Orthogonal polynomials with respect to the Abel weight

Abstract

While a lot is known about the classical orthogonal polynomials, their counterparts with respect to non-classical weights are not as well explored. Nevertheless, sometimes such weights come handy as well. For example, the famous Abel–Plana summation formula offers a convenient method of summing an infinite series, reducing the sum to an integral with the Abel weight function on the real line, w(x)=x2sinhπx. Orthogonal polynomials with respect to this weight naturally arise when we have to numerically evaluate this integral using the Gauss quadrature rule. These orthogonal polynomials are the object of study in this paper. We obtain a number of explicit formulas and algebraic relations between these and related polynomials, including the associated polynomials. In particular, for many of these polynomials we obtain Fourier expansions with the orthogonal polynomials as the basis. We also determine the weight functions whose orthogonal polynomials are the polynomials we discussed. At the end, we briefly discuss the asymptotic and perform numerical experiments.