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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 513821, 8 pages doi:10.1155/2011/513821 ResearchArticleAStudyonthep-Adicq-IntegralRepresentationonpAssociatedwiththeWeightedq-Bernsteinandq-Bernoulli Polynomials T. Kim, 1 A. Bayad, 2 and Y H. Kim 1 1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea 2 D ´ epartement de Ma th ´ ematiques, Universit ´ e d’Evry Val d’Essonne, Boulevard Franc¸ois Mitterrand, 91025 Evry Cedex, France Correspondence should be addressed to A. Bayad, abayad@maths.univ-evry.fr Received 6 December 2010; Accepted 15 January 2011 Academic Editor: Vijay Gupta Copyright q 2011 T. Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate some interesting properties of the w eighted q-Bernstein polynomials related to theweightedq-Bernoulli numbers and polynomials by using p-adicq-integralonp . 1. Introduction and Preliminaries Let p be a fixed prime number. Throughout this paper, p , p ,and p will denote the ring of p-adic integers, the field of p-adic rational numbers, andthe completion of the algebraic closure of p , respectively. Let be the set of natural numbers, and let ∪{0}.Letν p be the normalized exponential valuation of pwith |p| p p −ν p p 1/p.Letq be regarded as either a complex number q ∈ or ap-adic number q ∈ p .Ifq ∈ ,thenwealways assume |q| < 1. If q ∈ p , we assume that |1 − q| p < 1. In this paper, we define the q-number as x q 1 − q x /1 − qsee 1–13. Let C0, 1 be the set of continuous functions on 0, 1.Forα ∈ and n, k ∈ ,the weightedq-Bernstein operator of order n for f ∈ C0, 1 is defined by α n,q f | x n k0 f k n n k x k q α 1 − x n−k q −α n k0 f k n B α k,n x, q . 1.1 Here B α k,n x, q is called theweightedq-Bernstein polynomials of degree n see 2, 5, 6. 2 Journal of Inequalities and Applications Let UD p be the space of uniformly differentiable functions onp .Forf ∈ UD p , thep-adicq-integralonp , which is called the bosonic q-integralonp ,isdefinedby I q f p f x dμ q x lim N →∞ 1 p N q p N −1 x0 f x q x , 1.2 see 10. The Carlitz’s q-Bernoulli numbers are defined by β 0,q 1,q qβ 1 k − β k,q 1, if k 1, 0, if k>1, 1.3 withthe usual convention about replacing β k by β k,q see 3, 9, 10.In3, Carlitz also defined the expan sion of Carlitz’s q-Bernoulli numbers as follows: β h 0,q h h q ,q h qβ h 1 n − β h n,q 1, if n 1, 0, if n>1, 1.4 withthe usual convention about replacing β h n by β h n,q . Theweightedq-Bernoulli numbers are constructed in previous paper 6 as follows: for α ∈ , β α 0,q 1,q q α β α 1 n − β α n,q ⎧ ⎨ ⎩ α α q , if n 1, 0, if n>1, 1.5 withthe usual convention about replacing β α n by β α n,q .Letf n xfx n.Bythe definition 1.2 of p-adicq-integralonp , we easily get qI q f 1 q lim N →∞ 1 p N q p N −1 x0 f x 1 q x , lim N →∞ 1 p N q p N −1 x0 f x q x lim N →∞ f p N q p N − f 0 p N q p f x dμ q x q − 1 f 0 q − 1 log q f 0 , 1.6 Continuing this process, we obtain easily the relation q n p f n x dμ q x − p f x dμ q x q − 1 n−1 l0 q l f l q − 1 log q n−1 l0 q l f l , 1.7 where n ∈ and f ldf l/dx see 6. Journal of Inequalities and Applications 3 Then by 1.2, applying to the function x → x n q α ,wecanseethat β α n,q p x n q α dμ q x − nα α q ∞ m0 q mαm m n−1 q α 1 − q ∞ m0 q m m n q α . 1.8 Theweightedq-Bernoulli polynomials are also defined by the generating function as follows: F α q t, x −t α α q ∞ m0 q mαm e mx q α t 1 − q ∞ m0 q m e mx q α t ∞ n0 β α n,q x t n n! , 1.9 see6. Thus, we note that β α n,q x n l0 n l x n−l q α q αlx β α l,q − nα α q ∞ m0 q mαm m x n−1 q α 1 − q ∞ m0 q m m x n q α . 1.10 From 1.2 andthe previous equalities, we obtain the Witt’s formula for theweightedq-Bernoulli polynomials as follows: β α n,q x p x y n q α dμ q y n l0 n l q αlx x n−l q α p y l q α dμ q y . 