A Bayad

Byungje Lee

CS Ryoo

IN Cangul

L Carlitz

L-C Jang

Lee-Chae Jang

NK Govil

S Bernstein

S-H Rim

T Kim

Taekyun Kim

V Kurt

Y Simsek

Young-Hee Kim

Journal of Inequalities and Applications

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RESEARCH Open AccessSome new identities on the twisted carlitz’sq-bernoulli numbers and q-bernstein polynomialsLee-Chae Jang1, Taekyun Kim2*, Young-Hee Kim2 and Byungje Lee3* Correspondence: tkkim@kw.ac.kr2Division of General Education-Mathematics, KwangwoonUniversity, Seoul 139-701, Republicof KoreaFull list of author information isavailable at the end of the articleAbstractIn this paper, we consider the twisted Carlitz’s q-Bernoulli numbers using p-adic q-integral on ℤp. From the construction of the twisted Carlitz’s q-Bernoulli numbers, weinvestigate some properties for the twisted Carlitz’s q-Bernoulli numbers. Finally, wegive some relations between the twisted Carlitz’s q-Bernoulli numbers and q-Bernstein polynomials.Keywords: q-Bernoulli numbers, p-adic q-integral, twisted1. Introduction and preliminariesLet p be a fixed prime number. Throughout this paper, ℤp, Qp and Cp will denote thering of p-adic integers, the field of p-adic rational numbers and the completion of alge-braic closure of Qp, respectively. Let N be the set of natural numbers, and let ℤ+ = N ∪{0}. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp(p) = 1p. In thispaper, we assume that q ∈ Cp with |1 - q|p < 1. The q-number is defined by[x]q =1 − qx1 − q . Note that limq ® 1 [x]q = x.We say that f is a uniformly differentiable function at a point a Î ℤp, and denotethis property by f Î UD(ℤp), if the difference quotient Ff (x, y) =f (x)−f (y)x−y has a limitf’(a) as (x, y) ® (a, a). For f Î UD(ℤp), the p-adic q-integral on ℤp, which is called theq-Volkenborn integral, is defined by Kim as follows:Iq(f ) =∫Zpf (x)dμq(x) = limN→∞1[pN]qpN−1∑x=0f (x)qx, (see [1]). (1)In [2], Carlitz defined q-Bernoulli numbers, which are called the Carlitz’s q-Bernoullinumbers, byβ0,q = 1, and q(qβ + 1)n − βn,q ={1 if n = 1,0 if n > 1,(2)with the usual convention about replacing bn by bn, q.In [2,3], Carlitz also considered the expansion of q-Bernoulli numbers as follows:β(h)0,q =h[h]q, and qh(qβ(h) + 1)n − β(h)n,q ={1 if n = 1,0 if n > 1, (3)Jang et al. Journal of Inequalities and Applications 2011, 2011:52http://www.journalofinequalitiesandapplications.com/content/2011/1/52© 2011 Kim et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.with the usual convention about replacing (b(h))n by β(h)n,q.Let Cpn = {ξ |ξ pn = 1} be the cyclic group of order pn, and let Tp = limn→∞Cpn = Cp∞ =⋃n≥0Cpn(see [1-16]). Note that Tp is a locally constant space.For ξ Î Tp, the twisted q-Bernoulli numbers are defined bytξet − 1 = eBξ t =∞∑n=0Bn,ξtnn!, (4)(see [1-19]). From (4), we note thatB0,q = 0, and ξ(Bξ + 1)n − Bn,ξ ={1 if n = 1,0 if n > 1,(5)with the usual convention about replacing Bnξ by Bn,ξ (see [17-19]). Recently, severalauthors have studied the twisted Bernoulli numbers and q-Bernoulli numbers in thearea of number theory(see [17-19]).In the viewpoint of (5), it seems to be interesting to investigate the twisted propertiesof (3). Using p-adic q-integral equation on ℤp, we