An asymptotic formula for the zeros of the deformed exponential function

Abstract

We study the asymptotic representation for the zeros of the deformed exponential function ${\sum }_{n=0}^{\infty }\frac{1}{n!}{q}^{n(n-1)/2}{x}^n$, $q\in (0,1)$. 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 $g(q)={\sum }_{k=1}^{\infty }\sigma (k)q^k$ 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.

10 pages. To appear in Journal of Mathematical Analysis and Applications

References in corpus (1)

• The leading root of the partial theta function