The goal of this paper is to prove the following asymptotic formula Γ (x) ≈ √ 2πe−b (x+ b) x exp ( −x− 1 2 ψ (x+ c) ) as x ∈ N, x→∞, where Γ is the Euler Gamma function and ψ is the digamma function, namely, the logarithmic derivative of Γ. Moreover, optimal values of parameters b, c are calculated in such a way that this asymptotic convergence is the best possible.

In the paper, the authors establish integral representations of some functions related to the remainder of Burnside’s formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These results extend and generalize some known conclusions. 1. Motivation and main results A function f is said to be completely monotonic on an interval I if f ha...

where x = l , 2, . . . ; r > 0 , a > I (ar a + r > 0 , (a)~=a(a+l)...(a-kx--1), and r log F(z)/dz is the digamma or the psi function. The distribution (1.1) will be referred to as DGa (a, r). In this paper we extend DGa (a, r) to a multivariate distribution. The digamma distribution is closely related to the logarithmic series distribution LSr (0) defined by (1.2) P r [ X = x ; 0 ] = 0 ~ / ( l ...

We calculate some finite and infinite sums containing the digamma function in closed form. For this purpose, we differentiate selected reduction formulas of hypergeometric with respect to parameters applying derivative Pochhammer symbol. Additionally, compare two different differentiation generalized parameters. particular cases, recover results found literature. Finally, all have been numerica...