CS Ryoo

Hui-Young Lee

Jung-Yoog Kang

Nam-Soon Jung

NS Jung

SH Rim

T Kim

Y Simesk

Y Simsek

Journal of Inequalities and Applications

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RESEARCH Open AccessThe twisted (h, q)-Genocchi numbers andpolynomials with weight a and q-bernsteinpolynomials with weight aNam-Soon Jung*, Hui-Young Lee, Jung-Yoog Kang and CS Ryoo* Correspondence: jns4235@nate.comDepartment of Mathematics,Hannam University, Daejeon 306-791, KoreaAbstractIn this article, we give some identities on the twisted (h, q)-Genocchi numbers andpolynomials and q-Bernstein polynomials with weighted a.Keywords: Genocchi numbers and polynomials, twisted (h, q)-Genocchi numbersand polynomials with weight α, q-Bernstein polynomials1 IntroductionLet p be a fixed odd prime number. The symbol, Zp, Qp, and Cp denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic clo-sure of Qp, respectively. Let N be the set of natural numbers and Z+ = N ∪ {0}. As wellknown definition, the p-adic absolute value is given by |x|p = p-r, where x = prtswith (t,p) = (s, p) = (t, s) = 1. When one talks of q-extension, q is variously considered as anindeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp. In this article, weassume that q ∈ Cp with |1 - q|p < 1.For f ∈ UD(Zp) = {f /f : Zp → Cp is uniformly differentiable function}, Kim definedthe fermionic p-adic q-integral on Zp as follows:I−q(f ) =∫Zpf (x)dμ−q(x) = limN→∞1[pN]−qpN−1∑x=0f (x)(−q)x. (1:1)For n ∈ N, let fn(x) = f(x + n) be translation. As well known equation, by (1.1), wehaveqnI−q(fn) = (−1)nI−q(f ) + [2]qn−1∑l−0(−1)n−1−lqlf (l), . (1:2)Throughout this article we use the notation:[x]q =1 − qx1 − q .Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67© 2012 Jung 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.limq®1|x|q = x for any x with |x|p ≤ 1 in the present p-adic case. To investigate rela-tion of the twisted (h, q)-Genocchi numbers and polynomials with weight a and the q-Bernstein polynomials with weight a, we will use useful property for [x]qα as following;[x]qα = 1 − [1 − x]q−α[1 − x]q−α = 1 − [x]qα (1:3)The twisted (h, q)-Genocchi numbers and polynomials with weight a are defined bythe generating function, respectively:G(h,α)n,q,w = n∫Zpqx(h−1)φw(x)[x]n−1qα dμ−q(x). (1:4)G(h,α)n,q,w(x) = n∫Zpqy(h−1)φw(y)[y + x]n−1qα dμ−q(y). (1:5)In the special case, x = 0, G(h,α)n,q,w(0) = G(h,α)n,q,w are called the nth twisted (h, q)-Genocchinumbers with weight a (see [1]).Let Cpn = {w|wpn = 1} be the cyclic group of order pn and letTp = limn→∞Cpn = ∪n≥1Cpnsee [1-5].Kim defined the q-Bernstein polynomials with weight a of degree n as follows:B(α)k,n (x, q) =(nk)[x]kqα [1 − x]n−kq−α , where x ∈ [0, 1], n, k ∈ Z+ (1:6)cf [6-12].In this article, we investigate some properties