Key Words: Genocchi numbers and polynomals, q-genocchi numbers and polynomials, q-genocchi numbers and polynomials with weight α.

Transcription

1 Bol. Soc. Paran. Mat. 3s. v : c SPM ISSN on line ISSN in ress SPM: doi:0.5269/bsm.v3i.484 A Note On The Generalized q-genocchi measures with weight α Hassan Jolany, Serkan Araci, Mehmet Acikgoz and Jong-Jin Seo Key Words: Genocchi numbers and olynomals, q-genocchi numbers and olynomials, q-genocchi numbers and olynomials with weight α abstract: In this aer we investigate secial generalized q-genocchi measures. We introduce q-genocchi measures with weight α. The resent aer deals with q-extension of Genocchi measure. Some earlier results of T. Kim in terms of q- Genocchi olynomials can be deduced. We aly the method of generating function, which are exloited to derive further classes of q-genocchi olynomials and develo q-genocchi measures. To be more recise, we resent the integral reresentation of -adic q-genocchi measure with weight α which yields a deeer insight into the effectiveness of this tye of generalizations. Generalized q-genocchi numbers with weight α ossess a number of interesting roerties which we state in this aer. Contents Introduction, Definitions and Notations 7 2 -adic q-genocchi measure with weight α 2. Introduction, Definitions and Notations The study of q-genocchi measures and q-genocchi olynomials and their combinatorial relations has received much attention ], 2], 3], 5], 7], 9], ], 2], 6]. Genocchi numbers and secially q-genocchi numbers are the signs of very strong bond between elementary number theory, comlex analytic number theory, Homotoy theory stable Homotoy grous of sheres, differential toologydifferential structures on sheres, theory of modular forms Eisenstein series,-adic analytic number theory-adic L-functions, quantum hysicsquantum Grous. For showing the value of this tye of numbers, we list some of their alications.one of alications of Genocchi numbers that was investigated by JeffRemmel,In 24], is counting the number of u-down ascent sequences. Another alication of Genocchi numbers is in Grah theory. For instance, Boolean numbers of the associated Ferrers Grahs are the Genocchi numbers of the second kind. Also one of alication of q-genocchi numbers is in q-analysis. Actually there is an unexected connection of the literature of -adic analysis with q-analysis and quantum Grous, quantum to, and thus with non commutative Geometry, and q-analysis. For instance, sherical functions on quantum Grous are q-secial functions. The q-genocchi numbers can be defined in number of ways. The way in which it is defined is often determined by which sorts of alications they are intended to be 2000 Mathematics Subject Classification: R33 7 Tyeset by B S P M style. c Soc. Paran. de Mat.

2 8 H. Jolany, S. Araci, M. Acikgoz and J. Seo used for. There exist two imortant definitions of the q-genocchi numbers and olynomials. The generating function definition, which is the most commonly used definition, and q-analog of Seidel s triangle associated to Genocchi numbers. As such it makes it very aealing for use in combinatorial alications. This tye of definition have interesting combinatorial interret ions intheclassicalmodel for q-genocchi numbers. In the last decade, a surrising number of aersaeared roosing new generalizations of the Genocchi olynomials and q-genocchi olynomials to real and comlex variables or treating other toics related to q- Genocchi numbers. In 8], Carlitz defined q-extentions of Bernoulli numbers and olynomials and after several mathematicians develoed his definitionintermsof Genocchi numbers and defined q-genocchi numbers and olynomials. In 25], After Y. Simsek by alying a derivative oerator and the Mellin transformation for q-genocchi numbers defined q-analogue of the Genocchi zeta function, q-analogue Hurwitz tye Genocchi zeta function and q-genocchi tye l-function. By using this functions, he constructed -adic interolation functions which interolate generalized q-genocchi numbers at negative integers. Next, Professor T. Kim found some connections between q-genocchi numbers and q-volkenborn integral. In 2], Kim studied some families of multile Genocchi numbers and olynomials. By using the fermionic -adic invariant integral on,theyconstructed-adic Genocchi numbers and olynomials of higher order. Mehmet Cenkci, et al, in 7], defined q-extentions of -adic measures and obtained general systems of congruences, including Kummer-tye congruences for q-genocchi numbers. T.Kim, et al, in 22] 23], resented new concet of Bernoulli and Euler measure of weight α. Inthis aer we give another construction of q-genocchi numbers and showtheintegral reresentation of -adic q-genocchi measures with weight α. Assume that be a fixed odd rime number. Throughout this aer Z,, Q and C will denote by the ring of integers, the field of -adic rational numbers and the comletion of the algebraic closure of Q, resectively.also we denote N = N {0} and ex x =e x. Let v : C Q { } Q is the field of rational numbers denote the -adic valuation of C normalized so that v =.Theabsolutevalue on C will be denoted as.,and x = vx for x C. When one talks of q-extensions, q is considered in many ways, e.g. as an indeterminate, a comlex number q C, or a -adic number q C, If q C we assume that q <. If q C, we assume q <, so that q x = ex x log q for x. We use the following notation x] q = qx q, x] = x +q. Where lim q x] q = x; cf. -6,8,0-23,25]. For a fixed ositive integer d with d, f =, we set = d =lim N Z/d N Z, = 0<a<d a,= a + d

