Asymptotic expansions of several series and their application

Abstract

Asymptotic expansions of series ${\sum }_{k=0}^{\infty }{\epsilon }^k(k+a{)}^{\gamma }{e}^{-(k+a{)}^{\alpha }x}$ and $\sum_{k=0}^\infty ε^k(k+a)^γ/ (x(k+a)^α+1)^μ}$ 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.

21 pages

References in corpus (1)

• Mathieu's series: inequalities, asymptotics and positive definiteness

Cited by in corpus (3)

• Asymptotic expoansions of mathieu-Bessel series. I
• Asymptotics of a Mathieu-Gaussian series
• The asymptotic expansion of a Mathieu-exponential series