A New Proof of Stirling's Formula

A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented

A Note on Extended Binomial Coefficients

We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as sums of Hermite polynomials and Bernoulli numbers

Ap\'ery Polynomials and the multivariate Saddle Point Method

The Ap\'ery polynomials and in particular their asymptotic behavior play an essential role in the understanding of the irrationality of \zeta(3). In this paper, we present a method to study the asymptotic behavior of the sequence of the Ap\'ery polynomials ((B_{n})_{n=1}^{\infty}) in the whole complex plane as (n\rightarrow \infty). The proofs are based on a multivariate version of the complex saddle point method. Moreover, the asymptotic zero distributions for the polynomials ((B_{n})_{n=1}^{\infty}) and for some transformed Ap\'ery polynomials are derived by means of the theory of logarithmic potentials with external fields, establishing a characterization as the unique solution of a weighted equilibrium problem. The method applied is a general one, so that the treatment can serve as a model for the study of objects related to the Ap\'ery polynomials.Comment: 19 page

Asymptotic zero distribution of Jacobi-Pi\~neiro and multiple Laguerre polynomials

We give the asymptotic distribution of the zeros of Jacobi-Pi\~neiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio asymptotics of multiple orthogonal polynomials. We show how these asymptotic zero distributions are related to the Fuss-Catalan distribution.Comment: 19 pages, 2 figures. Some minor corrections and four new references adde

On the asymptotic normality of the Legendre-Stirling numbers of the second kind

For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers