Asymptotic expansions of several series and their application

TL;DR

This paper derives asymptotic expansions for specific infinite series as the variable approaches zero and applies these results to establish precise inequalities for Mathieu series.

Contribution

It introduces new asymptotic expansion formulas for series involving powers and exponentials, and uses them to improve inequalities for Mathieu series.

Findings

Derived asymptotic expansions for series as x approaches 0

Applied expansions to obtain precise Mathieu series inequalities

Enhanced understanding of series behavior near zero

Abstract

Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or $\epsilon=-1$. These expansions are applied to obtain precise inequalities for Mathieu series. Keywords: Asymptotic expansion, residues, generalized Mathieu series, inequalities.