Generalization of Binet's Gamma function formulas

Several representations for the logarithm of the Gamma function exist in the literature. There are four important expansions which bear the name of Binet. Hermite generalized Binet's first formula to the logarithm of the Gamma function with shifted argument. The generalization of Binet's second formula is apparently not known; however, it follows easily from another result of Hermite. The aim of this paper is to give possible generalizations of the third and fourth Binet formulas.