Some Sum Formulas of ( s , t )-Jacobsthal and ( s , t )-Jacobsthal Lucas Matrix Sequences

In this study, we first give the definitions of (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequence. By using these formulas we define (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas matrix sequences. After that we establish some sum formulas for these matrix sequences.


Introduction
There are so many studies in the literature that are concern about special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, and Padovan in [1] [2].They are widely used in many research areas as Engineering, Architecture, Nature and Art in [3]- [6].For example, microcontrollers (and other computers) use conditional instructions 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 winds up being useful for one case out of the four possibilities on 2 bits, 3 cases on 3 bits, 5 cases on 4 bits, 21 on 6 bits, 43 on 7 fits, 85 on 8 fits, ... , which are exactly the Jacosthal numbers [7].Jacobsthal and Jacobsthal Lucas numbers are given by the recurrence relations respectively in [7]- [9].Generalization of number sequences is studied in many articles.For example the generalization of Jacobsthal sequences is defined in [10].We can see any properties of these numbers in [7]- [9] [11] [12].Some properties of these sequences were deduced directly from elementary matrix algebra in [13] [14].By using matrix algebra H. Civciv and R. Turkmen defined ( ) , s t Fibonacci and ( ) , s t Lucas matrix sequences in [15] [16].Similarly K. Uslu and Ş. Uygun defined ( ) , s t Jacosthal and ( ) , s t Jacosthal Lucas matrix sequences and by using them found some properties of Jacobsthal numbers in [17].
Definition 1.The (s,t)-Jacobsthal sequence n n n j s t sj s t tj s t j s t j s t respectively, where 1, n ≥ 0, 0 s t > ≠ and 2 8 0 s t + > [10].Some basic properties of these sequences are given in the following: , , ˆ, ,   The following properties are hold: . .
For their proofs you can look at the Ref. [17].

The Generating Functions of Jacobsthal and Jacobsthal-Lucas Matrix Sequences
Theorem 4. For , n + ∈  , x ∈  we have the generating function of Jacobsthal matrix sequence in the following: Then we have we have the generating function of Jacobsthal-Lucas matrix sequence in the following: Proof.It can be seen easily by using theorem 4 and the property of .
> Then for (s,t)-Jacobsthal Lucas matrix sequence we have > Then for (s,t)-Jacobsthal Lucas sequence we have < let be r is odd positive integer and and for r is even positive integer Proof.By using proposition 3 (iv), the nth element of (s,t)-Jacobsthal matrix sequence can be written in the following: From this equality we have If r is an odd positive integer, then we have If r is an even positive integer, then we have

Partial Sums of Jacobsthal and Jacobsthal-Lucas Matrix Sequences
Theorem 11.The partial sum of (s,t)-Jacobsthal matrix sequence for J two sides of the equality, we get n n S J J J J + = + + +  By adding 1 J two sides of the equality, we get The inverse of 1 0 J J − is available for ( ) The partial sums of (s,t)-Jacobsthal sequence for 2 1 s t + ≠ are given in the following: It is proved by the equality of matrix sequences and from Theorem 11. ■ Theorem 13.The partial sum of (s,t)-Jacobsthal Lucas matrix sequence for .
and Theorem 11 we get ( ) ( ) ( ) If the product of matrices is made the desired result is found.■ Corollary 14.The partial sums of (s,t)-Jacobsthal Lucas sequence for 2 1 s t + ≠ are given in the following: ( ) J two sides of the equality, we get ( )

∑
In the following theorem we will show the partial sum of Jacobsthal Lucas matrix sequence of the elements of power of n.

Theorem 17 .
For (s,t)-Jacobsthal matrix sequence the equality is hold. , The odd and even elements sums of (s,t)-Jacobsthal sequence for