A multilinear operator for almost product evaluation of Hankel determinants
In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calcul...
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| Published in | Journal of combinatorial theory. Series A Vol. 117; no. 1; pp. 77 - 103 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
2010
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0097-3165 1096-0899 |
| DOI | 10.1016/j.jcta.2009.03.016 |
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| Abstract | In a recent paper we have presented a method to evaluate certain Hankel determinants as
almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end.
In this paper, we introduce a class of multilinear operators
γ acting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms, and can be combined easily with differentiation.
The paper is self contained. An example of an almost product evaluation using
γ-operators is worked out in detail and tables of the
γ-operator values on various forms of matrices are provided. We also present an explicit evaluation of a new class of Hankel determinants and conjectures. |
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| AbstractList | In a recent paper we have presented a method to evaluate certain Hankel determinants as
almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end.
In this paper, we introduce a class of multilinear operators
γ acting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms, and can be combined easily with differentiation.
The paper is self contained. An example of an almost product evaluation using
γ-operators is worked out in detail and tables of the
γ-operator values on various forms of matrices are provided. We also present an explicit evaluation of a new class of Hankel determinants and conjectures. |
| Author | Eğecioğlu, Ömer Ryavec, Charles Redmond, Timothy |
| Author_xml | – sequence: 1 givenname: Ömer surname: Eğecioğlu fullname: Eğecioğlu, Ömer email: omer@cs.ucsb.edu organization: Department of Computer Science, University of California, Santa Barbara, CA 93106, United States – sequence: 2 givenname: Timothy surname: Redmond fullname: Redmond, Timothy email: tredmond@stanford.edu organization: Stanford Medical Informatics, Stanford University, Stanford, CA 94305, United States – sequence: 3 givenname: Charles surname: Ryavec fullname: Ryavec, Charles email: ryavec@ccs.ucsb.edu organization: Department of Computer Science, University of California, Santa Barbara, CA 93106, United States |
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| Cites_doi | 10.37236/1580 10.37236/730 10.1016/j.laa.2005.06.042 10.37236/1079 |
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| Keywords | Almost product evaluation Binomial coefficients Hankel determinants Differential equations |
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| References | Eğecioğlu, Redmond, Ryavec (bib001) 2001; 8 or the package I. Gessel, personal communication, 2007 Macdonald (bib009) 1995 Muir (bib010) 1906 can be used, all available from Doron Zeilberger's website Eğecioğlu, Redmond, Ryavec (bib003) 2008; vol. AI K. Wegschaider, Computer generated proofs of binomial multi-sum identities, PhD thesis, Research Institute for Symbolic Computation, Institut für Mathematik, Johannes Kepler Universität, Linz, May 1997 with command Gessel, Xin (bib004) 2006; 13 Pólya, Szegö (bib012) 1972 D. Zeilberger C. Krattenthaler, personal communication, 2007 Krattenthaler (bib007) 2005; 411 Eğecioğlu, Redmond, Ryavec (bib002) 2008; 15 Krattenthaler (bib006) 1999; 42 Muir (bib011) 1960 Maple package, or after extracting coefficients 10.1016/j.jcta.2009.03.016_bib013 10.1016/j.jcta.2009.03.016_bib014 Eğecioğlu (10.1016/j.jcta.2009.03.016_bib002) 2008; 15 Gessel (10.1016/j.jcta.2009.03.016_bib004) 2006; 13 Pólya (10.1016/j.jcta.2009.03.016_bib012) 1972 Muir (10.1016/j.jcta.2009.03.016_bib010) 1906-1923 Macdonald (10.1016/j.jcta.2009.03.016_bib009) 1995 Eğecioğlu (10.1016/j.jcta.2009.03.016_bib003) 2008; vol. AI Eğecioğlu (10.1016/j.jcta.2009.03.016_bib001) 2001; 8 Krattenthaler (10.1016/j.jcta.2009.03.016_bib006) 1999; 42 Muir (10.1016/j.jcta.2009.03.016_bib011) 1960 10.1016/j.jcta.2009.03.016_bib005 Krattenthaler (10.1016/j.jcta.2009.03.016_bib007) 2005; 411 10.1016/j.jcta.2009.03.016_bib008 |
