An asymptotic formula for the zeros of the deformed exponential function

TL;DR

This paper derives an asymptotic formula for the zeros of the deformed exponential function, improving previous results by explicitly relating zeros to a series involving the sum-of-divisors function.

Contribution

It provides a new asymptotic expression for the zeros of the deformed exponential function, connecting them to the generating function of the sum-of-divisors function, using Jacobi's triple product identity.

Findings

Derived an explicit asymptotic formula for zeros

Connected zeros to the sum-of-divisors generating function

Improved upon earlier asymptotic results

Abstract

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.