BINOMINAL TRANSFORMS OF K-JACOBSTHAL SEQUENCES

In this paper, we define the binomial, k−binomial, rising, and falling transforms for k−Jacobsthal sequence. We investigate some properties of these sequences such as recurrence relations, Binet’s formula, generating functions and in the sequel of this paper denote Pascal Jacobsthal triangle for all binomial transformation sequences.


Introduction
Special integer sequences such as Fibonacci, Lucas, Jacobsthal, Pell, Horadam are very popular in the last decade.We can see abundant applications in Physics, Engineering, Architecture, Nature and Art.For instance, the ratio of two consecutive elements of Fibonacci sequence is called golden ratio.You can encounter it almost every area of science and art.And specially computers use conditional directives 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 brings out being useful for one case out of the four possibilities on 2 bits, 3 cases on 3 bits, 5 cases on 4 bits, 11 cases on 5 bits, 21 cases on 6 bits,..., which are exactly the Jacobsthal numbers [1].The Jacobsthal numbers j n are terms of the sequence {0, 1, 1, 3, 5, 11, ...}, defined by the recurrence relation for n ≥ 2, beginning with the values j 0 = 0, j 1 = 1.Because of the importance of special integer sequences, the researchers generalize them by the different methods.In this paper we introduce k-Jacobsthal sequence,depending only on one positive integer parameter k. k-Jacobsthal sequence, j k,n n∈N is defined recurrently by j k,n = k j k,n−1 + 2 j k,n−2 , j k,0 = 0, j k,1 = 1. (1.1) The Binet formula for the k-Jacobsthal sequence is denoted by j k,n = . x 1 and x 2 are the characteristic polynomial equation of recurrence formula (1.1).You can see detailed information about k-Jacobsthal sequence in [2].The main goal of this paper is to apply different binomial transforms to the k-Jacobsthal sequence and find some relations and properties of these new binomial transform sequences.In the literature Prodinger gave some information about the binomial transformation in [3].Chen investigated identities from the binomial transform in [4].Falcon and Plaza investigated the properties of k-Fibonacci sequence [5,6], and binomial transform of k-Fibonacci sequence in [7].The authors found the properties of the binomial transform of the k-Lucas sequence in [8].

Binomial Transform of k-Jacobsthal Sequences
Proof.We use the addition property of binomial numbers as ).
Now we denote the recurrence relation for the binomial transform of k-Jacobsthal sequence.
If we write the last equality again for n in place of n + 1,we get If we take into account that n−1 n = 0, we obtain By substituting the above equality (2.3) into (2.4),we get The proof is completed by some simple calculations.
From these three equalities and the recurrence relation (2.2), we obtain The generating function is denoted by (2.5)

Triangle of the binomial transform of the k-Jacobsthal sequence
In this part we introduce a new triangle of numbers for each k by using the following rules: • The elements of the left diagonal of the triangle consist of the elements of the k-Jacobsthal numbers.
• Any number off the left diagonal is the sum of the number to its left and the number diagonally above it to the left.
• On the right diagonal is the binomial transform of the k-Jacobsthal sequence.
For example following triangle is for 3-Jacobsthal sequence and its binomial transform.
You can see easily w Proof. ).
Proof.The initial conditions are found by (3.1) as w k,0 = 0 and w k,1 = k.By using Lemma (3.1), we obtain If we write th equality (3.4) again for n in place of n + 1 By substituting the above equality (3.4) into (3.5),we get The proof is found by doing some simple calculations as Theorem 3.2.(Binet Formula) The characteristic polynomial equation of recurrence for- It is demonstrated by where w 1 = k k+2+ . Proof.If we multiply the equality w k (x) with k(k + 2)x and k 2 (1 − k)x 2 , we get From these equalities and the recurrence relation (3.3), we have

Triangle of the k− binomial transform of the k-Jacobsthal sequence
In this part we introduce a new triangle of numbers for each k by using the following rules: • The left diagonal of the triangle consists of the elements of the k-Jacobsthal numbers.
• Any number of the left diagonal is k times the sum of the number to its left and the number diagonally above it to the left.
• On the right diagonal is the k−binomial transform of the k-Jacobsthal sequence.
For example following triangle is for 3-Jacobsthal sequence and its 3-binomial transform. where Theorem 4.2.For n 1, the rising k-binomial transform of the k−Jacobsthal sequence is a recurrence sequence such that with initial conditions r k,0 = 0 and r k,1 = k.
Proof.By using Binet Formulas

