Zeta functions, heat kernel expansions, and asymptotics for q-bessel functions

Analytic structure of the zeta functions zeta(nu)(z; q) = Sigma(n=1)(infinity)=[j(nu n)(q)](-z) of the zeros j(nu n) (q) of the q-Bessel functions J(nu)(x; q) and J(nu)((2))(x; q) is studied. All poles and corresponding residues of zeta(nu) are found. Explicit formulas for zeta(nu)(2n; q) at n = +/-1, +/-2,... are obtained. Asymptotics of the sum Z(nu)(t; q) = Sigma(n) exp[-tj(nu n)(2)(q)] as t down arrow 0 (''heat kernel expansion'') is derived. Asymptotics of the q-Bessel functions at large arguments are found. (C) 1995 Academic Press, Inc.