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

TL;DR

This work addresses deriving Raabe-type integral formulas for the Gamma function using left and right Riemann-Liouville fractional integrals. The authors establish explicit expressions for $I_{0^+}^\alpha log Gamma(1)$ and $I_{1^-}^\alpha log Gamma(0)$, expressed in terms of generalized Glaisher constants $D_k$ and related constants, and extend these to weighted forms such as $x^{\beta-1} log Gamma(x)$ and $(1-x)^{\gamma-1} log Gamma(x)$. They also derive corollaries including left- and right-sided repeated integration formulas for $log Gamma$ and connections to zeta derivatives, illustrating deep links between fractional calculus and special constants. The results broaden the toolkit for analytic representations of log Gamma and its relatives, with potential applications in analytic number theory and the theory of special 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.

Theorems & Definitions (32)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Corollary 1.1
• proof
• Remark 1