Five-Diagonal Toeplitz Determinants an Their Relation to Chebyshev Polynomials

Abstract

A five-diagonal Toeplitz (5DT) determinant is defined as having zeros everywhere except in its five principal diagonals, with each principal diagonal having the same element in all positions. Thus the determinant depends on five arbitrary parameters in addition to its order. The general 5DT determinant of order n is shown to be given by a simple closed expression involving Chebyshev polynomials of the second kind of order $n + 1$. An explicit generating function for the determinants is also derived such that the nth coefficient of a power series expansion of the function is the nth-order five-diagonal Toeplitz determinant.