for the twisted (h, q)-Genocchi num-bers and polynomials with weight a. By using these properties, we give some interest-ing identities on the twisted (h, q)-Genocchi polynomials with weight a and q-Bernstein polynomials with weight a.2 Twisted (h, q)-genocchi numbrs and polynomials with weight a and q-bernstein polynomials with weight aFrom (1.2), we can get the following form for the twisted (h, q)-Genocchi numberswith weight a:G(h,α)0,q,w = 0, and qhwG(h,α)n,q,w(1) + G(h,α)n,q,w ={[2]q, if n = 1,0, if n > 1,(2:1)G(h,α)0,q,w = 0, and qhw(1 + qαG(h,α)q,w )n + qαG(h,α)n,q,w ={qα[2]q, if n = 1,0, if n > 1,(2:2)qαxG(h,α)n+1,q,w(x) =([x]αq + qαxG(h,α)q,w)n+1(2:3)with usual convention about replacing (G(h,α)q,w )n by G(h,α)n,q,w.Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 2 of 9By (1.4), we can obtainG(h,α)n,q,w(x) = n[2]q(11 − qα)n−1 n−1∑l=0(n − 1l)(−1)l 11 + wqαl+h(2:4)By (2.4), we can getG(h,α)n,q−1,w−1 (1 − x) = n[2]q−1(11 − q−α)n−1 n−1∑l=0(n − 1l)(−1)l(q−1)αl(1−x) 11 + w−1(q−1)αl+h= n1q[2]q(11 − qα)n−1(−1)n−1qαn+αn−1∑l=0(n − 1l)(−1)lqαlx wqh1 + wqαl+h= n[2]q(11 − qα)n−1 n−1∑l=0(n − 1l)(−1)lqαlx 11 + wqαl+h1qqαn−α(−1)n−1wqh= (−1)n−1wqα(n−1)+(h−1)G(h,α)n,q,w(x).So, we get the following theorem.Theorem 1. Let n ∈ Z+. For w Î Tp, we haveG(h,α)n,q,w(x) = (−1)n−1w−1qα(1−n)+(1−h)G(h,α)n,q−1,w−1(1 − x).By (2.1), (2.2), and (2.3), we note thatG(h,α)n,q,w = −wqhG(h,α)n,q,w(1)= −wqh(q−α(1 + gαG(h,α)q,w )n)= −wqh−αn∑l=0(nl)(qα)lG(h,α)l,q,w= −wqh(n1)G(h,α)1,q,w − wqh−αn∑l=2(nl)qαl(−wqhG(h,α)l,q,w (1))= −wqh(n1)G(h,α)1,q,w − wqh−αn∑l=2(nl)qαl(−wqhq−α(1 + qαG(h,α)q,w )l)= −nwqhG(h,α)1,q,w + w2q2h−2αn∑l=2(nl)qαl(1 + qαG(h,α)q,w )l= −nwqhG(h,α)1,q,w + w2q2h−2α(1 + qα(1 + gαG(h,α)q,w ))n − nw2q2h−2αqα(1 + qαG(h,α)q,w )1= −nwqhG(h,α)1,q,w + w2q2h−2α([2]qα + q2αG(h,α)q,w )n − nw2q2h−α(qαG(h,α)1,q,w)= −nwqhG(h,α)1,q,w + w2q2h−2α([2]qα + q2αG(h,α)q,w )n − nw2q2hG(h,α)1,q,w= −nwqhG(h,α)1,q,w + w2q2hG(h,α)n,q,w(2) − nw2q2hG(h,α)1,q,w(2:5)Therefore, by (2.5), we obtain the theorem below.Theorem 2. For n ∈ N with n >1, we haveG(h,α)n,q,w(2) = w−2q−2hG(h,α)n,q,w + w−1q−hn[2]q1 + wqh+n[2]q1 + wqh.Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 3 of 9From Theorem 2,G(h,α)n+1,q,w(2)n + 1=1n + 1((n + 1)[2]q1 + wqh+(n + 1)w−1q−h[2]q1 + wqh)+ w−2q−2hG(h,α)n+1,q,wn + 1=[2]q1 + wqh+ w−1q−h[2]q1 + wqh+ w−2q−2hG(h,α)n+1,q,wn + 1Therefore, we obtain the Corollary 3 by (1.5) and Theorem 2.Corollary 3. For n ∈ N, we have∫Zpqy(h−1)φw(y)[y + 2]nqαqμ−q(y) =[2]q1 + wqh+ w−1q−h[2]q1 + wqh+ w−2q−2hG(h,α)n+1,q,wn + 1By Theorems 1, 2 and fermionic integral on Zp, we note that∫Zpqx(h−1)φw(x)[1 − x]nq−αqμ−q(x) = (−1)nqαn∫Zpqx(h−1)φw(x)[x − 1]nqαdμ−q(x)= (−1)nqαn G(h,α)n+1,q,w(−1)n + 1= w−1q1−hG(h,α)n+1,q−1 ,w−1(2)n + 1= w−1q1−h⎛⎝ [2]q−11 + q−hw−1+ wqh[2]q−11 + q−1w−1+ w2q2hG(h,α)n+1,q−1 ,w−1n + 1⎞⎠=[2]q1 + wqh+ wqh[2]q1 + wqh+ wqh+1G(h,α)n+1,q−1,w−1n + 1= [2]q + wqh+1G(h,α)n+1,q−1,w−1n + 1.