Zeros of the deformed exponential function

Let f (x) = Sigma(infinity)(n=0) 1/n!q(n(n-1)/2)x(n)(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 (x(k)) (k >= 1). We investigate the complete asymptotic expansion for x(k) and prove that for any n >= 1, as k -> infinity, x(k) = -kq(1-k) (1+Sigma(n)(i=1) C-i(q)k(-1-i) + o(k(-1-n))), where C-i(q) are some q series which can be determined recursively. We show that each C-i (q) is an element of Q[A(0), A(1), A(2)], where A(i) = Sigma(infinity)(m=1) m(i)sigma(m)q(m) and sigma(m) denotes the sum of positive divisors of m. When writing C-i as a polynomial in A(0), A(1) and A(2), we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that C-i(q) is an element of Q[E-2, E-4, E-6], where E-2, E-4 and E-6 are the classical Eisenstein series of weight 2, 4 and 6, respectively. (C) 2018 Elsevier Inc. All rights reserved.