Bi-periodic Jacobsthal polynomial sequences and binomial transformations

In this study, the definition of special number sequences such as Fibonacci, Lucas, Jacobsthal, Jacobsthal-Lucas and some basic properties of these number sequences are given. Binet formulas and generator functions of these special number sequences and generalised forms of these number sequences are given. After the definition of matrix sequences of generalised forms of Fibonacci, Lucas, Jacobsthal and Jacobsthal-Lucas number sequences, Binet formulas and generator functions of these matrix sequences are given. Then, some important relations and binomial transformation of the polynomial matrix sequence obtained by using the bi-periodic Fibonacci matrix polynomial are mentioned. In the last part of the thesis, a new polynomial matrix sequence is obtained by using the bi-periodic Jacobsthal polynomial matrix sequence. Binomial, k-binomial transformations of this polynomial matrix sequence were found. Then, using the property of binomial expansion, some equations for the new polynomial matrix sequence are mentioned and these equations are proved. After the definition of increasing and decreasing k-binomial transformation sequences of bi-periodic Jacobsthal polynomial matrix sequence, Binet formula and generator function of these transformation sequences are given. Translated with DeepL.com (free version) Key Words: Special number sequences, Generalized number and matrix sequences, Binomial Transformations, Generating Function, Binet Formula.