Asymptotic expansions for the gamma function

Asymptotic expansions for ratios of products of gamma functions

An asymptotic expansion for a ratio of products of gamma functions is derived. 2000 Mathematics Subject Classification: Primary 33B15; Secondary 33C20

Asymptotic Expansions of the Gamma Function Associated with the Windschitl and Smith Formulas

In this paper, we develop the Windschitl and Smith formulas for the gamma function to complete asymptotic expansions and provide explicit formulas for determining the coefficients of these asymptotic expansions. Furthermore, we establish new asymptotic expansions for the ratio of gamma functions Γ(x + 1)/Γ(x + 1 2 ).

Asymptotic Expansions for Integrals

Asymptotic Expansions

Asymptotic expansions of functions are useful in statistics in three main ways. Firstly, conventional asymptotic expansions of special functions are useful for approximate computation of integrals arising in statistical calculations. An example given below is the use of Stirling's approximation to the gamma function. Second, asymptotic expansions of density or distribution functions of estimato...

Asymptotic Expansions for Nonlocal Diffusion

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form ut = J ∗ u − u in the whole R with an initial condition u(x, 0) = u0(x). Under suitable hypotheses on J (involving its Fourier transform) and u0, it is proved an expansion of the form ∥∥u(u)− ∑ |α|≤k (−1)|α| α! ( ∫ u0(x)x dx ) ∂Kt ∥∥ Lq(Rd) ≤ Ct−A, where Kt is the regular part of the fundamental solution and...