Keywords: modified qGenocchi numbers with weight alpha and beta, modified qEuler numbers with weight alpha and beta, padic log gamma functions
Turkish Journal of Analysis and Number Theory, 2013 1 (1),
pp 912.
DOI: 10.12691/tjant113
Received September 14, 2013; Revised September 21, 2013; Accepted September 23, 2013
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
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 padic rational numbers and the completion of the algebraic closure of respectively. Also we denote and Let denote the padic valuation of normalized so that . The absolute value on will be denoted as , and for When one talks of qextensions, is considered in many ways, e.g. as an indeterminate, a complex number or a padic number If we assume that If we assume so that for We use the following notation.
 (1.1) 
where cf. [123]^{[1]}.
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 padic qintegral of the function is defined by
 (1.2) 
The bosonic integral is considered by Kim as the bosonic limit . Similarly, the padic fermionic integration on defined by Kim as follows:
Let then we have padic fermionic integral on as follows:
Stirling asymptotic series are known as
 (1.3) 
where are familiar nth Bernoulli numbers cf. ^{[6, 8, 9, 23]}.
Recently, Araci et al. defined modified qGenocchi numbers and polynomials with weight and in ^{[4, 5]} by the means of generating function:
 (1.4) 
So from the above, we easily get Witt's formula of modified qGenocchi numbers and polynomials with weight and as follows:
 (1.5) 
where are modified qextension of Genocchi numbers with weight and cf. ^{[4, 5]}.
In ^{[21]}, Rim and Jeong are defined modified qEuler numbers with weight as follows:
 (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 qanalogue of padic lod gamma type functions and derived some interesting identities. They were motivated from paper of T. Kim by "On a qanalogue of the padic log gamma functions and related integrals, J. Number Theory, 76 (1999), no. 2, 320329." By the same motivation, we introduce qanalogue of padic log gamma type function with weight and We derive in this paper some interesting identities including this type of functions.
2. On PAdic log Γ Function with Weight α and β
In this part, from (1.2), we start at the following nice identity:
 (2.1) 
where and (see ^{[4]}).
In particular for into (2.1), we easily see that
 (2.2) 
By simple an application, it is easy to indicate as follows:
 (2.3) 
where .
By expression of (2.3), we can derive
 (2.4) 
where is constant.
If we take so we get By expression of (2.3) and (2.4), we easily see that,
 (2.5) 
It is considered by T. Kim for qanalogue of padic locally analytic function on as follows:
 (2.6) 
(for detail, see ^{[5, 6]}).
By the same motivation of (2.6), in previous paper ^{[6]}, qanalogue of padic locally analytic function on with weight is considered
 (2.7) 
In particular in (2.7), we easily see that,
With the same manner, we introduce qanalogue of padic locally analytic function on p with weight and as follows:
 (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,
 (2.9) 
Substituting into (2.5) and by using (2.9), we get interesting formula:
 (2.10) 
If we substitute into (2.10), we get Kim's qanalogue of padic 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 qGenocchi numbers and polynomials with their interpolation function, Honam Mathematical J. 34 (2012), No. 1, pp. 1118. 
 In article  

[2]  S. Araci, D. Erdal and J. J. Seo, A study on the fermionic p adic q integral representation on Z_{p} associated with weighted qBernstein and qGenocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages. 
 In article  

[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. 
 In article  

[4]  S. Araci, M. Acikgoz, F. Qi and H. Jolany, A note on the modified qGenocchi numbers and polynomials with weight (α, β) and their interpolation function at negative integers, accepted for publication in Journal of Fasc. Math. 
 In article  

[5]  S. Araci and M. Acikgoz, A note on the values of the weighted qBernstein polynomials and modified qGenocchi numbers with weight α and β via the padic qintegral on Z_{p}, submitted. 
 In article  

[6]  S. Araci, M. Acikgoz and K. H. Park, A note on the qanalogue of Kim's padic log gamma functions associated with qextension of Genocchi and Euler polynomials with weight α, Bull. Korean Math. Soc. 50 (2013), No. 2, pp. 583588. 
 In article  

[7]  S. Araci, M. Acikgoz and E. Şen, On the extended Kim's padic qdeformed fermionic integrals in the padic integer ring, Journal of Number Theory 133 (2013) 33483361. 
 In article  CrossRef 

[8]  T. Kim, A note on the qanalogue of padic log gamma function, arXiv:0710.4981v1 [math.NT]. 
 In article  

[9]  T. Kim, On a qanalogue of the padic log gamma functions and related integrals, J. Number Theory, 76 (1999), no. 2, 320329. 
 In article  

[10]  T. Kim, On the qextension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 14581465. 
 In article  CrossRef 

[11]  T. Kim, On the multiple qGenocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008) 481486. 
 In article  CrossRef 

[12]  T. Kim, On the weighted qBernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 21(2011), no.2, p. 207215. 
 In article  

[13]  T. Kim, qVolkenborn integration, Russ. J. Math. phys. 9 (2002), 288299. 
 In article  

[14]  T. Kim, An invariant padic qintegrals on Z_{p}, Applied Mathematics Letters, vol. 21, pp. 105108,2008. 
 In article  CrossRef 

[15]  T. Kim, qEuler numbers and polynomials associated with padic qintegrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 1527. 
 In article  

[16]  T. Kim, New approach to qEuler polynomials of higher order, Russ. J. Math. Phys. 17 (2010), no. 2, 218225. 
 In article  

[17]  T. Kim, Some identities on the qEuler polynomials of higher order and qStirling numbers by the fermionic padic integral on Z_{p}, Russ. J. Math. Phys. 16 (2009), no.4, 484491. 
 In article  

[18]  T. Kim and S.H. Rim, On the twisted qEuler numbers and polynomials associated with basic qlfunctions, Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 738744, 2007. 
 In article  CrossRef 

[19]  T. Kim, On padic qlfunctions and sums of powers, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.07.071. 
 In article  CrossRef 

[20]  C. S. Ryoo, A note on the weighted qEuler numbers and polynomials, Advan. Stud. Contemp. Math. 21(2011), 4754. 
 In article  

[21]  S.H. Rim and J. Jeong, A note on the modified qEuler numbers and polynomials with weight α, International Mathematical Forum, Vol. 6, 2011, no. 65, 32453250. 
 In article  

[22]  Y. Simsek, Theorems on twisted Lfunction and twisted Bernoulli numbers, Advan. Stud. Contemp. Math., 11(2005), 205218. 
 In article  

[23]  D. Zill and M. R. Cullen, Advanced Engineering Mathematics, Jones and Bartlett, 2005. 
 In article  
