1. Introduction

The Fibonacci sequence and the Lucas sequence are the two shining stars in the vast array of integer sequences. They have fascinated both amateurs and professional mathematicians for centuries. Also they continue to charm us with their beauty, their abundant applications, their ubiquitous habit of occurring in totally surprising and unrelated places. Fibonacci numbers have been generalized by different authors. Some authors have generalized the Fibonacci sequence by preserving the recurrence relation and altering the first two terms of the sequence ^{[10, 12, 17]}, while others have generalized the Fibonacci sequence by preserving the first two terms but altering the recurrence relation slightly ^{[14, 15, 18]}.

More recently, Fibonacci, Lucas, Pell, Pell-Lucas, Modified Pell, Jacobhstal, Jacobsthal-Lucas sequences were generalized for any positive real number k. Also the study of the k-Fibonacci sequence, the k-Lucas sequence, the k-Pell sequence, the k-Pell-Lucas sequence, the Modified k-Pell sequence, the k-Jacobhstal sequence and the k-Jacobsthal Lucas sequence appeared (see [1-6,8,13]). Falcon and plaza ^[7] has given several formulae for the sum of k-Fibonacci numbers with indexes in an arithmetic sequence. Falcon ^[9] defines the k-Lucas number with indexes in an arithmetic sequence. Also, deduced generating function and several sum formulae for these numbers with indexes in an arithmetic sequence.

For any positive real number k, the k-Jacobsthal sequence say ^[13] is defined recurrently by

(1.1)

with initial conditions.

Particular cases of definition (1.1)

If we obtain the Jacobsthal sequence ^[11] (A001045) ^[16]

If we obtain the sequence A002605 ^[16].

First few terms of the k-Jacobsthal numbers (1.1) are

Some of the interesting properties that the k-Jacobsthal sequence satisfies are summarized as below ^[13]:

1.1.1.1. Binet’s Formula

The Binet formula for the n^th k-Jacobsthal numbers is

(1.2)

where , are the roots of the characteristic equation associated to the recurrence relation defined in equation (1.1) and ,, , .

1.1.1.2. Catalan’s Identity

(1.3)

1.1.1.3. D’Ocagne Identity

If m > n then

(1.4)

1.1.1.4. Convolution Product

(1.5)

Now, we introduce the k-Jacobsthal Lucas sequence, whose recurrence relation is the same as the k-Jacobsthal sequence.

For any positive real number k, the k-Jacobsthal-Lucas sequence say ^[2] is defined recurrently by

(1.6)

with initial conditions .

As particular cases:

If we obtain the Jacobsthal-Lucas sequence ^[11] (A014551) ^[16].

If we obtain the sequence A080040 ^[16].

First few terms of the k-Jacobsthal Lucas numbers equation (1.6) are

Some of the interesting properties that the k-Jacobsthal Lucas sequence satisfies are summarized as below ^[2].

The Binet formula for the k-Jacobsthal numbers is

(1.7)

where , are the roots of the characteristic equation , , and associated to the recurrence relation defined in equation (1.6).

(1.8)

If m > n then

(1.9)

Now, we prove some properties of the k-Jacobsthal numbers that we will be needed later.

(1.10)

Proof: Taking R.H.S. and applying the Binet’s formula for k-Jacobsthal numbers

That is

(1.11)

Proof: We will prove this by using the Mathematical induction method. For

we see that it is true for

Now for we have

again we see that it is true for

Now, suppose the formula is true until

(1.12)

Then,

That is .

2. On the k-Jacobsthal Numbers of Kind an+r

In this section, we shall derive some formulae for the sums of the k- Jacobsthal numbers with index in an arithmetic sequence, say for fixed integer a and r such that Also, we have discuss generating function for these numbers with index in an arithmetic sequence.

First we prove following lemmas that will be needed later.

For all integers

(2.1)

Proof: Taking R.H.S. and applying the Binet’s formula for k-Jacobsthal numbers and .

That is

(2.2)

Proof: Taking R.H.S. and applying Binet’s formula and Lemma 2.1.

now, since , then the above formula can be rewritten as

equation (2.2) gives the general term of the k-Jacobsthal sequence as a linear combination of the two preceding terms.

2.2.1. Generating Function for the Sequence

Suppose that be the generating function of the sequence with That is

(2.3)

Now multiplying both sides by the algebraic expression we get

Now, from equation (2.2), the summation of right hand side of the above equation vanishes. That is

(2.4)

from equation (1.10), we have

Hence equation (2.4) becomes

(2.5)

Particular cases:

For the different values of a and r, the generating function of the sequences are:

1) If and then,

which is the generating function of the k-Jacobsthal sequence ^[13].

