Gaussian (p, q)-Jacobsthal and Gaussian (p, q)-Jacobsthal Lucas numbers and their some interesting properties

Abstract

In the present article, we introduce two new notions, which are called Gaussian (p, q)-Jacobsthal numbers sequence GJp,q,nn=0∞ and Gaussian (p, q)-Jacobsthal Lucas numbers sequence Gjp,q,nn=0∞, and we present and prove our exciting properties and results, which relate these sequences. We first give recurrence relations, Binet’s formulas, explicit formulas, and negative extensions of them. We then obtain some important identities for Gaussian (p, q)-Jacobsthal and Gaussian (p, q)-Jacobsthal Lucas numbers and some connection formulas between these Gaussian numbers. After that, we give some summation formulas and the symmetric functions of Gaussian (p, q)-Jacobsthal and Gaussian (p, q)-Jacobsthal Lucas numbers. In addition, by using the symmetric functions, we derive a new class of generating functions for Gaussian (p, q)-Jacobsthal and Gaussian (p, q)-Jacobsthal Lucas numbers.