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q-Analogue of p-Adic log Γ Type Functions Associated with Modified q-Extension of Genocchi Numbers with Weight α and β

Erdoğan Şen*, Mehmet Acikgoz, Serkan Araci*

Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, Tekirdağ, TurkeyAtatürk Street, 31290 Hatay, TURKEY

Abstract: The p-adic log gamma functions associated with q-extensions of Genocchi and Euler polynomials with weight α were recently studied . By the same motivation, we aim in this paper to describe q-analogue of p-adic log gamma functions with weight alpha and beta. Moreover, we give relationship between p-adic q-log gamma functions with weight (α,β) and q-extension of Genocchi numbers with weight alpha and beta and modified q-Euler numbers with weight α.

Keywords: modified q-Genocchi numbers with weight alpha and beta, modified q-Euler numbers with weight alpha and beta, p-adic log gamma functions

Received September 14, 2013; Revised September 21, 2013; Accepted September 23, 2013

Citation: Erdoğan Şen, Mehmet Acikgoz, Serkan Araci, q-Analogue of p-Adic log Γ Type Functions Associated with Modified q-Extension of Genocchi Numbers with Weight α and β. Turkish Journal of Analysis and Number Theory, 1 (1). 9-12. January 2013. doi:10.12691/tjant-1-1-3

1. Introduction

Assume that p is a fixed odd prime number. Throughout this paper and will denote by the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of respectively. Also we denote and Let denote the p-adic valuation of normalized so that . The absolute value on will be denoted as , and for When one talks of q-extensions, is considered in many ways, e.g. as an indeterminate, a complex number or a p-adic number If we assume that If we assume so that for We use the following notation.

[picture] | (1.1)

where cf. [1-23].

For a fixed positive integer with we set

and

where satisfies the condition .

It is known that

is a distribution on for with

Let be the set of uniformly differentiable function on We say that is a uniformly differentiable function at a point if the difference quotient

has a limit as and denote this by . The p-adic q-integral of the function is defined by

[picture] | (1.2)

The bosonic integral is considered by Kim as the bosonic limit . Similarly, the p-adic fermionic integration on defined by Kim as follows:

Let then we have p-adic fermionic integral on as follows:

Stirling asymptotic series are known as

[picture] | (1.3)

where are familiar n-th Bernoulli numbers cf. [6,8,9,23].

Recently, Araci et al. defined modified q-Genocchi numbers and polynomials with weight and in [4,5] by the means of generating function:

[picture] | (1.4)

So from the above, we easily get Witt's formula of modified q-Genocchi numbers and polynomials with weight and as follows:

[picture] | (1.5)

where are modified q-extension of Genocchi numbers with weight and cf. [4,5].

In [21], Rim and Jeong are defined modified q-Euler numbers with weight as follows:

[picture] | (1.6)

From expressions of (1.5) and (1.6), we get the following Proposition 1.

Proposition 1. The following

is true.

In previous paper [6], Araci, Acikgoz and Park introduced weighted q-analogue of p-adic lod gamma type functions and derived some interesting identities. They were motivated from paper of T. Kim by "On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory, 76 (1999), no. 2, 320-329." By the same motivation, we introduce q-analogue of p-adic log gamma type function with weight and We derive in this paper some interesting identities including this type of functions.

2. On P-Adic logΓ Function with Weight α and β

In this part, from (1.2), we start at the following nice identity:

[picture] | (2.1)

where and (see [4]).

In particular for into (2.1), we easily see that

[picture] | (2.2)

By simple an application, it is easy to indicate as follows:

[picture] | (2.3)

where .

By expression of (2.3), we can derive

[picture] | (2.4)

where is constant.

If we take so we get By expression of (2.3) and (2.4), we easily see that,

[picture] | (2.5)

It is considered by T. Kim for q-analogue of p-adic locally analytic function on as follows:

[picture] | (2.6)

(for detail, see [5,6]).

