标题： 三阶Jacobsthal$3$-参数广义四元数 显示英文标题

标题： Third-order Jacobsthal $3$-Parameter Generalized Quaternions

摘要： 本文的目的在于将三阶Jacobsthal数与3参数广义四元数结合起来，其中3参数广义四元数是根据3个参数的四元数代数的一种一般形式。为此，我们引入并研究了一种新的特殊的较大数量系统，称为三阶Jacobsthal 3参数广义四元数（简称三阶Jacobsthal 3PGQs）。此外，我们计算了一些新的方程和经典的已知方程，如：线性递归、Binet公式、生成函数、求和公式、Cassini恒等式和d'Ocagne恒等式。 显示英文摘要

摘要： The purpose of this article is to bring together the third-order Jacobsthal numbers and 3-parameter generalized quaternions, which are a general form of the quaternion algebra according to 3-parameters. With this purpose, we introduce and examine a new type of quite big special numbers system, which is called third-order Jacobsthal 3-parameter generalized quaternions (shortly, third-order Jacobsthal 3PGQs). Further, we compute both some new equations and classical well-known equations such as: linear recurrence, Binet formulas, generating function, sum formulas, Cassini identity and d'Ocagne identity.