investigate the properties of thetwisted q-Bernoulli numbers and polynomials related to q-Bernstein polynomials. Fromthese properties, we derive some new identities for the twisted q-Bernoulli numbersand polynomials. Final purpose of this paper is to give some relations between thetwisted Carlitz’s q-Bernoulli numbers and q-Bernstein polynomials.2. On the twisted Carlitz ‘s q-Bernoulli numbersIn this section, we assume that n Î ℤ+, ξ Î Tp and q ∈ Cp with |1 - q|p < 1.Let us consider the nth twisted Carlitz’s q-Bernoulli polynomials using p-adic q-integral on ℤp as follows:βn,ξ ,q(x) =∫Zp[y + x]nqξydμq(y)=1(1 − q)nn∑l=0(nl)(−1)lqlx∫Zpξ yqlydμq(y)=1(1 − q)n−1n∑l=0(nl)(l + 11 − ξql+1)(−1)lqlx.(6)In the special case, x = 0, bn,ξ,q(0) = bn,ξ,q are called the nth twisted Carlitz’sq-Bernoulli numbers.From (6), we note thatβn,ξ ,q(x) =1(1 − q)n−1n−1∑l=0(nl)(−1)lqlx(11 − ξql+1)+1(1 − q)n−1n∑l=0(nl)(−1)lqlx(11 − ξql+1)= −n∞∑m=0ξmq2m+x[x +m]n−1q +∞∑m=0ξmqm(1 − q)[x +m]nq .(7)Therefore, by (7), we obtain the following theorem.Jang et al. Journal of Inequalities and Applications 2011, 2011:52http://www.journalofinequalitiesandapplications.com/content/2011/1/52Page 2 of 6Theorem 1. For n Î ℤ+, we haveβn,ξ ,q(x) = −n∞∑m=0ξmqm[x +m]n−1q + (1 − q)(n + 1)∞∑m=0ξmqm[x +m]nq .Let Fq, ξ (t, x) be the generating function of the twisted Carlitz’s q-Bernoulli poly-nomials, which are given byFq,ξ (t, x) = eβξ ,q(x)t =∞∑n=0βn,ξ ,q(x)tnn!, (8)with the usual convention about replacing (bξ,q(x))n by bn,ξ,q(x).By (8) and Theorem 1, we getFq,ξ (t, x) =∞∑n=0βn,ξ ,q(x)tnn!= −t∞∑m=0ξmq2m+xe[x+m]qt + (1 − q)∞∑m=0ξmqme[x+m]qt.(9)Let Fq,ξ(t, 0) = Fq,ξ(t). Then, we haveqξFq,ξ (t, 1) − Fq,ξ (t) = t + (q − 1). (10)Therefore, by (9) and (10), we obtain the following theorem.Theorem 2. For n Î ℤ+, we haveβ0,ξ ,q(x) =q − 1qξ − 1 , and qξβn,ξ ,q(1) − βn,ξ ,q ={1 if n = 1,0 if n > 1.From (6), we note thatβn,ξ ,q(x) =n∑l=0(nl)[x]n−lq qlx∫Zpξ y[y]lqdμq(y)=n∑l=0(nl)[x]n−lq qlxβl,ξ ,q=([x]q + qxβξ ,q)n,(11)with the usual convention about replacing (bξ,q)n by bn,ξ,q. By (11) and Theorem 2,we getqξ(qβξ ,q + 1)n − βn,ξ ,q =⎧⎨⎩q − 1 if n = 0,1 if n = 1,0 if n > 1.(12)It is easy to show thatβn,ξ−1,q−1 (1 − x) =∫Zpξ−y[1 − x + y]nq−1dμq−1 (y)=(−1)nqn(1 − q)nn∑l=0(nl)(−1)lq−l+lx∫Zpξ−yq−lydμq−1 (y)= ξqn(−1)n(1(1 − q)n−1n∑l=0(nl)(−1)lqlx( l + 11 − ξql+1 ))= ξqn(−1)nβn,ξ ,q(x).(13)Jang et al. Journal of Inequalities and Applications 2011, 2011:52http://www.journalofinequalitiesandapplications.com/content/2011/1/52Page 3 of 6Therefore, by (13), we obtain the following theorem.Theorem 3. For n Î ℤ+, we haveβn,ξ−1,q−1 (1 − x) = ξqn(−1)nβn,ξ ,q(x).From Theorem 3, we can derive the following functional equation:Fq−1,ξ−1 (t, 1 − x) = ξFq,ξ (−qt, x). (14)Therefore, by (14), we obtain the following corollary.Corollary 4. Let Fq,ξ (t, x) =∑∞n=0 βn,ξ ,q(x)tnn!. Then we haveFq−1,ξ−1 (t, 1 − x) = ξFq,ξ (−qt, x).By (11), we get thatq2ξ2βn,ξ ,q(2) = q2ξ2n∑l=0(nl)ql(1 + qβξ ,q)l= q2ξ2(1 − q1 − qξ ) +(n1)q2ξ(1 + β1,ξ ,q) + q2ξ2n∑l=0(nl)qlβl,ξ ,q(1)= (1 − q) q2ξ21 − qξ +(n1)q2ξ + qξn∑l=0(nl)qlβl,ξ ,q=1 − q1 − qξ q2ξ2 + nq2ξ − qξ 1 − q1 − qξ + βn,ξ ,q, if n > 1.