and then the generating function is obtained as
(4.4) Proof.By following same procedure with Theorem ( 3.3), we have By the above computations, we obtain the generating function as Triangle of the rising k− binomial transform of the k-Jacobsthal sequence In this part we introduce a new triangle of numbers for each k by using the following rules: • The left diagonal of the triangle consists of the elements of the k-Jacobsthal numbers.
• Any number of the left diagonal is the sum of the number diagonally above it to the left and k-times the number to its left.
• On the right diagonal is the rising k−binomial transform of the k-Jacobsthal sequence.
For example following triangle is for 3-Jacobsthal sequence and its rising 3-binomial transform.with initial conditions f k,0 = 0 and f k,1 = 1. Proof. ).
Proof.The initial conditions are found by the definition (5.1) as f k,0 = 0 and f k,1 = 1.By using Lemma 5.1 and (1.1), we obtain (5.4) If we write this equality again for n in place of n + 1 By substituting the above equality (5.4 into (5.5),we get The proof is completed by some simple calculations. (5.7) Proof.If we multiply the equality f k (x) with −3kx and 2(k 2 − 1)x 2 , we get From these equalities and the recurrence relation (5.3), we obtain

Triangle of the falling k− binomial transform of the k-Jacobsthal sequence
In this part we introduce a new triangle of numbers for each k by using the following rules: • The left diagonal of the triangle consists of the elements of the k-Jacobsthal numbers.
• Any number of the left diagonal is the sum of the number to its left and k times the number diagonally above it to the left.
• On the right diagonal is the falling k−binomial transform of the k-Jacobsthal sequence.
For example the following triangle is for 3-Jacobsthal sequence and its falling 3-binomial

Definition 2 . 1 .Lemma 2 . 1 .
The binomial transform of k-Jacobsthal sequence j k,n n∈N is indicated as b k,n n∈N where b k,n is given by b k,n = n ∑ i=0 n i j k,i .(2.1) for any positive integer parameter k.Let n ≥ 1 positive integer, then the binomial transform of k-Jacobsthal sequence verifies the following relation b

Theorem 2 . 1 .
The following recurrence relation is verified by the binomial transform of k-Jacobsthal sequence b k,n+1 = (k + 2)b k,n +(1 − k)b k,n−1 .(2.2) Proof.The initial conditions are found by the definition as b k,0 = 0 and b k,1 = 1.By using Lemma 2.1 and (1.1), we obtain b

Theorem 2 . 2 .
(Binet Formula) Any terms of the binomial transform of k-Jacobsthal sequence can be calculated by means of Binet formula.It is demonstrated by b

Theorem 2 . 3 .
(Generating function) The generating function of the binomial transform of the k-Jacobsthal sequence is a power series centered at the origin whose coefficients are the binomial transform of the k-Jacobsthal sequence.Let us demonstrate the generating function as b k (x) .So, b k (x) = b k,0 +b k,1 x + b k,2 x 2 +...And then, if we multiply the equality with -(k + 2)x and (1 − k)x 2 , we get

3 .
The k-Binomial Transforms Of The k-Jacobsthal Sequence Definition 3.1.The k-binomial transform of the k−Jacobsthal sequence w k,n n∈N is given by the following formula 1,n n∈N = b k,n n∈N , and w k,n = k n b k,n .Lemma 3.1.The k−binomial transform of the k−Jacobsthal sequence verifies the relation

Theorem 3 . 1 .
The following recurrence relation is verified by the k−binomial transform of the k−Jacobsthal sequence whose solutions are w 1 and w 2 .Any terms of the k−binomial transform of k-Jacobsthal sequence can be calculated by means of Binet formula.

Theorem 3 . 3 .
(Generating function) Let us demonstrate the generating function as w k (x) = w k,0 +w k,1 x + w k,2 x 2 +...The generating function for the k−binomial transform of k-Jacobsthal sequence is obtained as

4 . 1 )
The Rising k-Binomial Transform of the k-Jacobsthal Sequence Definition 4.1.The rising k-binomial transform of the k−Jacobsthal sequence r k,n n∈N is efined as the following formula r k,n = Theorem 4.1.(Binet Formula) The Binet formula for the rising k-binomial transform of the k−Jacobsthal sequence is

Theorem 4 . 3 .
(Generating function) The generating function of rising k-binomial transform of the k−Jacobsthal sequence is denoted by R

5 .Lemma 5 . 1 .
The Falling k-Binomial Transform of the k-Jacobsthal Sequence Definition 5.1.Let k is any positive integer.The falling k-binomial transform of the k−Jacobsthal sequence f k,n n∈N is given by f k,n = The falling k−binomial transform of the k−Jacobsthal sequence verifies the relai ( j k,i + j k,i+1 ).(5.2)

Theorem 5 . 1 .
The following recurrence relation is verified by the falling k−binomial transform of the k−Jacobsthal sequence