An asymptotic formula for the zeros of the deformed exponential function

We study the asymptotic representation for the zeros of the deformed exponential function ∑n=0∞1n!qn(n−1)/2xn, q∈(0,1)q∈(0,1). Indeed, we obtain an asymptotic formula for these zeros:xn=−nq1−n(1+g(q)n−2+o(n−2)),n≥1,xn=−nq1−n(1+g(q)n−2+o(n−2)),n≥1, where g(q)=∑k=1∞σ(k)qk is the generating function of the sum-of-divisors function σ(k)σ(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.