Taras Goy and Mark Shattuck

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 4, Pages 83-95

DOI: 10.7546/nntdm.2019.25.4.83-95

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## Details

### Authors and affiliations

Taras Goy

*Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University
57 Shevchenko St., 76018 Ivano-Frankivsk, Ukraine
*

Mark Shattuck

*Department of Mathematics, University of Tennessee
37996 Knoxville, TN, USA
*

### Abstract

In this paper, we evaluate several families of Toeplitz–Hessenberg matrices whose entries are generalized Fibonacci numbers. In particular, we find simple formulas for several determinants whose entries are translates of the Chebyshev polynomials of the second kind. Equivalently, these determinant formulas may also be rewritten as identities involving sums of products of generalized Fibonacci numbers and multinomial coefficients. Combinatorial proofs which make use of sign-reversing involutions and the definition of a determinant as a signed sum over the symmetric group S_{n} are given for our formulas in several particular cases, including those involving the Chebyshev polynomials.

### Keywords

- Toeplitz–Hessenberg matrix
- Horadam sequence
- Trudi’s formula
- Generating function
- Chebyshev polynomials

### 2010 Mathematics Subject Classification

- 11B39
- 05A15
- 15B05

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## Related papers

- Goy, T. & Shattuck, M. (2020). Fibonacci–Lucas identities and the generalized Trudi formula. Notes on Number Theory and Discrete Mathematics, 26 (3), 203-217, doi: 10.7546/nntdm.2020.26.3.203-217.

## Cite this paper

Goy, T., & Schattuck, Mark. (2019). Determinants of Toeplitz–Hessenberg matrices with generalized Fibonacci entries. Notes on Number Theory and Discrete Mathematics, 25(4), 83-95, doi: 10.7546/nntdm.2019.25.4.83-95.