Asymptotics of orthogonal polynomials for the Abel weight

Abstract

Orthogonal polynomials $P_n\left(x\right)$ with respect to the Abel weight function $w_A\left(x\right)=\frac{x}{2\sinh \left(\pi x\right)}$ are used for numerical summation of alternating series by means of the Abel-Plana formula. Also, they are known to be related to Bernouli and Euler numbers. In this paper we first give an integral representation for these polynomials. Then, using this representation, we investigate the asymptotic behavior of $P_n\left(x\right)$ when n is large.