The Generalized Bivariate Jacobsthal and Jacobsthal Lucas Polynomial Sequences

Abstract

From the definition of Fibonacci numbers (the first known special integer sequence), there are many studies on the integer sequences because of so many applications in science and art, and etc. For instance, the ratio of two consecutive elements of Fibonacci sequence is the golden ratio, is very important number almost every area of science and art. And the other integer sequence Jacobsthal numbers are met in computer science. It is well known that computers use conditional directives to change the flow of execution of a program. In addition to branch instructions, some microcontrollers use skip instructions which conditionally bypass the next instruction. This brings out being useful for one case out of the four possibilities on 2 bits, 3 cases on 3 bits, 5 cases on 4 bits, 11 cases on 5 bits, 21 cases on 6 bits, ..., which are exactly the Jacobsthal numbers. In this study, first of all we define and study the generalized bivariate Jacobsthal and generalized bivariate Jacobsthal Lucas polynomial sequences. Then the Binet formulae, some different types of generating functions, D’ Ocagne, Catalan, Cassini properties and some interesting properties of these sequences are given. The sum of the square of elements of these sequences and some generalized sum formulae are obtained for the generalized bivariate Jacobsthal sequence and the bivariate Jacobsthal Lucas sequences. Finally, a divisibility property of the generalized bivariate Jacobsthal sequence is given.