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 | |
| Publication status | Published - 2012 |
| Externally published | Yes |
| Event | 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 - Grenoble, France Duration: Jul 22 2012 → Jul 25 2012 |
Other
| Other | 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 |
|---|---|
| Country | France |
| City | Grenoble |
| Period | 7/22/12 → 7/25/12 |
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Keywords
- Computer proof
- Determinant
- Holonomic systems approach
- Minor
- Motzkin number
- Pfaffian
- Symbolic summation
- WZ theory
ASJC Scopus subject areas
- Mathematics(all)
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.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
TY - GEN
T1 - Zeilberger's holonomic ansatz for pfaffians
AU - Ishikawa, Masao
AU - Koutschan, Christoph
PY - 2012
Y1 - 2012
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
SP - 227
EP - 233
BT - ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
ER -