Zeilberger's holonomic ansatz for Pfaffians

Authors:
Masao Ishikawa;Christoph Koutschan
Affiliations:
University of the Ryukyus Nishihara, Okinawa, Japan;MSR-INRIA Joint Centre, INRIA-Saclay, Orsay Cedex, France
Venue:
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Year:
2012

Citing 3
Cited 0

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An extension of Zeilberger's fast algorithm to general holonomic functions

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Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

Journal of Combinatorial Theory Series A

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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.