3 A Note On The Generalized q-genocchi measures with weight α 9 And a + d N = { x x a mod d N}, Where a Z satisfies the condition 0 a<d N. It is known that µ q x + N = q x N ] q Is a distribution on for q C with q. Let UD be the set of uniformly differentiable function on. We say that f is a uniformly differentiable function at a oint a, if the difference quotient F f x, y = f x f y x y has a limit f a as x, y a, a and denote this by f UD. The -adic q-integral of the function f UD is defined by I q f = f x dµ q x = lim Z N ] q f x q x.2 The bosonic integral is considered as the bosonic limit q,i f=lim q I q f. Similarly we have -adic fermionic integration defined by T.Kim 7], on as follows I f = lim I q f = f x dµ x q Let q, then we have -adic fermionic integral on as follows I f = lim I q f = lim f x x, q So by alying f x =e tx, we get t e xt dµ x = 2t e t + = n=0 G n t n n!.3 Where G n are Genocchi numbers. By using.3, we have so from above, we obtain n=0 e tx dµ x = x n dµ x n=0 G n+ t n n +n! t n n! = Gn+ t n n + n! n=0

4 20 H. Jolany, S. Araci, M. Acikgoz and J. Seo So by comuting the coefficients of tn n! on both sides of the above equation we get G n+ n + = x n dµ x In 3], q-extension of Genocchi numbers are defined by { G 0,q =0,qqG q + n 2]q,n=0 + G n,q = 0, n 0.4 with the usual convention about relacing G q n by G n,q. The h, q-extension of Genocchi numbers are defined by { n G h 0,q =0,q qg h q + + G h 2]q,n=0 n,q = 0, n 0.5 n with the usual convention about relacing G h h q by G n,q see 6]. Recently, for n N, Araci et al. are defined weighted q-genocchi numbers by G n α 0,q =0,q α α q G q + + Gα n,q = { 2]q, if n =0 0, if n 0,.6 with the usual convention about relacing n Gα q by Gα n,q. From. and.6, the Witt s formula for the q-genocchi numbers and olynomials with weight α are defined by G α n,q n = y] n q dµ α y, where n N..7 The q-genocchi olynomials with weight α are also defined by G α n,q x =q αx n k=0 n kx x] n k q k Gα α.8 By.,.7and.8, we can derive the Witt s formula for G α n,q x as follows: G α n,q x = x + y] n q n dµ α y, where n N..9 For n N and d N, the distribution relation for the q-genocchi olynomials with weight α are known that n,q x = d]n d a x + a q a d] Gα, see 2]5].0 n,q d d G α

5 A Note On The Generalized q-genocchi measures with weight α 2 Obviously, a secial case of.0 whenα =is lim q G n,q x =G n x, where G n x are called Genocchi olynomials. Let U be any comact oen set of, by the same method of 22], it can be written as a finite disjoint union of sets U = k U j= aj + N, where N N and a,a 2,..., a k N with 0 a i < N. Lemma.. Every ma µ from the collection of comact-oen sets in to Q for which µ a + N = a + b N + d N+ b=0 holds whenever a+ N, extends to a -adic measure on. for roof see 22] In this aer, we derive our results by using Kim et al. method in 22]. The urose of this aer is to establish -adic q-genocchi measure with weight α on and to derive their integral reresantations. Finally, we obtain generalized q- Genocchi numbers with weight α and some interesting roerties of the generalized q-genocchi numbers with weight α. 2. -adic q-genocchi measure with weight α In this section, by using -adic q-integral on and the following Eq. 2., we derive some interesting roerties concerning the q-genocchi numbers ans olynomials with weight α. Now we define a ma µ α on the balls in as follows: a + n = n ] k n a q a f α {a}n k, ] n n µ α where {a} n = a mod n. 2. Theorem 2.. Let α, k N. Then we see that µ α is -adic measure on if and only if ] k a q n α bn q bn f α + b n ] n k,q n b=0 = f α a n n. Proof: For each n N and 0 a< n.fromexression2.,then b=0 µ α a + b n + n+ ] n+ k q = α n+ a+b n α a + b n f ] n+ b=0 n+ = a q a ]k q n α n ] k a ] n n bn q bn f α n ] k,q n b=0 + b 2.2

