On Some Asymptotic Expansions for the Gamma Function

Abstract

Inequalities play a fundamental role in both theoretical and applied mathematics and contain many patterns of symmetries. In many studies, inequalities have been used to provide estimates of some functions based on the properties of their symmetry. In this paper, we present the following new asymptotic expansion related to the ordinary gamma function Γ(1+w)∼2πw(w/e)ww2+760w2−120w/2exp∑r=1∞μrwr,w→∞, with the recurrence relation of coefficients μr. Furthermore, we use Padé approximants and our new asymptotic expansion to deduce the new bounds of Γ(w) better than some of its recent ones.

Two Approximation Formulas for Gamma Function with Monotonic Remainders

“…Our new bounds are, of course, superior to those in [31], which deduced from the bounds of the remainder function σ 9 (r). Also, even though the bounds in [30] are superior to our new bounds, what strengthens our results and gives them an advantage is proving that the remainders θ 1 (r) and θ 2 (r) are monotonic and bounded in the new approximation formulas, which was not discussed in [30].…”