(15)Therefore, by (15), we obtain the following theorem.Theorem 5. For n Î N with n > 1, we haveβn,ξ ,q(2) =1 − q1 − qξ +nξ− 1qξ(1 − q1 − qξ ) + (1qξ)2βn,ξ ,q.By a simple calculation, we easily setξ∫Zp[1 − x]nq−1 ξ xdμq(x) = ξ(−1)nqn∫Zp[x − 1]nqξ xdμq(x)= ξ(−1)nqnβn,ξ ,q(−1) = βn,ξ−1,q−1 (2).(16)For n Î ℤ+ with n > 1, we haveξ∫Zp[1 − x]nq−1ξ xdμq(x) = βn,ξ−1,q−1 (2)= ξ(1 − q1 − qξ ) + nξ − qξ2(1 − q1 − qξ ) + (qξ)2βn,ξ−1,q−1= ξ(1 − q) + nξ + (qξ)2βn,ξ−1,q−1 .(17)Therefore, by (16) and (17), we obtain the following theorem.Theorem 6. For n Î ℤ+ with n > 1, we have∫Zp[1 − x]nq−1ξ xdμq(x) = (1 − q) + n + q2ξβn,ξ−1,q−1 .Jang et al. Journal of Inequalities and Applications 2011, 2011:52http://www.journalofinequalitiesandapplications.com/content/2011/1/52Page 4 of 6For x Îℤp and n, k Î ℤ+, the p-adic q-Bernstein polynomials are given byBk,n(x, q) =(nk)[x]kq[1 − x]n−kq−1 , (18)(see [8,20]).In [8], the q-Bernstein operator of order n is given byBn,q(f |x) =n∑k=0f (nk)Bk,n(x, q) =n∑k=0f (nk)(nk)[x]kq[1 − x]n−kq−1 .Let f be continuous function on ℤp. Then, the sequence Bn,q(f |x) converges uniformlyto f on ℤp (see [8]). If q is same version in (18), we cannot say that the sequenceBn,q(f |x) converges uniformly to f on ℤp.Let s Î N with s ≥ 2. For n1, ..., ns, k Î ℤ+ with n1 + · · · + ns >sk + 1, we take thep-adic q-integral on ℤp for the multiple product of q-Bernstein polynomials as follows:∫Zpξ xBk,n1(x, q) · · · Bk,ns(x, q)dμq(x)=(n1k). . .(nsk)∫Zp[x]kq[1 − x]n1+···+ns−skq−1 ξ xdμq(x)=(n1k). . .(nsk) sk∑l=0(skl)(−1)l+sk∫Zp[1 − x]n1+···+ns−lq−1 ξ xdμq(x)=(n1k). . .(nsk) sk∑l=0(skl)(−1)l+sk×(q2ξβn1+···+ns−l,ξ−1,q−1 + n1 + · · · + ns − l + 1 − q)dμq(x)={q2ξβn1+···+ns ,ξ−1,q−1 + n1 + · · · + ns + (1 − q) if k = 0,q2ξ(n1k) · · · (nsk )∑skl=0 (skl )(−1)l+skβn1+···+ns−l,ξ−1.q−1 if k > 0,(19)and we also have∫Zpξ xBk,n1 (x, q) · · ·Bk,ns (x, q)dμq(x)=(n1k). . .(nsk) n1+···+ns−sk∑l=0(n1 + · · · + ns − skl)(−1)lβl+sk,ξ ,q.(20)By comparing the coefficients on the both sides of (19) and (20), we obtain the fol-lowing theorem.Theorem 7. Let s Î N with s ≥ 2. For n1, ..., ns, k Î ℤ+ with n1 + ... + ns >sk + 1, wehaven1+···+ns−sk∑l=0(n1 + · · · + ns − skl)(−1)lβl+sk,ξ ,q={q2ξβn1+···+ns ,ξ−1,q−1 + n1 + · · · + ns + (1 − q) if k = 0,q2ξ∑skl=0(skl)(−1)l+skβn1+···+ns−l,ξ−1 .q−1 if k > 0.Jang et al. Journal of Inequalities and Applications 2011, 2011:52http://www.journalofinequalitiesandapplications.com/content/2011/1/52Page 5 of 6AcknowledgementsThe authors express their sincere gratitude to referees for their valuable suggestions and comments. This paper wassupported by the research grant Kwangwoon University in 2011.Author details1Department of Mathematics and Computer Science, Konkuk University, Chungju 380-701, Republic of Korea 2Divisionof General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea 3Department of WirelessCommunications Engineering, Kwangwoon University, Seoul 139-701, Republic of KoreaCompeting interestsThe authors declare that they have no competing interests.Received: 21 February 2011 Accepted: 13 September 2011 Published: 13 September 2011References1. Kim, T: On a q-analogue of the p-adic log gamma functions and related integrals. J Number Theory. 