We study the asymptotic representation for the zeros of the deformed exponential function $\sum\nolimits_{ n = 0}^\infty { \frac1{ n!}{ q^{ n(n - 1)/2}{ x^n}}} $, $q\in (0,1)$. Indeed, we obtain an asymptotic formula for these zeros: \[x_n=- nq^{ 1-n}(1 + g(q)n^{ -2}+o(n^{ -2})),n\ge1,\] where $g(q)=\sum\nolimits_{ k = 1}^\infty { \sigma (k){ q^k}}$ is the generating function of the sum-of-divisors function $\sigma(k)$. This improves earlier results by Langley and Liu. 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.
Categories: math.CA