2024/12/03 by Alexey Kuznetsov, Kuznetsov, Alexey · 1 citation
Mathematics · #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Iterative Methods for Nonlinear Equations
paper · pdf · doi:10.48550/arxiv.2412.02462
For q ∈ (0, 1), the deformed exponential function f(x) = ∑n ≥ 1 xn qn(n-1)/2/n! is known to have infinitely many simple and negative zeros \xk(q)\k ≥ 1. In this paper, we analyze the series expansions of -xk(q)/k and k/xk(q) in powers of q. We prove that the coefficients of these expansions are rational functions of the form Pn(k)/Qn(k) and \widehatPn(k)/Qn(k), where Qn(k) ∈ \mathbb Z[k] is explicitly defined and the polynomials Pn(k), \widehatPn(k)∈ \mathbb Z[k] can be computed recursively. We provide explicit formulas for the leading coefficients of Pn(k) and \widehatPn(k) and compute the coefficients of these polynomials for n ≤ 300. Numerical verification shows that Pn(k) and \widehatPn(k) take non-negative values for all k ∈ ℕ and n≤ 300, offering further evidence in support of conjectures by Alan Sokal.