On the zeros of the q-analogue exponential function

Abstract

An asymptotic formula for the zeros, zn, of the entire function eq(x) for q<<1 is obtained. As q increases above the first collision point at q1* approximately=0.14, these zeros collide in pairs and then move off into the complex z plane. They move off as (and remain) a complex conjugate pair. The zeros of the ordinary higher derivatives and of the ordinary indefinite integrals of eq(x) vary with q in a similar manner. Properties of eq(z) for z complex and for arbitrary q are deduced. For 0<or=q<1,eq(z) is an entire function of order 0. By the Hadamard-Weierstrass factorization theorem, infinite product representations are obtained for eq(z) and for the reciprocal function eq-1(z). If q not=1, the zeros satisfy the sum rule Sigma n=1infinity (1/zn)=-1.