1.11 By using 1.2 andtheweightedq-Bernoulli polynomials, we easily get q n β α m,q n − β α m,q q − 1 n−1 l0 q l l m q α mα α q n−1 l0 q αll l m−1 q α , 1.12 where n, α ∈ and m ∈ see 6. In this paper, we consider theweightedq-Bernstein polynomials to express the bosonic q-integralonpand investigate some properties of theweightedq-Bernstein polynomials associatedwiththeweightedq-Bernoulli polynomials by using the expression of p-adic q- integral onp of those polynomials. 2. Weightedq-Bernstein Polynomials andq-Bernoulli Polynomials In this section, we assume that α ∈ and q ∈ pwith |1 − q| p < 1. Now we consider thep-adicweightedq-Bernstein operator as follows: α n,q f | x fx n k0 f k n n k x k q α 1 − x n−k q −α n k0 f k n B α k,n x, q . 2.1 4 Journal of Inequalities and Applications Thep-adicq-Bernstein polynomials with weight α of degree n are given by B α k,n x, q n k x k q α 1 − x n−k q −α , 2.2 where x ∈ p , α ∈ ,andn, k ∈ see 6, 7.NotethatB α k,n x, qB α n−k,n 1 − x, 1/q.That is, theweightedq-Bernstein polynomials are symmetric. From the definition of theweightedq-Bernoulli polynomials, we have β α n,q −1 1 − x −1 n q αn β α n,q x . 2.3 By the definition of p-adicq-integralonp ,weget p 1 − x n q −α dμ q x q αn −1 n p −1 x n q α dμ q x p 1 − x q α n dμ q x . 2.4 From 2.3 and 2.4,wehave p 1 − x n q −α dμ q x n l0 n l −1 l β α l,q q αn −1 n β α n,q −1 β α n,q 2 . 2.5 Therefore, we obtain the following lemma. Lemma 2.1. For n ∈ ,onehas p 1 − x n q −α dμ q x n l0 n l −1 l β α l,q q αn −1 n β α n,q −1 β α n,q 2 , β α n,q −1 1 − x −1 n q αn β α n,q x . 2.6 By 2.2, 2.3,and2.4,weget q 2 β α n,q 2 n α α q q 1α q 2 − q β α n,q , if n>1. 2.7 Thus, we have β α n,q 2 1 q 2 β α n,q nα α q q α−1 1 − 1 q , if n>1. 2.8 Therefore, by 2.8, we obtain the following proposition. Journal of Inequalities and Applications 5 Proposition 2.2. For n ∈ with n>1,onehas β α n,q 2 1 q 2 β α n,q nα α q q α−1 1 − 1 q . 2.9 By using Proposition 2.2 and Lemma 2.1, we obtain the following corollary. Corollary 2.3. For n ∈ with n>1,onehas p 1 − x n q −α dμ q x q 2 β α n,q −1 nα α q 1 − q, 2.10 p 1 − x n q −α dμ q x nα α q 1 − q q 2 p x n q −α dμ q −1 x p 1 − x q α n dμ q x . 2.11 Taking the bosonic q-integralonp for one weightedq-Bernstein polynomials in 2.1, we have p B α k,n x, q dμ q x n k p x k q α 1 − x n−k q −α dμ q x n k n−k l0 n − k l −1 l p x kl q α dμ q x n k n−k l0 n − k l −1 l β α kl,q . 2.12 By the symmetry of q-Bernstein polynomials, we get p B α k,n x, q dμ q x p B α n−k,n 1 − x, 1 q dμ q x n k k l0 k l −1 kl p 1 − x n−l q −α dμ q x . 2.13 6 Journal of Inequalities and Applications For n>k 1, by 2.11 and 2.13,wehave p B α k,n x, q dμ q x n k k l0 k l −1 kl nα α q 1 − q q 2 p x n−l q −α dμ q −1 x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ nα α q 1 − q q 2 β α n,q −1 , if k 0, ⎛ ⎝ n k ⎞ ⎠ q 2 k l0 ⎛ ⎝ k l ⎞ ⎠ −1 kl β α n−l,q −1 , if k>0. 2.14 By comparing the coefficients onthe both sides of 2.12 and 2.14,weobtainthe following theorem. Theorem 2.4. For n, k ∈ with n>k 1,onehas n−k l0 n − k l −1 l β α kl,q q 2 k l0 k l −1 kl β α n−l,q −1 , if k / 0. 2.15 In particular, when k 0,onehas nα α q 1 − q q 2 β α n,q −1 n l0 n l −1 l β α l,q . 2.16 Let m, n, k ∈ with m n>2k 1. Then we see that p B α k,n x, q B α k,m x, q dμ q x n k m k p x 2k q α 1 − x nm−2k q −α dμ q x n k m k 2k l0 2k l −1 l2k p 1 − x nm−l q −α dμ q x . n k m k 2k l0 2k l −1 l2k nα α q 1 − q q 2 p x nm−l q −α dμ q −1 x n k m k 2k l0 2k l −1 l2k nα α q 1 − q q 2 β α nm−l,q −1 . 2.17 Therefore, by 2.17, we obtain the following theorem. Journal of Inequalities and Applications 7 Theorem 2.5. For m, n, k ∈ with m n>2k 1,onehas p B α k,n x, q B α k,m x, q dμ q x ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ nα α q 1 − q q 2 β α nm,q −1 , if k 0, ⎛ ⎝ n k ⎞ ⎠ ⎛ ⎝ m k ⎞ ⎠ q 2 2k l0 ⎛ ⎝ 2k l ⎞ ⎠ −1 l2k β α nm−l,q −1 , if k / 0. 