Convolution identities for Tribonacci numbers via the diagonal of a bivariate generating function

TL;DR

This paper derives convolution identities for Tribonacci numbers using bivariate generating functions and diagonalization, providing rational generating functions and extending methods to Tetranacci numbers.

Contribution

It introduces a novel approach using diagonalization of bivariate generating functions to obtain convolution identities for Tribonacci numbers, with potential extensions to Tetranacci numbers.

Findings

Established rational generating functions for Tribonacci convolutions

Derived explicit formulas for convolution coefficients

Outlined extension to Tetranacci and similar sequences

Abstract

Convolutions for Tribonacci numbers involving binomial coefficients are treated with ordinary generating functions and the diagonalization method of Hautus and Klarner. In this way, the relevant generating function can be established, which is rational. The coefficients can also be expressed. It is sketched how to extend this to Tetranacci numbers and similar quantities.