# Generalized Stieltjes constants and integrals involving the log-log function: Kummer's Theorem in action

## TL;DR

This paper explores the use of Kummer's Fourier series expansion to derive closed-form expressions for series related to generalized Stieltjes constants and integrals involving the log-log function, enhancing analytical tools in special functions.

## Contribution

It introduces new closed-form formulas for series and integrals involving the generalized Stieltjes constants and the log-log function using Kummer's theorem.

## Key findings

- Derived closed-form expressions for series related to Stieltjes constants
- Obtained explicit integrals involving the log-log function
- Applied Fourier series expansion to special functions

## Abstract

In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related to the generalized Stieltjes constants, and some integrals involving the function $x\mapsto \ln \ln(1/x)$.