On P-Adic Analogue Of Q-Bernstein Polynomials And Related .

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2010, Article ID 179430, 9 pagesdoi:10.1155/2010/179430Research ArticleOn p-Adic Analogue of q-Bernstein Polynomialsand Related IntegralsT. Kim,1 J. Choi,1 Y. H. Kim,1 and L. C. Jang212Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of KoreaDepartment of Mathematics and Computer Science, KonKuk University,Chungju 380-701, Republic of KoreaCorrespondence should be addressed to T. Kim, tkkim@kw.ac.krReceived 17 September 2010; Accepted 22 December 2010Academic Editor: Binggen ZhangCopyright q 2010 T. Kim et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.Recently, Kim’s work in press introduced q-Bernstein polynomials which are different Phillips’q-Bernstein polynomials introduced in the work by Phillips, 1996; 1997 . The purpose of thispaper is to study some properties of several type Kim’s q-Bernstein polynomials to express thep-adic q-integral of these polynomials on Zp associated with Carlitz’s q-Bernoulli numbers andpolynomials. Finally, we also derive some relations on the p-adic q-integral of the products ofseveral type Kim’s q-Bernstein polynomials and the powers of them on Zp .1. IntroductionLet C 0, 1 denote the set of continuous functions on 0, 1 . For 0 q 1 and f C 0, 1 , Kimintroduced the q-extension of Bernstein linear operator of order n for f as follows:Bn,q n nn kkkn kBk,n x, q ,fff x x q 1 x 1/q nnkk 0k 0 1.1 where x q 1 qx / 1 q see 1 . Here Bn,q f x is called Kim’s q-Bernstein operatorof order n for f. For k, n Z N {0} , Bk,n x, q nk x kq 1 x n k1/q are called the Kim’sq-Bernstein polynomials of degree n see 2–6 .

2Discrete Dynamics in Nature and SocietyIn 7 , Carlitz defined a set of numbers ξk ξk q inductively byξ0 1, 1 kqξ 1 ξk 0if k 1,if k 1, 1.2 with the usual convention of replacing ξk by ξk . These numbers are q-analogues of ordinaryBernoulli numbers Bk , but they do not remain finite for q 1. So he modified the definitionas follows:β0,q 1, 1 kq qβ 1 βk,q 0if k 1,if k 1, 1.3 with the usual convention of replacing βk by βk,q see 7 . These numbers βn,q are called thenth Carlitz q-Bernoulli numbers. And Carlitz’s q-Bernoulli polynomials are defined byβk,q x q β x qxk k kβi,q qix x k i q .ii 0 1.4 As q 1, we have βk,q Bk and βk,q x Bk x , where Bk and Bk x are the ordinaryBernoulli numbers and polynomials, respectively.Let p be a fixed prime number. Throughout this paper, Z, Q, Zp , Qp , and Cp will denotethe ring of rational integers, the field of rational numbers, the ring of p-adic integers, the fieldof p-adic rational numbers and the completion of algebraic closure of Qp , respectively. Let νpbe the normalized exponential valuation of Cp such that p p p νp p 1/p.Let q be regarded as either a complex number q C or a p-adic number q Cp . Ifq C, we assume q 1, and if q Cp , we normally assume 1 q p 1.We say that f is a uniformly differentiable function at a point a Zp and denote thisproperty by f UD Zp if the difference quotient Ff x, y f x f y / x y has a limitf a as x, y a, a see 1, 3, 8–13 .For f UD Zp , let us begin with the expression 1qx f x f x μq x pN Zp , N p q 0 x pNN0 x p 1.5 representing a q-analogue of the Riemann sums for f see 11 . The integral of f on Zp isdefined as the limit as n of the sums if exists . The p-adic q-integral on a functionf UD Zp is defined by Iq f see 11 . Zp1f x dμq x lim N N pqN 1p f x qx ,x 0 1.6

Discrete Dynamics in Nature and Society3As was shown in 3 , Carlitz’s q-Bernoulli numbers can be represented by p-adicq-integral on Zp as follows: Zp x mq dμq x βm,q ,for m Z . 1.7 Also, Carlitz’s q-Bernoulli polynomials βk,q x can be represented βm,q x Zp m x y q dμq y ,for m Z , 1.8 see 3 .In this paper, we consider the p-adic analogue of Kim’s q-Bernstein polynomials on Zpand give some properties of the several type Kim’s q-Bernstein polynomials to represent thep-adic q-integral on Zp of these polynomials. Finally, we derive some relations on the p-adicq-integral of the products of several type Kim’s q-Bernstein polynomials and the powers ofthem on Zp .2. q-Bernstein Polynomials Associated with p-Adic q-Integral on ZpIn this section, we assume that q Cp with 1 q p 1.From 1.5 , 1.7 and 1.8 , we note that n nqnl 1, 1 x 1 l qlx l 1 n 1q 1lZpq 1l 0 n nl 11n. x x1 q dμq x1 1 l qlx n 11 ql 1lZp1 ql 0x1 n1/q dμ1/q x1 2.1 By 2.1 , we get 1 n qn Zp x x1 nq dμq x1 Zp 1 x x1 n1/q dμ1/q x1 . 2.2 Therefore, we obtain the following theorem.Theorem 2.1. For n Z , one has Zp 1 x x1 n1/q dμ1/q x1 1 n qnZp x x1 nq dμq x1 . 2.3

