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Determinant identities for Toeplitz-Hessenberg matrices with tribonacci entries | ||
| Transactions on Combinatorics | ||
| مقاله 24، دوره 9، شماره 2، شهریور 2020، صفحه 89-109 اصل مقاله (302.92 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2020.116257.1631 | ||
| نویسندگان | ||
| Taras Goy1؛ Mark Shattuck* 2 | ||
| 1Faculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 76018, Ivano-Frankivsk, Ukraine | ||
| 2Department of Mathematics University of Tennessee Knoxville, TN, 37996-1300 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| tribonacci numbers؛ Toeplitz-Hessenberg matrix؛ determinant؛ multinomial coefficient | ||
| مراجع | ||
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