Persymmetric determinants

Abstract

The object of this paper is to show that three determinantal identities which appeared in the author's joint paper with Dale [3], and which at the time were thought to be independent, are in fact simple illustrations of a general family of identities.

Let M ₁ be a persymmetric matrix of modified Appell polynomials ψ_m(x) and let M ₂ be a symmetric matrix of the form

where A is a persymmetric array of the constants ψ_m(0)Pis an array of x-elements with binomial coefficients and O is an array of zeros. Then to each submatrix of M ₁ there corresponds a family of distinct submatrices of M ₂ the determinants of which represent identical polynomials. The result can be extended to functions of two variables. The paper ends with two determinantal representations of ψ_m(x) and an identity relating three determinants.