Negative moments of orthogonal polynomials

TL;DR

This paper introduces methods to compute negative moments of orthogonal polynomials, linking combinatorial models with algebraic reciprocity, and extends the understanding of negative index sequences.

Contribution

It provides two novel methods for calculating negative moments of orthogonal polynomial sequences and establishes combinatorial models for these negative versions.

Findings

Combinatorial model for negative bounded Motzkin paths

Proof of two conjectures on determinant reciprocity

Extension of negative moments to orthogonal polynomial sequences

Abstract

If a sequence indexed by nonnegative integers satisfies a linear recurrence without constant terms, one can extend the indices of the sequence to negative integers using the recurrence. Recently, Cigler and Krattenthaler showed that the negative version of the number of bounded Dyck paths is the number of bounded alternating sequences. In this paper we provide two methods to compute the negative versions of sequences related to moments of orthogonal polynomials. We give a combinatorial model for the negative version of the number of bounded Motzkin paths. We also prove two conjectures of Cigler and Krattenthaler on reciprocity between determinants.