6 22 H. Jolany, S. Araci, M. Acikgoz and J. Seo By 2., we note that µ α is -adic measure on if and only if f α a n n = ]k a q n α bn q bn f α n ] n k,q n Thus, we comlete the roof. b=0 + b. Now, we set as follows: f α α n x = G n x. 2.3 From 2. and 2.3, wesimlyseethat µ α a + n = n ] k n a a q a ] Gα n n By using.2,.0 and2.4, we get the following theorem: Theorem 2.2. For α, k N,we have dµ α α x = G. 2.4 Proof: For each k N, Thus d dµ α x = lim µ α x + d N = lim = G α We arrive at the desired result. d N ] k d N ] d a a q a Gα dn d N Let χ be Dirichlet s character with conductor d N. Then we define the generalized q-genocchi numbers attached to χ as follows: G α n,χ,q = n χ xx] n q dµ α x = lim = d]n d] = d]n d] n d d N ] d a q a χ a d a q a χ a x q x χ xx] n n lim N ] d d x a ] n d + x d α a G. 2.5 n,q d d

7 A Note On The Generalized q-genocchi measures with weight α 23 From 2.4 and2., we can derive the following theorem. Theorem 2.3. For each k N, we get χ x dµ α α x = G k,χ,q. Proof: For k N and by using 2.4 and2., then = d]k d] d χ x dµ α x = lim χ x µ α x + d N d = lim a q a χ a lim d N ] k d N ] N ] k d N ] d d χ x x x q x Gα dn d N a dx dx d q + x Gα dn N = d]k d] we obtain the desired result. d a q a χ a α a G d d Theorem 2.4. For each k N, we get χ x dµ α ]k q x =χ α ] Proof: From 2.4 and2., Then = ]k ] χ x dµ α x = lim d] k d d] χ a a = ]k ] = χ ]k ] ] d N+ k d N+ ] lim d] k d d] d ] N k q dα N ] d Gα k,χ,q. Gα k,χ,q. χ x x x q Gα dx Gα dn dn+ a a a χ χ a q Gα d d x d N+ d x + a d d N Thus, we get the desired result. So, we get the following theorems with the same method of Theorem 2.2, Theorem 2.3 and Theorem 2.4

8 24 H. Jolany, S. Araci, M. Acikgoz and J. Seo Theorem 2.5. For, we get χ x dµ α x =χ k ] ] Gα k,χ,q Theorem 2.6. For, we get χ x dµ α x =χ G α k,χ,q We can define the following equation, µ α k,,q U =µα U ] k ] µ U 2.6 Theorem 2.7. For, we get χ x dµ α k,,q x = χ χ Gα k,χ,q. Proof: By the definition of µ α 2.6, we obtain = G α k,χ,q +χ k,,q ]k q χ α ] ] k ], from Theorem 2.4, Theorem 2.5 and Theorem Gα k,χ,q G α k,χ,q = χ χ Gα k,χ,q ] k ] χ G α k,χ,q where the oerator χ y = χ y,k,α;q on f q defined by χ y f q =χ y,k,α;q f q = y]k y] χ y f q y 2.7