By the same motivation of (2.6), in previous paper [6], q-analogue of p-adic locally analytic function on with weight is considered

[picture] | (2.7)

In particular in (2.7), we easily see that,

With the same manner, we introduce q-analogue of p-adic locally analytic function on p with weight and as follows:

[picture] | (2.8)

From expressions of (2.2) and (2.8), we state the following Theorem:

Theorem 1. The following identity holds:

It is easy to show that,

[picture] | (2.9)

Substituting into (2.5) and by using (2.9), we get interesting formula:

[picture] | (2.10)

If we substitute into (2.10), we get Kim's q-analogue of p-adic log gamma function (for detail, see [8]). From expression of (1.2) and (2.10), we obtain the following worthwhile and interesting theorems.

Theorem 2. For the following

is true.

Corollary 1. Taking in Theorem 2, we get nice identity:

where are called famous Genocchi numbers.

Theorem 3. The following nice identity

is true.

Corollary 2. Putting into Theorem 3, we have the following identity:

where are familiar Euler numbers.

References

[1]   | S. Araci, M. Acikgoz and J. J. Seo, A note on the weighted q-Genocchi numbers and polynomials with their interpolation function, Honam Mathematical J. 34 (2012), No. 1, pp. 11-18.
[2]   | S. Araci, D. Erdal and J. J. Seo, A study on the fermionic p adic q- integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
[3]   | S. Araci, J. J. Seo and D. Erdal, New construction weighted (h, q)-Genocchi numbers and polynomials related to Zeta type function, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 487490, 7 pages.
[4]   | S. Araci, M. Acikgoz, F. Qi and H. Jolany, A note on the modified q-Genocchi numbers and polynomials with weight (α, β) and their interpolation function at negative integers, accepted for publication in Journal of Fasc. Math.
[5]   | S. Araci and M. Acikgoz, A note on the values of the weighted q-Bernstein polynomials and modified q-Genocchi numbers with weight α and β via the p-adic q-integral on Zp, submitted.
[6]   | S. Araci, M. Acikgoz and K. H. Park, A note on the q-analogue of Kim's p-adic log gamma functions associated with q-extension of Genocchi and Euler polynomials with weight α, Bull. Korean Math. Soc. 50 (2013), No. 2, pp. 583-588.
[7]   | S. Araci, M. Acikgoz and E. Şen, On the extended Kim's p-adic q-deformed fermionic integrals in the p-adic integer ring, Journal of Number Theory 133 (2013) 3348-3361.
[8]   | T. Kim, A note on the q-analogue of p-adic log gamma function, arXiv:0710.4981v1 [math.NT].
[9]   | T. Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory, 76 (1999), no. 2, 320-329.
[10]   | T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465.
[11]   | T. Kim, On the multiple q-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008) 481-486.
[12]   | T. Kim, On the weighted q-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 21(2011), no.2, p. 207-215.
[13]   | T. Kim, q-Volkenborn integration, Russ. J. Math. phys. 9 (2002), 288-299.
[14]   | T. Kim, An invariant p-adic q-integrals on Zp, Applied Mathematics Letters, vol. 21, pp. 105-108,2008.
[15]   | T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15-27.
[16]   | T. Kim, New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys. 17 (2010), no. 2, 218-225.
[17]   | T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp, Russ. J. Math. Phys. 16 (2009), no.4, 484-491.
[18]   | T. Kim and S.-H. Rim, On the twisted q-Euler numbers and polynomials associated with basic q-l-functions, Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 738-744, 2007.
[19]   | T. Kim, On p-adic q-l-functions and sums of powers, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.07.071.
[20]   | C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, Advan. Stud. Contemp. Math. 21(2011), 47-54.
[21]   | S.-H. Rim and J. Jeong, A note on the modified q-Euler numbers and polynomials with weight α, International Mathematical Forum, Vol. 6, 2011, no. 65, 3245-3250.
[22]   | Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers, Advan. Stud. Contemp. Math., 11(2005), 205-218.
[23]   | D. Zill and M. R. Cullen, Advanced Engineering Mathematics, Jones and Bartlett, 2005.