Zeilberger's holonomic ansatz for pfaffians

Masao Ishikawa, Christoph Koutschan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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.

Original languageEnglish
Title of host publicationISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Pages227-233
Number of pages7
DOIs
Publication statusPublished - 2012
Externally publishedYes
Event37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 - Grenoble, France
Duration: Jul 22 2012Jul 25 2012

Other

Other37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012
CountryFrance
CityGrenoble
Period7/22/127/25/12

Fingerprint

Pfaffian
Determinant
Cofactor
Motzkin Paths
Hankel Determinant
Skew symmetric matrix
Summation Formula
Computer Algebra
Laplace
Minor
Corollary
Partition
Analogue
Decompose
Evaluation
Demonstrate

Keywords

  • Computer proof
  • Determinant
  • Holonomic systems approach
  • Minor
  • Motzkin number
  • Pfaffian
  • Symbolic summation
  • WZ theory

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Ishikawa, M., & Koutschan, C. (2012). Zeilberger's holonomic ansatz for pfaffians. In ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (pp. 227-233) https://doi.org/10.1145/2442829.2442863

Zeilberger's holonomic ansatz for pfaffians. / Ishikawa, Masao; Koutschan, Christoph.

ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. 2012. p. 227-233.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Ishikawa, M & Koutschan, C 2012, Zeilberger's holonomic ansatz for pfaffians. in ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. pp. 227-233, 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012, Grenoble, France, 7/22/12. https://doi.org/10.1145/2442829.2442863
Ishikawa M, Koutschan C. Zeilberger's holonomic ansatz for pfaffians. In ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. 2012. p. 227-233 https://doi.org/10.1145/2442829.2442863
Ishikawa, Masao ; Koutschan, Christoph. / Zeilberger's holonomic ansatz for pfaffians. ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. 2012. pp. 227-233
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