IJAAMM Int. J. Adv. Appl. Math. and Mech. 6(3) (2019) 14 – 20 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics The binomial transforms of the generalized (s , t )-Jacobsthal matrix sequence Research Article Sukran UYGUN * Mathematics Department, Science and Art Faculty, Gaziantep University, Turkey Received 11 November 2018; accepted (in revised version) 11 January 2019 Abstract: In this paper, we study the binomial transforms of the generalized (s , t )Jacobsthal matrix sequence ℜ n+1 (s , t ) n∈N ; (s , t )-Jacobsthal J n+1 (s , t ) n∈N and, (s , t )-Jacobsthal Lucas C n+1 (s , t ) n∈N matrix sequences. After thatby using recur- rence relations of them, the generating functions have beenfounded for these transforms. Finally the relations among these transforms have been demonstrated with deriving new equalities. MSC: 15A24 • 11B39 • 15B36 Keywords: Generalized (s , t )-sequence • Generalized (s , t )- matrix sequence • (s , t )-Jacobsthal matrix sequence • (s , t )- Jacobsthal Lucas matrix sequence. © 2019 The Author(s). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/). 1. Introduction and preliminary There are so many studies in the literature that are concern about special integer sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Padovan. You can encounter the generalizations of these sequences in all of the references. In [1] the author wrote a book about these integer sequences You can see the generalized number and matrix sequences for Fibonacci and Lucas sequences in [3, 5, 6]. Similarly the author defined number and matrix sequences which generalizes Jacobsthal and Jacobsthal Lucas sequences in [7, 8]. Some authors introduced matrix based transforms for these special sequences. Binomial transform is one of most popular transforms. You can have detailed informa- tion about binomial transform in [9, 10]. Falcon defined different binomial transforms of thek -Fibonacci sequence such as falling, rising binomial transforms in [4]. The authors gave binomial transform for generalized (s , t )-matrix sequences in [11] .The authors introduced binomial transforms for the Padovan and Perrin numbers in [12] . And in [13] binomial transforms of the k -Jacobsthal sequence a reintroduced. In [14] the authors gave some properties of Lucas numbers with binomial coefficients.The goal of this paper is to apply the binomial transforms to the generalized Jacobsthal and Jacobsthal Lucas matrix sequences. Also, the generating function of this transform is found by recur- rence relations. Finally the relations among these transforms have been demonstrated with deriving new equalities. In [2] , the author defined the Jacobsthal and Jacobsthal Lucas sequence as follows respectively j n+1 = j n + 2 j n-1 n ≥ 1, ( j 0 = 0, j 1 = 1) c n+1 = c n + 2c n-1 n ≥ 1, (c 0 = 0, c 1 = 1) Now we give some preliminaries related to our study. For a given integer sequence X = {x 1 , x 2 , ..., x n , ...} the binomial transform Y of the sequence X;Y ( X ) = ' y n “ is defined by y n = n X i =0 ˆ n i ! x i * E-mail address: suygun@gantep.edu.tr. 14