q-ANALOGUE OF p-ADIC log Γ TYPE FUNCTIONS ASSOCIATED WITH MODIFIED q-EXTENSION OF GENOCCHI NUMBERS WITH WEIGHT α AND β 2 1 0 SERKANARACIANDMEHMETACIKGOZ 2 n Abstract. The fundamental aim of this paper is to describe q-Analogue of a p-adic log gamma functions with weight alpha and beta. Moreover, we give J relationship between p-adic q-log gamma funtions with weight (α,β) and q- 0 extension of Genocchi numbers with weight alpha and beta and modified q- 2 Eulernumberswithweightα ] T N 1. Introduction . h t Assume that p be a fixed odd prime number. Throughout this paper Z, Zp, Qp a and C will denote by the ring of integers,the field of p-adic rationalnumbers and m p the completion of the algebraic closure of Q , respectively. Also we denote N∗ = p [ N 0 andexp(x)=ex.Letvp :Cp Q (Q is the field of rational numbers) ∪{ } → ∪{∞} 2 denotethep-adicvaluationofCp normalizedsothatvp(p)=1. Theabsolutevalue 9v on Cp will be denoted as |.|p, and |x|p = p−vp(x) for x ∈ Cp. When one talks of q-extensions, q is considered in many ways, e.g. as an indeterminate, a complex 0 number q C, or a p-adic number q C , If q C we assume that q < 1. If 3 p ∈ ∈ ∈ | | 1.1 tqh∈e Cfopll,owweinagssnuomtaeti|o1n−q|p <p−p−11, so that qx =exp(xlogq) for |x|p ≤1. We use 0 1 qx 1 ( q)x 2 (1.1) [x] = − , [x] = − − q 1 q −q 1+q 1 − v: where limq→1[x]q =x; cf. [1-24]. i For a fixed positive integer d with (d,f)=1, we set X X = X =limZ/dpNZ, r d ←− a N X∗ = a+dpZ p 0<a∪<dp (a,p)=1 and a+dpNZ = x X x a moddpN , p ∈ | ≡ where a Z satisfies the condition 0 a<dpN. (cid:8) (cid:0) (cid:1)(cid:9) ∈ ≤ It is known that qx µ x+pNZ = q p [pN] q (cid:0) (cid:1) Date:January20,2012. 2000 Mathematics Subject Classification. Primary46A15,Secondary41A65. Key words and phrases. Modifiedq-Genocchi numbers withweightalpha andbeta, Modified q-Eulernumberswithweightalphaandbeta, p-adicloggammafunctions. 1 2 SERKANARACIANDMEHMETACIKGOZ is a distribution on X for q C with 1 q 1. ∈ p | − |p ≤ Let UD(Z ) be the set of uniformly differentiable function on Z . We say that p p f is a uniformly differentiable function at a point a Z , if the difference quotient p ∈ f(x) f(y) F (x,y)= − f x y − has a limit f´(a) as (x,y) (a,a) and denote this by f UD(Z ). The p-adic p → ∈ q-integral of the function f UD(Z ) is defined by p ∈ pN−1 1 (1.2) I (f)= f(x)dµ (x)= lim f(x)qx q ZZp q N→∞[pN]q x=0 X The bosonic integral is considered by Kim as the bosonic limit q 1, I (f) = 1 → limq→1Iq(f). Similarly, the p-adic fermionic integration on Zp defined by Kim as follows: I−q(f)=q→lim−qIq(f)= Z f(x)dµ−q(x) Z p Let q 1, then we have p-adic fermionic integral on Z as follows: p → pN−1 x I−1(f)= lim Iq(f)= lim f(x)( 1) . q→−1 N→∞ − x=0 X Stirling asymptotic series are defined by Γ(x+1) 1 ∞ ( 1)n+1 B n+1 (1.3) log = x logx+ − x √2π − 2 n(n+1) xn − (cid:18) (cid:19) (cid:18) (cid:19) n=1 X where B are familiar n-th Bernoulli numbers cf. [6, 8, 9, 25]. n Recently, Araci et al. defined modified q-Genocchi numbers and polynomials with weight α and β in [4, 5] by the means of generating function: ∞ tn (1.4) gn(α,q,β)(x) n! =t Z q−βξe[x+ξ]qαtdµ−qβ(ξ) n=0 Z p X Sofromabove,weeasilygetWitt’sformulaofmodifiedq-Genocchinumbersand polynomials with weight α and β as follows: g(α,β) (x) (1.5) nn+1+,q1 = Z q−βξ[x+ξ]nqαdµ−qβ (ξ) Z p where g(α,β)(0) := g(α,β) are modified q extension of Genocchi numbers with n,q n,q weight α and β cf. [4,5]. In [21], Rim and Jeong are defined modified q-Euler numbers with weight α as follows: (1.6) ξ(α) = q−t[t] dµ (t) n,q qα −q Z Z p e From expressions of (1.5) and (1.6), we get the following Proposition 1: ON THE q-ANALOGUE OF p-ADIC log GAMMA TYPE FUNCTIONS 3 Proposition 1. The