Binet's second formula, Hermite's generalization, and two related identities

Abstract

Legendre was the first to evaluate two well-known integrals involving sines and exponentials. One of these integrals can be used to prove Binet’s second formula for the logarithm of the gamma function. Here, we show that the other integral leads to a specific case of Hermite’s generalization of Binet’s formula. From the analogs of Legendre’s integrals, with sines replaced by cosines, we obtain two integration identities involving logarithms and trigonometric functions. Using these identities, we then subsequently derive generalizations of Binet’s and Hermite’s formulas involving the integral of a complex logarithm.

Keywords

• gamma and digamma functions
• binet’s second formula
• hermite’s generalization of binet’s second formula
• 30e20
• 33b10
• 33b15