Zeilberger's holonomic ansatz for Pfaffians

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.

[1]  Robert Donaghey,et al.  Motzkin Numbers , 1977, J. Comb. Theory, Ser. A.

[2]  Doron Zeilberger,et al.  Proof of George Andrews’s and David Robbins’s q-TSPP conjecture , 2010, Proceedings of the National Academy of Sciences.

[3]  Naiomi T. Cameron,et al.  Hankel determinants of sums of consecutive Motzkin numbers , 2011 .

[4]  Doron Zeilberger.  The Holonomic Ansatz II. Automatic Discovery(!) And Proof(!!) of Holonomic Determinant Evaluations , 2007 .

[5]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

[6]  Alain Lascoux,et al.  Hankel Pfaffians, Discriminants and Kazhdan-Lusztig bases , 2011, 1103.4971.

[7]  Christoph Koutschan,et al.  Advanced Computer Algebra for Determinants , 2011, ArXiv.

[8]  Christoph Koutschan,et al.  A Fast Approach to Creative Telescoping , 2010, Math. Comput. Sci..

[9]  Christoph Koutschan,et al.  Advanced applications of the holonomic systems approach , 2010, ACCA.

[10]  Masato Wakayama,et al.  Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities , 2006, J. Comb. Theory, Ser. A.

[11]  Jiang Zeng,et al.  A q-analogue of Catalan Hankel determinants , 2010, 1009.2004.

[12]  D. Zeilberger.  A holonomic systems approach to special functions identities , 1990 .

[13]  Jiang Zeng,et al.  Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants , 2010, J. Comb. Theory A.

[14]  Frédéric Chyzak,et al.  An extension of Zeilberger's fast algorithm to general holonomic functions , 2000, Discret. Math..