ZETA-FUNCTIONS, HEAT KERNEL EXPANSIONS, AND ASYMPTOTICS FOR Q-BESSEL FUNCTIONS

Analytic structure of the zeta functions zeta(nu)(z; q) = Sigma(n=1)(i nfinity)=[j(nu n)(q)](-z) of the zeros j(nu n) (q) of the q-Bessel fun ctions J(nu)(x; q) and J(nu)((2))(x; q) is studied. All poles and corr esponding residues of zeta(nu) are found. Explicit formulas for zeta(n u)(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 k ernel expansion'') is derived. Asymptotics of the q-Bessel functions a t large arguments are found. (C) 1995 Academic Press, Inc.