2.18 For m, n, k ∈ ,wehave p B α k,n x, q B α k,m x, q dμ q x n k m k p x 2k q α 1 − x nm−2k q −α dμ q x n k m k nm−2k l0 n m − 2k l −1 l p x 2kl q α dμ q x n k m k nm−2k l0 n m − 2k l −1 l β α l2k,q . 2.19 Therefore, by 2.18 and 2.19, we obtain the following theorem. Theorem 2.6. For m, n, k ∈ with m n>2k 1,onehas nα α q 1 − q q 2 β α nm−l,q −1 nm l0 n m l −1 l β α l,q . 2.20 Furthermore, for k / 0,onehas nm−2k l0 n m − 2k l −1 l β α l2k,q q 2 2k l0 2k l −1 l2k β α nm−l,q −1 . 2.21 By the induction hypothesis, we obtain the fo llowing theorem. Theorem 2.7. For s ∈ and k, n 1 , ,n s ∈ with n 1 n 2 ··· n s >sk 1,onehas p s i1 B α k,n i x, q dμ q x ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ nα α q 1 − q q 2 β α n 1 ···n s ,q −1 , if k 0, ⎛ ⎝ s i1 ⎛ ⎝ n i k ⎞ ⎠ ⎞ ⎠ sk l0 ⎛ ⎝ sk l ⎞ ⎠ −1 lsk β α n 1 ···n s −l,q −1 , if k / 0. 2.22 8 Journal of Inequalities and Applications For s ∈ ,letk, n 1 , ,n s ∈ with n 1 n 2 ··· n s >sk 1. Then we show that p s i1 B α k,n i x, q dμ q x s i1 n i k n 1 ···n s −sk l0 n 1 ··· n s − sk l −1 l β α lsk,q . 2.23 Therefore, by Theorem 2.7 and 2.23, we obtain the following theorem. Theorem 2.8. For s ∈ ,letk, n 1 , ,n s ∈ with n 1 n 2 ··· n s >sk 1. Then one sees that for k 0 n 1 ···n s l0 n 1 ··· n s l −1 l β α l,q nα α q 1 − q q 2 β α n 1 ···n s ,q −1 . 2.24 For k / 0,onehas sk l0 sk l −1 lsk β α n 1 ···n s −l,q −1 n 1 ···n s −sk l0 n 1 ··· n s − sk l −1 l β α lsk,q . 2.25 References 1 M. Acikgoz and Y. Simsek, “On multiple interpolation functions of the N ¨ orlund-type q-Euler polynomials,” Abstract and Applied Analysis, vol. 2009, Arti cle ID 382574, 14 pages, 2009. 2 A.Bayad,J.Choi,T.Kim,Y H.Kim,andL.C.Jang,“q-extension of Bernstein polynomials withweighted α;β,” Journal of Computational and Applied Mathematics. In press. 3 L. Ca rlitz, “Expansions of q-Bernoulli numbers,” Duke Mathematical Journal, vol. 25, pp. 355–364, 1958. 4 A. S. Hegazi and M. Mansour, “A note onq-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 13, no. 1, pp. 9–18, 2006. 5 L C. Jang, W J. Kim, and Y. Simsek, “A studyonthep-adic integral representationonpassociatedwith Bernstein and Bernoulli polynomials,” Advances in Difference Equations, vol. 2010, Article ID 163217, 6 pages, 2010. 6 T. Kim, “On theweightedq-Bernoulli numbers a nd polynomials,” http://arxiv.or g /abs/1011.5305. 7 T. Kim, “A note onq-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 1, 2011. 8 T. Kim, “Barnes-type multiple q-zeta fun ctions and q-Euler polynomials,” Journal of Physics A,vol.43, no. 25, Article ID 255201, 11 pages, 2010. 9 T. Kim, “q-Bernoulli numbers and polynomials associatedwith Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008. 10 T. Kim, “On a q-analogue of thep-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999. 11 B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of Nonlinear Mathematical Physics, vol. 12, supplement 1, pp. 412–422, 2005. 12 H. Ozden, I. N. Ca ngul, and Y. Simsek, “Remarks onq-Bernoulli numbers associatedwith Daehee numbers,” Advanced Studies in Contemp orary Mathematics, vol. 18, no. 1, pp. 41–48, 2009. 13 S H. Rim, J H. Jin, E J. Moon, and S J. Lee, “On multiple interpolation functions of the q-Genocchi polynomials,” Journal of Inequalities and Applications, vol. 2010, Article ID 351419, 13 pages, 2010. . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 513821, 8 pages doi:10.1155/2011/513821 Research Article A Study on the p- Adic q-Integral Representation. Let be the set of natural numbers, and let ∪{0}.Letν p be the normalized exponential valuation of p with |p| p p −ν p p 1 /p. Letq be regarded as either a complex number q ∈ or a p- adic. fixed prime number. Throughout this paper, p , p ,and p will denote the ring of p- adic integers, the field of p- adic rational numbers, and the completion of the algebraic closure of p , respectively.