Minor summation formula of hyperpfaffians and Selberg integrals

TL;DR

This paper introduces a new method using de Bruijn type formulas and Selberg integrals to evaluate hyperpfaffians, extending previous conjectures and connecting to classical q-orthogonal polynomials.

Contribution

It presents a novel approach for evaluating hyperpfaffians via Selberg integrals, surpassing earlier algebraic methods and proving more generalized identities.

Findings

Derived new hyperpfaffian identities using Selberg integrals.

Extended previous conjectures with more general formulas.

Explored Pfaffians related to q-orthogonal polynomials.

Abstract

In the previous paper (J. Combin. Theory Ser. A, 120, 2013, 1263--1284) H. Tagawa and the two authors proposed an algebraic method to compute certain Pfaffians whose form resemble to Hankel determinants associated with moment sequences of the classical orthogonal polynomials. At the end of the paper they offered several conjectures. In this work we employ a completely different method to evaluate this type of Pfaffians. The idea is to apply certain de Bruijn type formula and convert the evaluation of the Pfaffians to the certain Selberg type integrals. This approach works not only for Pfaffians but also for hyperpfaffians. Hence it enables us to establish much more generalized identities than those conjectured in the previous paper. We also investigate some Pfaffians related to classical $q$-orthogonal polynomials.