9 A Note On The Generalized q-genocchi measures with weight α 25 From exression 2.7, we get χ x,k,α;q χ y,k,α;q y]k x,k,α;q q f q = χ α χ y f q y y] = y]k y] χ y χ x y] k y y] y = xy]k χ xy f q xy xy] = χ xy,k,α;q f q = χ xy f q. Assume that define χ x.χ y = χ x,k,α;q.χ y,k,α;q. Then we have χ x.χ y = χ xy. χ y f q xy By the definition χ x, we can simly derive the following equation: χ χ = χ χ + χ Assume that f q = α G k,χ,q. Then we get χ χ Gα k,χ,q Thus, we arrive at the desired result. References. Araci, S., Seo, J-J., and Erdal, D., New construction weighted h, q-genocchi numbers and olynomials related to zeta tye function, Discrete Dynamics in Nature and Society, Volume 20, Article ID , 7 ages, doi:0.55/20/ Araci, S., Erdal, D., and Seo, J-J., A study on the fermionic -adic q-integral reresentation on Associated with weighted q-bernstein and q-genocchi olynomials, Abstract and Alied Analysisin ress. 3. Araci, S., Erdal, D., and Kang, D-J., Some New Proerties on the q-genocchi numbers and Polynomials associated with q-bernstein olynomials, Honam Mathematical J no. 2, Araci, S., Aslan, N., and Seo, J-J., A Note on the weighted Twisted Dirichlet s tye q-euler numbers and olynomials, Honam Mathematical Journalin ress. 5. Araci, S., Acikgoz, M., and Seo, J.J., A study on the weighted q-genocchi numbers and olynomials with their Interolation Function,Submitted 6. Araci, S., and Acikgoz, M., Some identities concerning the h, q-genocchi numbers and olynomials via the -adic q-integral on and q-bernstein olynomials,submitted. 7. Cenkci, M., Strong versions of Kummer-tye congruences for Genocchi numbers and olynomials and tangent coefficients, Acta Mathematica Universitatis Ostraviensis, Vol , No., 3-.

10 26 H. Jolany, S. Araci, M. Acikgoz and J. Seo 8. Carlitz, L., Exansions of q-bernoulli numbers, Duke Mathematical Journal, vol. 25, , Jolany, H., and Sahrifi, H., Some Results for the Aostol-Genocchi olynomials of Higher Order, Acceted in Bulletin of the Malaysian Mathematical Sciences Society, ariv: 04.50v math. NT]. 0. Kim, T., A New Aroach to q-zeta Function, Adv. Stud. Contem. Math Kim, T., On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Al Kim, T., On the multile q-genocchi and Euler numbers, Russian J. Math. Phys ariv: v math.nt] 3. Kim, T., q-volkenborn integration, Russ. J. Math. hys , Kim, T., q-bernoulli numbers and olynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys , Kim, T., An invariant -adic q-integrals on, Alied Mathematics Letters, vol. 2, , Kim, T., A Note on the q-genocchi Numbers and Polynomials, Journal of Inequalities and Alications, Article ID 7452, 8 ages, doi:0.55/2007/ Kim, T., q-euler numbers and olynomials associated with -adic q-integrals, J. Nonlinear Math. Phys., , no., Kim, T., New aroach to q-euler olynomials of higher order, Russ. J. Math. Phys., 7 200, no. 2, Kim, T., Some identities on the q-euler olynomials of higher order and q-stirling numbers by the fermionic -adic integral on, Russ. J. Math. Phys., , no.4, Kim, T., q-bernoulli numbers and olynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys., , no., Kim, T., On the weighted q-bernoulli numbers and olynomials, Advanced Studies in Contemorary Mathematics 220, no.2, , htt://arxiv.org/abs/ Kim, T., Lee, S. H., Dolgy, D. V., Ryoo, C. S., A note on the generalized q-bernoulli measures with weight α, Abstract and Alied Analysis, Article ID 86727, 9 ages, doi: 0.55/20/86727, ariv: v math.nt]. 23. Kim, T., Choi, J., Kim, Y. H., Ryoo, C. S., A note on the weighted -adic q-euler measure on,advn.stud.contem.math.220, Remmel, J., Ascent sequences, 2+2-free osets, Uer triangular Matrices, and Genocchi numbers, Worksho on Combinatorics, Enumeration, and Invariant Theory, George Mason University, Virginia, Y. Simsek, I. N. Cangul, V. Kurt, and D. Kim, q-genocchi numbers and olynomials associated with q-genocchi-tye l-functions, Adv. Difference Equ., doi: /85750

11 A Note On The Generalized q-genocchi measures with weight α 27 Hassan Jolany School of Mathematics, Statistics and Comuter Science University of Tehran,Iran address: and Serkan Araci University of Gaziante, Faculty of Science and Arts Deartment of Mathematics 2730 Gaziante, Turkey address: and Mehmet Acikgoz University of Gaziante, Faculty of Science and Arts Deartment of Mathematics 2730 Gaziante, Turkey address: and Jong-Jin Seo Deartment of Alied Mathematics Pukyong National University Busan , Reublic of Korea address: