Determinants of some Hessenberg matrices with generating functions

: In this paper, we derive some relationships between the determinants of some special lower Hessenberg matrices whose entries are the terms of certain sequences and the generating functions of these sequences. Moreover, our results are generalizations of the earlier results from previous researches. Furthermore, interesting examples of the determinants of some special lower Hessenberg matrices are presented.


Introduction
Hessenberg matrices play an important role in both computational and applied mathematics (see [1-5]). For examples, Hessenberg matrix decomposition is the important key of computing the eigenvalue matrix [4] and the rule of the Hessenberg matrix for computing the determinant of general centrosymmetric matrix [5].
A lower Hessenberg matrix ( ) = A a n i j is an × n n matrix whose entries above the superdiagonal are all zero but the matrix is not lower triangular, that is, = a 0 ij for all > + j i 1, Similarly, the × n n upper Hessenberg matrix is considered as transpose of the lower Hessenberg matrix A n . Throughout this paper, we are interested in a lower Hessenberg matrix so in fact our results will be also valid for an upper Hessenberg matrix.
Getu [6] computed determinants of a class of Hessenberg matrices by using a generating function method. The author considered an infinite matrix with 1s in the super diagonal Then the author showed that if the following equation holds are the generating functions. Janjic [7] considered a particular case of upper Hessenberg matrices, in which all subdiagonal elements are −1 and showed its relationship with a generalization of the Fibonacci numbers.
Merca [8] showed that determinant of an × n n Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n by using generating function method.
Ramirez [9] derived some relations between the generalized Fibonacci-Narayana sequences, and permanents and determinants of one type of upper Hessenberg matrix.
In 2017, Kılıç and Arıkan [10] obtained the relationships between determinants of three classes of Hessenberg matrices whose entries are terms of certain sequences, and the generating functions of these sequences.
In this paper, we use the generating function method to determine the relationships between determinants of some special lower Hessenberg matrices whose entries are terms of certain sequences, and generating functions of these sequences. Moreover, we also find interesting examples of determinants of such lower Hesseberg matrices.  where + H n 1 is the ( ) ( ) + × + n n 1

Main results
1 lower Hessenberg matrix defined as follows: Proof. For each nonnegative integer n, we consider the linear system of equations . Using Cramer's rule yields a unique solution to (2.1):  Now, we consider the infinite linear system of equations:  By summing both sides of the aforementioned equalities, we obtain   By comparing the coefficients, we have 1 lower Hessenberg matrix + H n 1 is defined as follows: which is the generating function for the sequence of the Lucas numbers.   1 matrix + H n 1 is defined as follows: for all nonnegative integers n.
The proof of this corollary is analogous to the proof Theorem 2.1. Note that the matrix + H n 1 in this corollary is included in [10, Theorem 2.12].
Example 2.5. For any integer ≥ n 0, we have where the ( ) ( ) + × + n n 1 1 matrix + H n 1 is defined as follows: for all nonnegative integers n.
Proof. For each nonnegative integer n, we consider the linear system of equations: which is a linear differential equation of order 1 having the integrating factor ( ) = − I x xe .
x 2 Hence, the general solution to this linear differential equation is expressed as follows: where c is an arbitrary constant. For = x 0, we have = c 0, and hence,