Determinant Identities for Toeplitz-Hessenberg Matrices with Tribonacci Entries

Document Type: Research Paper

Authors

1 Department of Mathematics University of Tennessee Knoxville, TN, 37996-1300

2 Faculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 76018, Ivano-Frankivsk, Ukraine

10.22108/toc.2020.116257.1631

Abstract

In this paper, we evaluate determinants of some families of Toeplitz--Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz--Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying a recurrence of the form
T_n^{(r)}=T_{n-1}^{(r)}+T_{n-2}^{(r)}+T_{n-r}^{(r)} for n >= r, with the appropriate initial conditions, where r >= 3 is arbitrary.

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