# Asymptotics of some generalized Mathieu series

## TL;DR

This paper derives asymptotic estimates for generalized Mathieu series with power-logarithmic and factorial sequences, using Mellin transforms and Dirichlet series analysis, providing precise first-order asymptotics.

## Contribution

It introduces a novel approach to asymptotic analysis of Mathieu series with complex sequences via Mellin transform techniques.

## Key findings

- Power-logarithmic sequences yield precise first-order asymptotics.
- Factorial sequences are analyzed despite natural boundary challenges.
- Elementary estimates provide reasonably accurate asymptotics for factorial cases.

## Abstract

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics.