Determinant identities for toeplitz-hessenberg matrices with tribonacci entries

Document Type: Research Paper

Authors

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

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

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 $n \geq r$‎, ‎with the appropriate initial conditions‎, ‎where $r \geq 3$ is arbitrary‎.

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