New asymptotic expansion and error bound for Stirling formula of reciprocal Gamma function

Abstract

Studying the problem about if certain probability measures are determinate by its moments [4, 8, 10] is useful to know the asymptotic behavior of the probability densities for large values of argument. This requires, previously, the knowledge of the asymptotic expansion of reciprocal Gamma function 1/Γ(z) when ℜz is large and ℑz is fixed [8]. Then, the well known Stirling formula for large |z| of the Gamma function Γ(z) or its reciprocal 1/Γ(z) is not appropriate for this problem. So, the main aim of this paper is to obtain a new asymptotic expansion for reciprocal Gamma function valid for large ℜz and establish a new explicit error bound for the first term of this expansion, that is, the Stirling formula.