A Multilinear Operator for Almost Product Evaluation of Hankel Determinants

TL;DR

This paper introduces multilinear b3-operators as a new method for evaluating Hankel determinants as almost products, simplifying calculations and avoiding extraneous nonlinear terms.

Contribution

The paper presents b3-operators as an alternative to trace methods for Hankel determinant evaluation, enabling more straightforward calculations and new explicit evaluations.

Findings

b3-operators avoid extraneous nonlinear terms

Explicit evaluation of a new class of Hankel determinants

Tables of b3-operator values provided

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 \gamma 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 \gamma-operators is worked out in detail and tables of the \gamma-operator values on various forms of matrices are provided. We also present an explicit evaluation of a new class of Hankel determinants and conjectures.