Zeros of the deformed exponential function

Abstract

Let f(x)=∑n=0∞1n!qn(n−1)/2xn (0<q<1) be the deformed exponential function. It is known that the zeros of f(x) are real and form a negative decreasing sequence (xk) (k≥1). We investigate the complete asymptotic expansion for xk and prove that for any n≥1, as k→∞,xk=−kq1−k(1+∑i=1nCi(q)k−1−i+o(k−1−n)), where Ci(q) are some q series which can be determined recursively. We show that each Ci(q)∈Q[A0,A1,A2], where Ai=∑m=1∞miσ(m)qm and σ(m) denotes the sum of positive divisors of m. When writing Ci as a polynomial in A0,A1 and A2, we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that Ci(q)∈Q[E2,E4,E6], where E2, E4 and E6 are the classical Eisenstein series of weight 2, 4 and 6, respectively.