76, 320–329(1999). doi:10.1006/jnth.1999.23732. Carlitz, L: q-Bernoulli numbers and polynomials. Duke Math J. 15, 987–1000 (1948). doi:10.1215/S0012-7094-48-01588-93. Kim, T: q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ J Math Phys. 15,51–57 (2008)4. Bernstein, S: Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilities. Commun Soc MathKharkow. 13, 1–2 (1912)5. Cangul, IN, Kurt, V, Ozden, H, Simsek, Y: On the higher-order w-q-Genocchi numbers. Adv Stud Contemp Math. 19,39–57 (2009)6. Govil, NK, Gupta, V: Convergence of q-Meyer-König-Zeller-Durrmeyer operators. Adv Stud Contemp Math. 19, 97–108(2009)7. Jang, L-C: A study on the distribution of twisted q-Genocchi polynomials. Adv Stud Contemp Math. 19, 181–189 (2009)8. Kim, T: A note on q-Bernstein polynomials. Russ J Math Phys. 18, 41–50 (2011)9. Kim, T: q-Volkenborn integration. Russ J Math Phys. 9, 288–299 (2002)10. Kim, T, Choi, J, Kim, Y-H: Some identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers.Adv Stud Contemp Math. 20, 335–341 (2010)11. Kim, T: Barnes type multiple q-zeta functions and q-Euler polynomials. J Physics A: Math Theor 43, 11 (2010). 25520112. Kurt, V: further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchipolynomials. Appl Math Sci (Ruse). 3, 2757–2764 (2008)13. Rim, S-H, Moon, E-J, Lee, S-J, Jin, J-H: Multivariate twisted p-adic q-integral on ℤp associated with twisted q-Bernoullipolynomials and numbers. J Inequal Appl 2010, Art ID 579509 (2010). 6 pp14. Ryoo, CS, Kim, YH: A numericla investigation on the structure of the roots of the twisted q-Euler polynomials. Adv StudContemp Math. 19, 131–141 (2009)15. Ryoo, CS: On the generalized Barnes’ type multiple q-Euler polynomials twisted by ramified roots of unity. ProcJangjeon Math Soc. 13, 255–263 (2010)16. Ryoo, CS: A note on the weighted q-Euler numbers and polynomials. Adv Stud Contemp Math. 21, 47–54 (2011)17. Simsek, Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolationfunctions. Adv Stud Contemp Math. 16, 251–278 (2008)18. Kim, T: Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials. Russ J Math Phys. 10,91–98 (2003)19. Simsek, Y: Theorems on twisted L-function and twisted Bernoulli numbers. Adv Stud Contemp Math. 11, 205–218(2005)20. Bayad, A, Kim, T: Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials. Russ J Math Phys. 18,133–143 (2011). doi:10.1134/S1061920811020014doi:10.1186/1029-242X-2011-52Cite this article as: Jang et al.: Some new identities on the twisted carlitz’s q-bernoulli numbers and q-bernsteinpolynomials. Journal of Inequalities and Applications 2011 2011:52.Submit your manuscript to a journal and benefi t from:7 Convenient online submission7 Rigorous peer review7 Immediate publication on acceptance7 Open access: articles freely available online7 High visibility within the fi eld7 Retaining the copyright to your article    Submit your next manuscript at 7 springeropen.comJang et al. Journal of Inequalities and Applications 2011, 2011:52http://www.journalofinequalitiesandapplications.com/content/2011/1/52Page 6 of 6

Some new identities on the twisted carlitz's q-bernoulli numbers and q-bernstein polynomials

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Some new identities on the twisted carlitz's q-bernoulli numbers and q-bernstein polynomials

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