions at large arguments are found.