An asymptotic formula for the zeros of the deformed exponential function

2015/01/12 by Zhang, Cheng
#33E20 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1501.02700

Abstract

We study the asymptotic representation for the zeros of the deformed exponential function ∑\nolimits_{n = 0}^∞ \frac1n!q^{n(n - 1)/2}xⁿ , q∈ (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)=∑\nolimits_{k = 1}^∞ σ(k)q^k is the generating function of the sum-of-divisors function σ(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.

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