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 -