Evaluation of Hessenberg determinants via generating function approach

Abstract

In this paper, we will present various results on computing of wide classes of Hessenberg matrices whose entries are the terms of any sequence. We present many new results on the subject as well as our results will cover and generalize earlier many results by using generating function method. Moreover, we will present a new approach on computing Hessenberg determinants, whose entries are general higher order linear recursions with arbitrary constant coefficients, based on finding an adjacency-factor matrix. We will give some interesting showcases to show how to use our new method.

A periodic determinantal property for (0,1) double banded matrices

“…There are several methods to compute the determinant of such matrices (cf. for example, [16][17][18][19][20] and references therein). Due to the (0, 1) pattern above the lower band, we develop a useful method for the calculation of the determinant of these matrices.…”

A note on Pell-Padovan numbers and their connection with Fibonacci numbers

“…where a 0 = 0 and a k = 0 for at least one k > 0, is called a lower Toeplitz-Hessenberg matrix. This class of matrix have been encountered in many scientific and engineering applications (see, among others, [1,[13][14][15][16] and related references therein).…”

Fibonacci–Lucas identities and the generalized Trudi formula

“…In [28], Tangboonduangjit and Thanatipanonda considered determinants of matrices whose entries are powers of the Fibonacci numbers, while Civciv [11] studied the determinant of a five-diagonal matrix with Fibonacci entries. For further examples of combinatorial determinants, we refer the reader to [18,23].…”