Raabe's Formula For Gamma Function Via Riemann-Liouville Fractional Integrals And Generalized Glaisher Constants

TL;DR

This paper derives Raabe-type integral formulas for the gamma function using Riemann-Liouville fractional integrals, explores their implications for log-gamma functions, and investigates connections with generalized Glaisher constants.

Contribution

It introduces new Raabe-type formulas for the gamma function via fractional integrals and links these to generalized Glaisher constants, expanding theoretical understanding.

Findings

Derived Raabe-type integral formulas for gamma function

Established relationships between Glaisher constants and fractional integrals

Provided new integral formulas for log-gamma functions

Abstract

In this paper, we prove Raabe-type integral formulas for gamma function via left and right sided Riemann-Liouville fractional integrals. As corollaries, we give the left and right sided repeated integration formulas for the log-gamma and related functions. The relationship between the generalized Glaisher constants and aforementioned integrals are investigated.