following (α) gn(α+,11),q (1.7) ξ = n,q n+1 is true. e In previous paper [6], Araci, Acikgoz and Park introduced weighted q-Analogue of p-Adic log gamma type functions and they derived some interesting identities in Analytic Numbers Theory and in p-Adic Analysis. They were motivated from paperofT.Kimby”Ona q-analogueofthe p-adicloggammafunctionsandrelated integrals, J. Number Theory, 76 (1999), no. 2, 320-329.” We also introduce q- Analogue of p-Adic log gamma type function with weight α and β. We derive in this paper some interesting identities this type of functions. On p-adic log Γ function with weight α and β In this part, from (1.2), we begin with the following nice identity: n−1 (1.8) I(β) q−βxf +( 1)n−1I(β) q−βxf =[2] ( 1)n−1−lf(l) −q n − −q qβ − l=0 (cid:0) (cid:1) (cid:0) (cid:1) X where f (x)=f(x+n) and n N (see [4]). n ∈ In particular for n=1 into (1.8), we easily see that (1.9) I(β) q−βxf +I(β) q−βxf =[2] f(0). −q 1 −q qβ With the simple appli(cid:0)cation, i(cid:1)t is easy(cid:0)to ind(cid:1)icate as follows: ∞ ( 1)n+1 (1.10) ((1+x)log(1+x))´=1+log(1+x)=1+ − xn n(n+1) n=1 X where ((1+x)log(1+x))´= d ((1+x)log(1+x)) dx By expression of (1.10), we can derive ∞ ( 1)n+1 (1.11) (1+x)log(1+x)= − xn+1+x+c, where c is constant. n(n+1) n=1 X If we take x = 0, so we get c = 0. By expression of (1.10) and (1.11), we easily see that, ∞ ( 1)n+1 (1.12) (1+x)log(1+x)= − xn+1+x. n(n+1) n=1 X It is considered by T. Kim for q-analogue of p adic locally analytic function on C Z as follows: p p \ (1.13) G (x)= [x+ξ] log[x+ξ] 1 dµ (ξ) (for detail, see[5,6]). p,q Z q q− −q Z p (cid:16) (cid:17) By the same motivation of (1.13), in previous paper [6], q-analogue of p-adic locally analytic function on C Z with weight α is considered p p \ (1.14) Gp(α,q)(x)= Z [x+ξ]qα log[x+ξ]qα −1 dµ−q(ξ) Z p (cid:16) (cid:17) (1) In particular α=1 into (1.14), we easily see that, G (x)=G (x). p,q p,q 4 SERKANARACIANDMEHMETACIKGOZ Withthesamemanner,weintroduceq-Analogeofp-adiclocallyanalyticfunction on C Z with weight α and β as follows: p p \ (1.15) G(α,β)(x)= q−βξ[x+ξ] log[x+ξ] 1 dµ (ξ) p,q Z qα qα − −qβ Z p (cid:16) (cid:17) From expressions of (1.9) and (1.16), we state the following Theorem: Theorem 1. The following identity holds: G(α,β)(x+1)+G(α,β)(x)=[2] [x] log[x] 1 . p,q p,q qβ qα qα − (cid:16) (cid:17) It is easy to show that, 1 qα(x+ξ) (1.16) [x+ξ] = − qα 1 qα − 1 qαx+qαx qα(x+ξ) = − − 1 qα − 1 qαx 1 qαξ = − +qαx − 1 qα 1 qα (cid:18) − (cid:19) (cid:18) − (cid:19) = [x] +qαx[ξ] qα qα Substituting x qαx[ξ]qα into (1.12) and by using (1.16), we get interesting → [x]qα formula: (1.17) [x+ξ] log[x+ξ] 1 = [x] +qαx[ξ] log[x] + ∞ (−qαx)n+1[ξ]nqα+1 [x] qα qα − qα qα qα n(n+1) [x]n − qα (cid:16) (cid:17) (cid:16) (cid:17) nX=1 qα Ifwesubstitute α=1into(1.17), wegetKim’sq-Analogueofp-adicloggamma fuction (for detail, see[8]). From expressionof (1.2) and (1.17), we obtain worthwhile and interesting theo- rems as follows: Theorem 2. For x C Z the following p p ∈ \ (1.18) G(α,β)(x)= [2]qβ [x] +qαxg2(α,q,β) log[x] + ∞ (−qαx)n+1 gn(α+,β1,)q [x] [2]qβ p,q 2 qα 2 ! qα n=1n(n+1)(n+2) [x]nqα − qα 2 X is true. Corollary 1. Taking q 1 into (1.18), we get nice identity: → G ∞ ( 1)n+1 G (α,β) 2 n+1 G (x)= x+ logx+ − x p,1 2 n(n+1)(n+2) x − (cid:18) (cid:19) n=1 X where G are called famous Genocchi numbers. n Theorem 3. The following nice identity (1.19) G(α,1)(x)= [2]q [x] +qαxξ(α) log[x] + ∞ (−qαx)n+1 ξ(nα,q) [2]q [x] p,q 2 qα 1,q qα n(n+1) [x]n − 2 qα (cid:18) (cid:19) nX=1 e qα is true. e ON THE q-ANALOGUE OF p-ADIC log GAMMA TYPE FUNCTIONS 5 Corollary 2. Putting q 1 into (1.19), we have the following identity: → ∞ ( 1)n+1 E G(α,β)(x)=(x+E )logx+ − n x