2) If , then

At ,

At ,

3) If , then

At ,

At ,

At ,

2.2.2. Sum of k-Jacobsthal numbers with arithmetic index

In this section, we have discuss the sum formula for the k-Jacobsthal numbers with arithmetic index, where a and r are fixed integer such that

Sum of k-Jacobsthal number of kind is

(2.6)

Proof: Applying Binet’s formula for the k- Jacobsthal numbers, we have

Formula for sum of odd k-Jacobsthal numbers

If then equation (2.6) gives

(2.7)

For example: (1) If , then and, we have

(2) If, then, we have

If , then

If , then

If , then

(3) If , then , we have

If , then

If , then

If , then

If , then

If , then

Sum of even k-Jacobsthal numbers.

If , then equation (2.6) is

(2.8)

For example: (1) If, then , we have

If , then

If , then

(2) If , then , we have

If , then

If , then

If , then

If , then

Now, we have considered the alternating sequence . By the previous method we can also find the sum formula for this sequence.

Alternating sum of the k-Jacobsthal numbers with index is given by

(2.9)

Proof: Taking L.H.S. and applying the binet formula for the k-Jacobsthal numbers equation (1.2) we have

that is

For different values of a and r, the above sum can be written as

For we have

For,

a) If then

b) If then

For

If then

If then

If then

Reference

[1]	Bolat C., Kose H., On the Properties of k-Fibonacci Numbers. Int. J.Contemp. Math. Sciences 2010, 22 (5), 1097-1105.
	In article
[2]	Campos H., Catarino P., Aires A.P., Vasco P. and Borges A., On Some Identities of k-Jacobsthal-Lucas Numbers, Int. Journal of Math. Analysis, 2014, 8 (10), 489-494.
	In article
[3]	Catarino P., On Some Identities and Generating Functions for k-Pell Numbers. Int. Journal of Math. Analysis 2013, 7 (38), 1877-1884.
	In article
[4]	Catarino P., Vasco P., Modified k-Pell Sequence: Some Identities and Ordinary Generating Function. Applied Mathematical Sciences 2013, 7 (121), 6031-6037.
	In article
[5]	Catarino P., Vasco P., On some Identities and Generating Functions for k-Pell-Lucas Sequence. Applied Mathematical Sciences 2013, 7 (98), 4867-4873.
	In article
[6]	Catarino P., On Some Identities for k-Fibonacci Sequence. International Journal of Contemporary mathematical Sciences 2014, 9 (1), 37-42.
	In article
[7]	Falcon S. and Plaza A., On k-Fibonacci Numbers of Arithmetic Indexes, Applied Mathematics and Computation, 2009, 208, 180-185.
	In article		CrossRef
[8]	Falcon S., On the k-Lucas Numbers, International Journal of Contemporary mathematical Sciences 2011, 6 (21), 1039-1050.
	In article
[9]	Falcon S., On the k-Lucas Numbers of Arithmetic Indexes, Applied Mathematics, 2012, 3, 1202-1206.
	In article		CrossRef
[10]	Horadam A. F., A Generalized Fibonacci Sequence, The American Mathematical Monthly, 1961, 68, 455-459.
	In article		CrossRef
[11]	Horadam A.F., Jacobsthal Representation Numbers. Fibonacci Quarterly, 1996, 34 (1), 40-54.
	In article
[12]	Jaiswal D. V., On a Generalized Fibonacci Sequence, Labdev Journal of Science and Technology, Part A, 1969, 7, 67-71.
	In article
[13]	Jhala D., Sisodiya K., Rathore G.P.S., On Some Identities for k-Jacobsthal Numbers. Int. Journal of Math. Analysis 2013, 7 (2), 551-556.
	In article
[14]	Lee G. Y., Lee S. G., Kim J. S., Shin H. K., The Binet Formula and the Representation of k-Generalized Fibonacci Numbers, Fibonacci Quarterly, 2001, 39 (2), 158-164.
	In article
[15]	Lee G. Y., Lee S. G., Shin H. K., On the k-Ggeneralized Fibonacci Matrix Qk, Linear Algebra Applications, 1997, 251, 73-88.
	In article		CrossRef
[16]	Slone NJA, The On-Line Encyclopedia of Integer Sequences, (2006), www.research.att.com/~njas/sequences/.
	In article
[17]	Walton J. E. and Horadam A. F., Some Further Identities for the Generalized Fibonacci sequence, Fibonacci Quarterly, 1974, 12 (3), 272-280.
	In article
[18]	Zai L. J. and Sheng L. J., Some Properties of the Generalization of the Fibonacci Sequence, Fibonacci Quarterly, 1987, 25 (2), 111-117.
	In article