Turkish Journal of Analysis and Number Theory, 2013, Vol. 1, No. 1, 9-12 Available online at http://pubs.sciepub.com/tjant/1/1/3 © Science and Education Publishing DOI:10.12691/tjant-1-1-3 q-Analogue of p-Adic log Γ Type Functions Associated with Modified q-Extension of Genocchi Numbers with Weight α and β Erdoğan Şen 1 , Mehmet Acikgoz 2 , Serkan Araci 3,* 1 Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, Tekirdağ, Turkey 2 Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, Turkey 3 Atatürk Street, 31290 Hatay, TURKEY *Corresponding author: mtsrkn@hotmail.com Received August 14, 2013; Revised September 17, 2013; Accepted September 25, 2013 Abstract The p-adic log gamma functions associated with q-extensions of Genocchi and Euler polynomials with weight α were recently studied [6]. 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 Cite This Article: Erdoğan Şen, Mehmet Acikgoz, and 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, no. 1 (2013): 9-12. doi: 10.12691/tjant-1-1-3. 1. Introduction Assume that p is a fixed odd prime number. Throughout this paper Z, Z, p Q p and C p will denote by the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Q , p respectively. Also we denote * N N {0} = ∪ and ( ) exp . x x e = Let { } :C Q p p v → ∪∞ ( ) Q is the field of rational numbers denote the p-adic valuation of C p normalized so that ( ) 1 p v p = . The absolute value on C p will be denoted as . p , and ( ) v x p p x p − = for C. p x ∈ When one talks of q- extensions, q is considered in many ways, e.g. as an indeterminate, a complex number C, q ∈ or a p-adic number C. p q ∈ If C, q ∈ we assume that 1. q < If C , p q ∈ we assume 1 1 1 , p p q p − − − < so that ( ) exp log x q x q = for 1. p x ≤ We use the following notation. [] [] ( ) 1 1 , 1 1 x x q q q q x x q q − −− − = = − + (1.1) where [] 1 lim ; q q x x → = cf. [1-23]. For a fixed positive integer d with ( ) , 1, d f = we set ( ) 0 , 1 lim Z / Z, Z N d N p a dp ap X X dp X a dp ∗ << = = = = ∪ +  and ( ) { } Z | mod , N N p a dp x X x a dp + = ∈ ≡ where Z a ∈ satisfies the condition 0 N a dp ≤ < . It is known that ( ) Z x N q p N q q x p p µ + =     is a distribution on X for C p q ∈ with 1 1. p q − ≤ Let ( ) Z p UD be the set of uniformly differentiable function on Z. p We say that f is a uniformly differentiable function at a point Z, p a ∈ if the difference quotient ( ) ( ) ( ) , f f x f y F xy x y − = −