Evaluations of certain Catalan-Hankel Pfaffians via classical skew orthogonal polynomials

TL;DR

This paper evaluates specific Catalan-Hankel Pfaffians using skew orthogonal polynomials, providing explicit constructions and examples related to hypergeometric orthogonal polynomials within the Askey scheme.

Contribution

It introduces explicit constructions of classical skew orthogonal polynomials and applies them to evaluate Catalan-Hankel Pfaffians for various hypergeometric polynomial cases.

Findings

Explicit skew orthogonal polynomials constructed for different cases.

Evaluation of Catalan-Hankel Pfaffians with continuous and q-moment sequences.

Examples illustrating the application of the theory.

Abstract

This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew orthogonal polynomials and then give different examples of Catalan-Hankel Pfaffians with continuous and $q$-moment sequences.