<abstract xmlns="http://www.w3.org/1999/xhtml">

Let <italic>un </italic>= <italic>un</italic>(<italic>k</italic>) denote the <italic>generalized Leonardo</italic> number defined recursively by <italic>un </italic>= <italic>un−</italic>1 + <italic>un−</italic>2 + <italic>k</italic> for <italic>n</italic> ≥ 2, where <italic>u</italic>0 = <italic>u</italic>1 = 1. Terms of the sequence <italic>un</italic>(1) are referred to simply as <italic>Leonardo</italic> numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ <italic>k</italic> ≤ 7, where several connections are made to entries in the <italic>On-Line Encyclopedia of Integer Sequences</italic>. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case <italic>k</italic> = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.
</abstract>

Let un = un(k) denote the generalized Leonardo number defined recursively by un = un−1 + un−2 + k for n ≥ 2, where u0 = u1 = 1. Terms of the sequence un(1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ k ≤ 7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Vasyl Stefanyk Precarpathian National University

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

Let un = un(k) denote the generalized Leonardo number defined recursively by un = un−1 + un−2 + k for n ≥ 2, where u0 = u1 = 1. Terms of the sequence un(1) are...

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Published Online: Jan 10, 2024

Page range: -

Received: Aug 22, 2023

Accepted: Dec 11, 2023

DOI: https://doi.org/10.2478/amsil-2023-0027

Keywords
Leonardo number, Fibonacci number, Toeplitz–Hessenberg matrix, Trudi’s formula, combinatorial proof, generating function

© 2023 Taras Goy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Published Online: Jan 10, 2024

Page range: -

Received: Aug 22, 2023

Accepted: Dec 11, 2023

DOI: https://doi.org/10.2478/amsil-2023-0027

KeywordsLeonardo number, Fibonacci number, Toeplitz–Hessenberg matrix, Trudi’s formula, combinatorial proof, generating function

© 2023 Taras Goy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Leonardo number, Fibonacci number, Toeplitz–Hessenberg matrix, Trudi’s formula, combinatorial proof, generating function