Bi Periodic Jacobsthal Polynomial Sequence and Its Various Binomial Transformations

Abstract

In this study, we first introduce the bi-periodic Jacobsthal polynomial sequence. Using the elements of the bi-periodic Jacobsthal polynomial sequence, we find the generating function and Binet formula for this sequence. We also obtain some properties and sum formulas for this matrix sequence. In the main part of the study, the binomial transformations of the bi-periodic Jacobsthal polynomial sequence are obtained as a result of using the bi-periodic Jacobsthal polynomial sequence. The binomial and k-binomial transformations of the bi-periodic Jacobsthal polynomial sequence are denoted. Some relations for the binomial transformations of the bi-periodic Jacobsthal polynomial sequence are provided. In addition, the increasing and decreasing k-binomial transformations of the bi-periodic Jacobsthal polynomial sequence are given, and the recurrence formulas, Binet formulas, and generating functions of these transformation sequences are investigated.