On a Nesbitt — Carlitz determinant

Abstract

A matrix whose component are binomial coefficients and determinant was calculated earlier by L. Carlitz is investigated. It is shown that Carlitz matrix is the result of binomal specialization for dual Jacobi — Trudi determinant presentation of certain Schur function. It leads to another way to calculate Carlitz determinant based upon symmetric function theory. The eigenvalues of Carlitz matrix are shown to be powers of two as well. In order to calculate these eigenvalues the author uses suitable linear operator on the space of polynomials whose degree does not exceed given number. It is shown that in suitable basis matrix of that linear operator has triangular form with powers of two on its diagonal. Main result
is generalised from quadratic to cubic case corresponding to a certain matrix, consisted of trinomial coefficients.

Keywords:

linear algebra, binomial coefficients, symmetric functions, matrix eigenvalues