Zeilberger's holonomic ansatz for pfaffians

Masao Ishikawa; Christoph Koutschan

doi:10.1145/2442829.2442863

Zeilberger's holonomic ansatz for pfaffians

Masao Ishikawa, Christoph Koutschan

Research output: Chapter in Book/Report/Conference proceeding › Conference 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 language	English
Title of host publication	ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Pages	227-233
Number of pages	7
DOIs	https://doi.org/10.1145/2442829.2442863
Publication status	Published - Dec 1 2012
Externally published	Yes
Event	37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 - Grenoble, France Duration: Jul 22 2012 → Jul 25 2012

Publication series

Name	Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Other

Other	37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012
Country	France
City	Grenoble
Period	7/22/12 → 7/25/12

Keywords

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

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1145/2442829.2442863

Fingerprint Dive into the research topics of 'Zeilberger's holonomic ansatz for pfaffians'. Together they form a unique fingerprint.

Cite this

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 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).

Research output: Chapter in Book/Report/Conference proceeding › Conference 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. Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, 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

@inproceedings{8decc0656197425bb79d9e29efccfaaf,

title = "Zeilberger's holonomic ansatz for pfaffians",

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.",

keywords = "Computer proof, Determinant, Holonomic systems approach, Minor, Motzkin number, Pfaffian, Symbolic summation, WZ theory",

author = "Masao Ishikawa and Christoph Koutschan",

year = "2012",

month = dec,

day = "1",

doi = "10.1145/2442829.2442863",

language = "English",

isbn = "9781450312691",

series = "Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC",

pages = "227--233",

booktitle = "ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation",

note = "37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 ; Conference date: 22-07-2012 Through 25-07-2012",

}

TY - GEN

T1 - Zeilberger's holonomic ansatz for pfaffians

AU - Ishikawa, Masao

AU - Koutschan, Christoph

PY - 2012/12/1

Y1 - 2012/12/1

N2 - 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.

AB - 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.

KW - Computer proof

KW - Determinant

KW - Holonomic systems approach

KW - Minor

KW - Motzkin number

KW - Pfaffian

KW - Symbolic summation

KW - WZ theory

UR - http://www.scopus.com/inward/record.url?scp=84874966068&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874966068&partnerID=8YFLogxK

U2 - 10.1145/2442829.2442863

DO - 10.1145/2442829.2442863

M3 - Conference contribution

AN - SCOPUS:84874966068

SN - 9781450312691

T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

SP - 227

EP - 233

BT - ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

T2 - 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012

Y2 - 22 July 2012 through 25 July 2012

ER -