The Gelin-Ces\`aro identity in some third-order Jacobsthal sequences

TL;DR

This paper investigates third-order Jacobsthal sequences, proving the Gelin-Cesàro identity for them, and introduces a generalized family with explicit formulas, demonstrating that key identities hold for these sequences.

Contribution

It introduces a new generalized third-order Jacobsthal sequence with explicit formulas and proves that classical identities like Gelin-Cesàro and Cassini hold for it.

Findings

Gelin-Cesàro identity is satisfied by the sequences.

Derived generating function and Binet's formula for the generalized sequence.

Proved Cassini and Gelin-Cesàro identities for the generalized sequence.

Abstract

In this paper, we deal with two families of third-order Jacobsthal sequences. The first family consists of generalizations of the Jacobsthal sequence. We show that the Gelin-Ces\`aro identity is satisfied. Also, we define a family of generalized third-order Jacobsthal sequences $\{\mathbb{J}_{n}^{(3)}\}_{n\geq 0}$ by the recurrence relation $$\mathbb{J}_{n+3}^{(3)}=\mathbb{J}_{n+2}^{(3)}+\mathbb{J}_{n+1}^{(3)}+2\mathbb{J}_{n}^{(3)},\ n\geq0,$$ with initials conditions $\mathbb{J}_{0}^{(3)}=a$, $\mathbb{J}_{1}^{(3)}=b$ and $\mathbb{J}_{2}^{(3)}=c$, where $a$, $b$ and $c$ are non-zero real numbers. Many sequences in the literature are special cases of this sequence. We find the generating function and Binet's formula of the sequence. Then we show that the Cassini and Gelin-Ces\`aro identities are satisfied by the indices of this generalized sequence.