p,1 1 n(n+1) xn − n=1 X where E are familiar Euler numbers. n References [1] Araci, S., Acikgoz, M., and Seo, J-J., A note on the weighted q-Genocchi numbers and polynomialswithTheirInterpolationFunction, Accepted inHonamMathematical Journal. [2] Araci, S., Erdal, D., and Seo., J-J., A study on the fermionicp adic q- integral representa- tiononZp associated withweighted q-Bernsteinandq-Genocchi polynomials,Abstract and AppliedAnalysis,Volume2011,ArticleID649248,10pages. [3] Araci, S., Seo, J-J., and Erdal, D.,New construction weighted (h,q)-Genocchi numbers and polynomialsrelatedtoZetatypefunction,DiscreteDynamicsinnatureandSociety,Volume 2011,ArticleID487490,7pages. [4] Araci,S.,Acikgoz,M.,Qi,Feng.,andJolany,H.,Anoteonthemodifiedq-Genocchinumbers and polynomials with weight (α,β) and Their Interpolation function at negative integer, submitted. [5] Araci, S., Acikgoz, M., and Ryoo, C-S., A note on the values of the weighted q-Bernstein PolynomialsandModifiedq-GenocchiNumberswithweightαandβviathep-adicq-integral onZp,submitted [6] Araci, S., Acikgoz, M., Park, K-H., 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, submitted. [7] Acikgoz,M.,andSimsek,Y.,OnmultipleinterpolationfunctionsoftheN¨orlundtypeq-Euler polynomials,AbstractandAppliedAnalysis,Volume2009,ArticleID382574, 14pages. [8] Kim, T., A note on the q-analogue of p-adic log gamma function, arXiv:0710.4981v1 [math.NT]. [9] Kim,T.,Onaq-analogueofthep-adicloggammafunctionsandrelatedintegrals,J.Number Theory,76(1999), no.2,320-329. [10] Kim,T.,Ontheq-extensionofEulerandGenocchinumbers,J.Math.Anal.Appl.326(2007) 1458-1465. [11] Kim, T., On the multiple q-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008)481-486. arXiv:0801.0978v1[math.NT] [12] Kim,T., Onthe weighted q-Bernoullinumbers and polynomials, Advanced Studies inCon- temporaryMathematics 21(2011), no.2,p.207-215,http://arxiv.org/abs/1011.5305. [13] Kim,T.,q-Volkenbornintegration,Russ.J.Math.phys.9(2002),288-299. [14] Kim, T., An invariant p-adic q-integrals on Zp, Applied Mathematics Letters, vol. 21, pp. 105-108,2008. [15] Kim, T., q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math.Phys.,14(2007), no.1,15–27. [16] Kim, T., New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no.2,218–225. [17] Kim,T., Someidentities onthe q-Eulerpolynomialsof higher orderand q-Stirlingnumbers bythefermionicp-adicintegralonZp,Russ.J.Math.Phys.,16(2009), no.4,484–491. [18] Kim, T. and Rim, S.-H., 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] Kim, T., On p-adic q-l-functions and sums of powers, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.07.071 [20] Ryoo. C. S., A note on the weighted q-Euler numbers and polynomials, Advan. Stud. Con- temp.Math.21(2011), 47-54. [21] Rim, S-H., and Jeong, J., A note on the modified q-Euler numbers and Polynomials with weightα,International Mathematical Forum,Vol.6,2011, no.65,3245-3250. [22] Simsek, Y., Theorems on twisted L-function and twisted Bernoulli numbers, Advan. Stud. Contemp.Math.,11(2005), 205-218. 6 SERKANARACIANDMEHMETACIKGOZ [23] Simsek,Y.,Twisted(h,q)-Bernoullinumbersandpolynomialsrelatedtotwisted(h,q)−zeta functionandL-function,J.Math.Anal.Appl.,324(2006), 790-804. [24] Simsek, Y., On p-Adic Twisted q-L-Functions Related to Generalized Twisted Bernoulli Numbers,RussianJ.Math.Phys.,13(3)(2006), 340-348. [25] Zill,D.,andCullen,M.R.,AdvancedEngineeringMathematics, JonesandBartlett,2005. UniversityofGaziantep,FacultyofScienceandArts,DepartmentofMathematics, 27310Gaziantep,TURKEY E-mail address: [email protected] UniversityofGaziantep,FacultyofScienceandArts,DepartmentofMathematics, 27310Gaziantep,TURKEY E-mail address: [email protected]