_{1}

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.

There are so many studies in the literature that are concern about special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, and Padovan in [

Definition 1. The (s,t)-Jacobsthal sequence

respectively, where

Some basic properties of these sequences are given in the following:

In the following definition, (s,t)-Jacosthal

Definition 2. The (s,t)-Jacobsthal matrix sequence

respectively, where

Throughout this paper, for convenience we will use the symbol

Proposition 3. Let us consider

1)

2) For

3) For

4) For

For their proofs you can look at the Ref. [

Theorem 4. For

Proof. By using the expansion of geometric series and proposition 3, we can write

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Corollary 5. Let

and

Corollary 6. Let

Corollary 7. Let

Proof. It can be seen easily by using theorem 4 and the property of

Corollary 8. Let

Corollary 9. Let

and

Theorem 10. For

Then we have

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

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Theorem 11. The partial sum of (s,t)-Jacobsthal matrix sequence for

Proof. Let

By adding

The inverse of

By using following equalities

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Corollary 12. The partial sums of (s,t)-Jacobsthal sequence for

and

Proof. 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

ing

Proof. By using

If the product of matrices is made the desired result is found. ■

Corollary 14. The partial sums of (s,t)-Jacobsthal Lucas sequence for

and

Proof. It is proved by the equality of matrix sequences and from Theorem 11. ■

Theorem 15. Let

Proof. By multiplying

By adding

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Corollary 16. The odd and even elements sums of (s,t)-Jacobsthal sequence for

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.

Proof. By using the equality of

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Thank you very much to the editor and the referee for their valuable comments.

ŞükranUygun, (2016) Some Sum Formulas of ( s , t )-Jacobsthal and ( s , t )-Jacobsthal Lucas Matrix Sequences. Applied Mathematics,07,61-69. doi: 10.4236/am.2016.71005