4Discrete Dynamics in Nature and SocietyBy the definition of Carlitz’s q-Bernoulli numbers and polynomials, we get nq2 βn,q 2 n 1 q2 q q qβ 1 βn,qif n 1. 2.4 Thus, we have the following proposition.Proposition 2.2. For n N with n 1, one hasβn,q 2 11βn,q n 1 .qq2 2.5 It is easy to show that 1 x n1/q 1 x qn 1 n qn x 1 nq . 2.6 Hence, we have Zp 1 x n1/q dμq x 1 n qnZp x 1 nq dμq x . 2.7 By 1.8 , we get Zp 1 x n1/q dμq x 1 n qn βn,q 1 . 2.8 By Theorem 2.1 and 2.8 , we see that Zp 1 x n1/q dμq x 1 n qn βn,q 1 βn,1/q 2 . 2.9 From 2.9 and Proposition 2.2, we have Zp 1 x n1/q dμq x βn,1/q 2 q2 βn,1/q n 1 q. 2.10 By 1.7 and 2.10 , we obtain the following theorem.Theorem 2.3. For n N with n 1, one has Zp 1 x n1/q dμq x q2Zp x n1/q dμ1/q x n 1 q. 2.11

Discrete Dynamics in Nature and Society5Taking the p-adic q-integral on Zp for one Kim’s q-Bernstein polynomials, we get Zp Bk,n x, q dμq x nkZp x kq 1 x n k1/q dμq x n k n kn kl 0ll 0l 1 lZp x k lq dμq x 2.12 n k n kn k 1 l βk l,q ,and, by the q-symmetric property of Bk,n x, q , we see that Zp Bk,n x, q dμq x 1dμq x Bn k,n 1 x,qZp kkn kl 0 1 l 2.13 k lZp 1 x n l1/q dμq x .For n k 1, by Theorem 2.3 and 2.13 , one has ZpBk,n kkn k ln l2n l 1 q qx, q dμq x 1 x 1/q dμ1/q x lk l 0Zp k n k k l n l 1 q q2 βn l,1/q . 1 k l 0l 2.14 Let m, n, k Z with m n 2k 1. Then the p-adic q-integral for the multiplicationof two Kim’s q-Bernstein polynomials on Zp can be given by the following relation: Zp Bk,n x, q Bk,m x, q dμq x nmkkZpn m 2kdμq x x 2kq 1 x 1/q 2k2knm 1 x n m l 1 l 2k q1/q dμq x .kk l 0 lZp 2.15

6Discrete Dynamics in Nature and SocietyBy Theorem 2.3 and 2.15 , we get Zp Bk,n x, q Bk,m x, q dμq x 2k2knm kkll 0 2knm 2k kkll 0 1 l 2k n m l 1 q q x n m l1/q dμ1/q x 2Zp 2.16 1 l 2k n m l 1 q q2 βn m l,1/q .By the simple calculation, we easily get Zp Bk,n x, q Bk,m x, q dμq x nmn m 2kdμq x x 2kq 1 x 1/qkkZp nm n m 2kn m 2kkkl 0ll 0l 1 lZpdμq x x l 2kq nm n m 2kn m 2kkk 1 l βl 2k,q . 2.17 Continuing this process, we obtain sZp Bk,ni x, q dμq x i 1 s nii 1 kZpn1 ··· ns skdμq x x skq 1 x 1/q s sk niski 1kl 0l 1 sk lZp1 ··· ns ldμq x . 1 x n1/q 2.18 Let s N and n1 , . . . , ns , k Z with n1 n2 · · · ns sk 1. By Theorem 2.3 and 2.18 , we get s Bk,ni x, q dμq x Zpi 1 s sk niski 1 kl 0l ssk niski 1kl 0l 1 sk l s 2ni l 1 q qi 1 1 sk lZp 1 ··· ns ldμ1/q x x n1/q s 2ni l 1 q q βn1 ··· ns l,1/q .i 1 2.19

