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On Generalized Bihyperbolic Third-order Jacobsthal Polynomials

Gamaliel Cerda-Morales

TL;DR

Introducing the generalized bihyperbolic third order Jacobsthal polynomials $BJ_n^{(a,b,c)}(x)$ defined by $BJ_n^{(a,b,c)}(x)=J_n^{(3)}(x)+J_{n+a}^{(3)}(x) j1+J_{n+b}^{(3)}(x) j2+J_{n+c}^{(3)}(x) j3$, the paper extends the classic $J_n^{(3)}$ to a bihyperbolic context. It proves a recurrence $BJ_n^{(a,b,c)}(x)=(x-1)BJ_{n-1}^{(a,b,c)}(x)+(x-1)BJ_{n-2}^{(a,b,c)}(x)+xBJ_{n-3}^{(a,b,c)}(x)$ and derives a Binet-type closed form, a generating function, and Vadja-type identities with $\Theta$, $\Phi_1$, $\Phi_2$. The results yield Catalan, Cassini, and d'Ocagne identities and include a matrix representation, establishing a compact algebraic framework for bihyperbolic polynomial families and suggesting directions for further generalizations.

Abstract

In this paper, a new generalization of third-order Jacobsthal bihyperbolic polynomials is introduced. Some of the properties of presented polynomials are given. A Vadja formula for the generalized bihyperbolic third-order Jacobsthal polynomials is obtained. This result implies the Catalan, Cassini and d'Ocagne identities. Moreover, generating function and matrix generators for these polynomials are presented.

On Generalized Bihyperbolic Third-order Jacobsthal Polynomials

TL;DR

Introducing the generalized bihyperbolic third order Jacobsthal polynomials defined by , the paper extends the classic to a bihyperbolic context. It proves a recurrence and derives a Binet-type closed form, a generating function, and Vadja-type identities with , , . The results yield Catalan, Cassini, and d'Ocagne identities and include a matrix representation, establishing a compact algebraic framework for bihyperbolic polynomial families and suggesting directions for further generalizations.

Abstract

In this paper, a new generalization of third-order Jacobsthal bihyperbolic polynomials is introduced. Some of the properties of presented polynomials are given. A Vadja formula for the generalized bihyperbolic third-order Jacobsthal polynomials is obtained. This result implies the Catalan, Cassini and d'Ocagne identities. Moreover, generating function and matrix generators for these polynomials are presented.
Paper Structure (3 sections, 14 theorems, 39 equations)

This paper contains 3 sections, 14 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Generalized bihyperbolic third-order Jacobsthal polynomials
  3. Conclusions

Key Result

Theorem 2.1

Let $n\geq 3$, $a\geq1$, $b\geq1$ and $c\geq1$ integers. Then, where $\mathcal{BJ}_{0}^{(a,b,c)}(x)$, $\mathcal{BJ}_{1}^{(a,b,c)}(x)$ and $\mathcal{BJ}_{2}^{(a,b,c)}(x)$ are given by Eq. (e3).

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4: Binet formula for the sequence $\mathcal{BJ}_{n}^{(a,b,c)}(x)$
  • proof
  • Corollary 2.5
  • Theorem 2.6
  • proof
  • ...and 11 more