A multilinear operator for almost product evaluation of Hankel determinants

• Published in: Journal of combinatorial theory. Series A Vol. 117; no. 1; pp. 77 - 103
• Main Authors: Eğecioğlu, Ömer, Redmond, Timothy, Ryavec, Charles
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 2010
• Subjects: Almost product evaluation; Binomial coefficients; Differential equations; Hankel determinants; Almost product evaluation; Binomial coefficients; Hankel determinants; Differential equations
• ISSN: 0097-3165; 1096-0899
• DOI: 10.1016/j.jcta.2009.03.016

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.