摘要

We study the asymptotic representation for the zeros of the deformed exponential function Sigma(infinity)(n=0)1/n!q(n(n-1))/2x(n), q is an element of(0,1). Indeed, we obtain an asymptotic formula for these zeros: x(n) = -nq(1-n)(1+g(q)n(-2)+o(n(-2))), n >= 1, where g(q) = Sigma(infinity)(k=1) sigma(k)q(k) is the generating function of the sum-of-divisors function sigma(k). This improves earlier results by Langley [3] and Liu [4]. The proof of this formula is reduced to estimating the sum of an alternating series, where the Jacobi's triple product identity plays a key role.

  • 出版日期2016-9-15