Discrete Dynamics in Nature and Society7From the definition of binomial coefficient, we note that s Bk,ni x, q dμq x Zpi 1 s niki 1Zpn1 ··· ns skdμq x x skq 1 x 1/q n ··· n sk s1 s nin1 · · · ns skldμq x 1 x sk lqlkZpi 1l 0 n ··· n sk s1 s nin1 · · · ns sk 1 l βsk l,q ,lki 1l 0 2.20 where s N and n1 , . . . , ns , k Z .By 2.19 and 2.20 , we obtain the following theorem.Theorem 2.4. I For s N and n1 , . . . , ns , k N with n1 n2 · · · ns sk 1, one has s Bk,ni x, q dμq x Zpi 1 ssk niskki 1l 0 s sk l2ni l 1 q q βn1 ··· ns l,1/q . 1 l 2.21 i 1 II For s N and n1 , . . . , ns , k Z , one has sZp Bk,ni x, q dμq x i 1 n ··· n sk s1 s nin1 · · · ns ski 1kl 0 1 l βsk l,q .l 2.22 By Theorem 2.4, we obtain the following corollary.Corollary 2.5. For s N and n1 , . . . , ns , k N with n1 n2 · · · ns sk 1, one has sk skl 0l 1 sk ls ni l 1 q q2 βn1 ··· ns l,1/qi 1 n1 ··· n s sk n1 · · · ns sk 1 l βsk l,q .ll 0 2.23

8Discrete Dynamics in Nature and SocietyLet s N and m1 , . . . , ms , n1 , . . . , ns , k Z with m1 n1 · · · ms ns m1 · · · ms k 1.Then one has s mi k s mii 1 s k mi ni mi Bk,nx, q dμq x i 1 1 k i 1 mi liki 1i 1l 0l sn m l 1 x q i 1 i i dμq x sZp s Zp s mi k s misi 1 s mknii i 1 1 k i 1 mi lki 1l 0l s s 2i 1 ni mi l mi ni l 1 q qdμ1/q x x 1/qi 1Zp s s mi k s mii 1 s nimki i 1 1 k i 1 mi lki 1l 0l s 2 mi ni l 1 q q βn1 m1 ··· ns ms l,1/q .i 1 2.24 From the definition of binomial coefficient, one has sZpi 1 mi Bk,nx, qidμq x s s mi s ni mi k s mi si 1 i 1 ni mi k mi ni i 1 1 li 1ki 1l 0l 1 ··· ms k l dμq x x mqZp s s mi s ni mi k s mi si 1 i 1 ni mi k mi ni i 1 i 1ki 1l 0l 1 l β m1 ··· ms k l,q .By 2.24 and 2.25 , we obtain the following theorem. 2.25

Discrete Dynamics in Nature and Society9Theorem 2.6. For s N and m1 , . . . , ms , n1 , . . . , ns , k Z with m1 n1 · · · ms ns m1 · · · ms k 1, one hask s si 1 mi mki k si 1 mi l2mi ni l 1 q q βn1 m1 ··· ns ms l,1/q i 1 1 sl 0i 1l ss kmiii 1 ni mi 1 ni mi k mi i 1 1 l β m1 ··· ms k l,q .i 1 s sl 0 2.26 lAcknowledgmentThis paper was supported by the research grant of Kwangwoon University in 2010.References 1 T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics. In press. 2 M. Acikgoz and S. Araci, “A study on the integral of the product of several type Bernsteinpolynomials,” IST Transaction of Applied Mathematics-Modelling and Simulation, vol. 1, no. 1, pp. 10–14, 2010. 3 T. Kim, “On a q-analogue of the p-adic log gamma functions and related integrals,” Journal of NumberTheory, vol. 76, no. 2, pp. 320–329, 1999. 4 G. M. Phillips, “On generalized Bernstein polynomials,” in Numerical Analysis, pp. 263–269, WorldScientific, River Edge, NJ, USA, 1996. 5 G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Analysis, vol. 4,pp. 511–514, 1997. 6 Y. Simsek and M. Acikgoz, “A new generating function of q- Bernstein-type polynomials and theirinterpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010. 7 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000,1948. 8 M. Cenkci, V. Kurt, S. H. Rim, and Y. Simsek, “On i, q Bernoulli and Euler numbers,” AppliedMathematics Letters, vol. 21, no. 7, pp. 706–711, 2008. 9 L.-C. Jang, “A new q-analogue of Bernoulli polynomials associated with p-adic q-integrals,” Abstractand Applied Analysis, vol. 2008, Article ID 295307, 6 pages, 2008. 10 T. Kim, “Barnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics A, vol. 43,no. 25, Article ID 255201, 11 pages, 2010. 11 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,2002. 12 T. Kim, L.-C. Jang, and H. Yi, “A note on the modified q-Bernstein polynomials,” Discrete Dynamics inNature and Society, vol. 2010, Article ID 706483, 12 pages, 2010. 13 B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of NonlinearMathematical Physics, vol. 12, supplement 1, pp. 412–422, 2005.