Zeilberger's Holonomic Ansatz for Pfaffians

TL;DR

This paper extends Zeilberger's holonomic approach to evaluate Pfaffians and skew-symmetric determinants, enabling proofs of conjectures and deriving new formulas using computer algebra.

Contribution

It introduces a novel holonomic method tailored for Pfaffians, expanding the applicability of Zeilberger's approach beyond determinants.

Findings

Proved conjectures on Pfaffian decomposition using computer algebra.

Derived a minor summation formula related to partitions and Motzkin paths.

Extended the holonomic ansatz to skew-symmetric matrices.

Abstract

A variation of Zeilberger's holonomic ansatz for symbolic determinant evaluations is proposed which is tailored to deal with Pfaffians. The method is also applicable to determinants of skew-symmetric matrices, for which the original approach does not work. As Zeilberger's approach is based on the Laplace expansion (cofactor expansion) of the determinant, we derive our approach from the cofactor expansion of the Pfaffian. To demonstrate the power of our method, we prove, using computer algebra algorithms, some conjectures proposed in the paper "Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants" by Ishikawa, Tagawa, and Zeng. A minor summation formula related to partitions and Motzkin paths follows as a corollary.