An Asymptotic Expansion for the Generalized Gamma Function

Abstract

The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function Γμ(v) and study the completely monotonic (CM) property of a function involving Γμ(v) and the generalized digamma function ψμ(v). As a consequence, we establish some bounds for Γμ(v), ψμ(v) and polygamma functions ψμ(r)(v), r≥1.

Two Approximation Formulas for Gamma Function with Monotonic Remainders

“…The derivative of ln Γ(r), denoted by ψ(r) = Γ ′ (r) Γ(r) , is called the digamma function and the derivatives ψ (n) (r) for n ≥ 0 are referred to as the polygamma functions. For more information about gamma function and polygamma functions see [2][3][4][5][6], as well as the closely linked references therein, for further details.…”