| References_xml | – year: 1906 ident: bib010 article-title: The Theory of Determinants in the Historical Order of Development, vols. 1–4 – volume: 411 start-page: 68 year: 2005 end-page: 166 ident: bib007 article-title: Advanced determinant calculus: A complement publication-title: Linear Algebra Appl. – reference: C. Krattenthaler, personal communication, 2007 – year: 1972 ident: bib012 article-title: Problems and Theorems in Analysis, vol. I – year: 1995 ident: bib009 article-title: Symmetric Functions Hall Polynomials – year: 1960 ident: bib011 article-title: A Treatise on the Theory of Determinants – reference: D. Zeilberger, – reference: with command – volume: vol. AI start-page: 251 year: 2008 end-page: 268 ident: bib003 article-title: Evaluation of a special Hankel determinant of binomial coefficients publication-title: Proc. Fifth Colloquium on Mathematics and Computer Science – reference: Maple package, or after extracting coefficients – reference: can be used, all available from Doron Zeilberger's website – volume: 13 start-page: R53 year: 2006 ident: bib004 article-title: The generating function of ternary trees and continued fractions publication-title: Electron. J. Combin. – volume: 15 start-page: #R6 year: 2008 ident: bib002 article-title: Almost product evaluation of Hankel determinants publication-title: Electron. J. Combin. – reference: I. Gessel, personal communication, 2007 – volume: 42 year: 1999 ident: bib006 article-title: Advanced determinant calculus publication-title: The Andrews Festschrift – reference: K. Wegschaider, Computer generated proofs of binomial multi-sum identities, PhD thesis, Research Institute for Symbolic Computation, Institut für Mathematik, Johannes Kepler Universität, Linz, May 1997 – volume: 8 start-page: #R36 year: 2001 ident: bib001 article-title: From a polynomial Riemann hypothesis to alternating sign matrices publication-title: Electron. J. Combin. – reference: , or the package – volume: 42 year: 1999 ident: 10.1016/j.jcta.2009.03.016_bib006 article-title: Advanced determinant calculus publication-title: Sem. Lothar. Combin. – year: 1972 ident: 10.1016/j.jcta.2009.03.016_bib012 – year: 1995 ident: 10.1016/j.jcta.2009.03.016_bib009 – volume: vol. AI start-page: 251 year: 2008 ident: 10.1016/j.jcta.2009.03.016_bib003 article-title: Evaluation of a special Hankel determinant of binomial coefficients – ident: 10.1016/j.jcta.2009.03.016_bib005 – ident: 10.1016/j.jcta.2009.03.016_bib008 – year: 1906-1923 ident: 10.1016/j.jcta.2009.03.016_bib010 – volume: 8 start-page: #R36 issue: 1 year: 2001 ident: 10.1016/j.jcta.2009.03.016_bib001 article-title: From a polynomial Riemann hypothesis to alternating sign matrices publication-title: Electron. J. Combin. doi: 10.37236/1580 – year: 1960 ident: 10.1016/j.jcta.2009.03.016_bib011 – volume: 15 start-page: #R6 issue: 1 year: 2008 ident: 10.1016/j.jcta.2009.03.016_bib002 article-title: Almost product evaluation of Hankel determinants publication-title: Electron. J. Combin. doi: 10.37236/730 – volume: 411 start-page: 68 year: 2005 ident: 10.1016/j.jcta.2009.03.016_bib007 article-title: Advanced determinant calculus: A complement publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2005.06.042 – volume: 13 start-page: R53 issue: 1 year: 2006 ident: 10.1016/j.jcta.2009.03.016_bib004 article-title: The generating function of ternary trees and continued fractions publication-title: Electron. J. Combin. doi: 10.37236/1079 – ident: 10.1016/j.jcta.2009.03.016_bib014 – ident: 10.1016/j.jcta.2009.03.016_bib013 |
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| Snippet | In a recent paper we have presented a method to evaluate certain Hankel determinants as
almost products; i.e. as a sum of a small number of products. The... |
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| SubjectTerms | Almost product evaluation Binomial coefficients Differential equations Hankel determinants |
| Title | A multilinear operator for almost product evaluation of Hankel determinants |
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