(2:6)Hence, we get the following theorem.Theorem 4. for n ∈ N with n >1, we have∫Zpqx(h−1)φw(x)[1 − x]nq−αdμ−q(x) = [2]q + wqh+1G(h,α)n+1,q−1,w−1n + 1. (2:7)Corollary 5.From (1.3) and Theorem 4, we take the fermionic p-adic invariant integral on Zp forq-Bernstein polynomials as follows:∫Zpqx(h−1)φw(x)Bk,n(x, q)dμ−q(x) =∫Zpqx(h−1)φw(x)(nk)[x]kqα [1 − x]n−kq−αdμ−q(x)=(nk)∫Zpqx(h−1)φw(x)[x]kqα (1 − [x]qα )n−kdμ−q(x)=(nk)∫Zpqx(h−1)φw(x)[x]kqα (1 − [x]qα )n−kdμ−q(x)=(nk) n−k∑l=0(nn − k)(−1)l∫Zpqx(h−1)φw(x)[x]k+lqα dμ−q(x)=(nk) n−k∑l=0(n − kl)(−1)lG(h,α)k+l+1,q,wk + l + 1(2:8)Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 4 of 9And we get the following formula;∫Zpqx(h−1)φw(x)Bk,n(x, q)dμ−q(x)=∫Zpqx(h−1)φw(x)(nk)[x]n−kqα [1 − x]kq−αdμ−q(x)=∫Zpqx(h−1)φw(x)(nk)[1 − x]kq−α (1 − [1 − x]q−α )n−kdμ−q(x)=(nk)n−k∑l=0(n − kl)(−1)n−k−l[1 − x]n−k−lq−α∫Zpqx(h−1)φw(x)[1 − x]kq−αdμ−q(x)=(nk)n−k∑l=0(n − kl)(−1)n−k−l∫Zpqx(h−1)φw(x)[1 − x]n−lq−α dμ−q(x)=(nk)n−k∑l=0(n − kl)(−1)n−k−l⎛⎝[2]q + wq1+h G(h,α)n−l+1,q−1,w−1n − l + 1⎞⎠(2:9)Hence, we can get the following theorem by (2.8) and (2.9).Theorem 5. for n ∈ N with n >1, we haven−k∑l=0(n − kl)(−1)lG(h,α)k+l+1,q,wk + l + 1=n−k∑l=0(n − kl)(−1)n−k−l⎛⎝[2] + wq1+h G(h,α)n−l+1,q−1,w−1n − l + 1⎞⎠=n−k∑l=0(n − kl)(−1)n−k−l⎛⎝w−1q1−h G(h,α)n−l+1,q−1 ,w−1 (2)n − l + 1⎞⎠(2:10)Also, we can see that∫Zpqx(h−1)φw(x)Bk,n(x, q)dμ−q(x)=(nk) n−k∑l=0(n − kl)(−1)lG(h,α)k+l+1,q,wk + l + 1=(nk)∫Zpqx(h−1)φw(x)[1 − x]n−kq−α [x]kqαdμ−q(x)=(nk)∫Zpqx(h−1)φw(x)[1 − x]n−kq−α(1 − [1 − x]q−α)kdμ−q(x)=(nk) k∑l=0(kl)(−1)k−l∫Zpqx(h−1)φw(x)[1 − x]n−lq−α dμ−q(x)=(nk) k∑l=0(kl)(−1)k−l⎛⎝[2]q + wq1+h G(h,α)n−l+1,q−1 ,w−1n − l + 1⎞⎠.(2:11)Therefore, we have the theorem below.Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 5 of 9Theorem 6. For n, k ∈ Z+ with n > k + 1, we have∫Zpqx(h−1)φw(x)Bk,n(x, q)dμ−q(x)=(nk)k∑l=0(kl)(−1)k−l⎛⎝[2]q + wq1+h G(h,α)n−l+1,q−1,w−1n − l + 1⎞⎠.(2:12)By (2.7) and Theorem 6, we can get the theorem below.Theorem 7. Let n, k ∈ Z+ with n > k + 1. Then we haven−k∑l=0(n − kl)(−1)lG(h,α)k+l+1,q,wk + l + 1=k∑l=0(kl)(−1)k−1⎛⎝[2]q + wq1+h G(h,α)n−l+1,q−1 ,w−1n − l + 1⎞⎠.Let n1,n2, k ∈ Z+ with n1 + n2 >2k + 1. Then we get∫Zpqx(h−1)φw(x)B(α)k,n1(x, q)B(α)k,n2(x, q)dμ−q(x)=∫Zpqx(h−1)φw(x)(n1k)[x]kqα [1 − x]n1−kq−α(n2k)[x]kqα [1 − x]n2−kq−α dμ−q(x)=(n1k)(n2k)∫Zpqx(h−1)φw(x)[x]2kqα [1 − x]n1+n2−kq−α dμ−q(x)=(n1k)(n2k)2k∑l=0(2kl)(−1)2k−l∫Zpqx(h−1)φw(x)[1 − x]n1+n2−lq−α dμ−q(x)=(n1k)(n2k) 2k∑l=0(2kl)(−1)2k−l⎛⎝[2]q + wq1+h G(h,α)n1+n2−l+1,q−1,w−1n1 + n2 − l + 1⎞⎠.Therefore, we obtain the theorem below.Theorem 8. For n1,n2, k ∈ Z+, we have∫Zpqx(h−1)φw(x)B(α)k,n1(x, q)B(α)k,n2(x, q)dμ−q(x)=(n1k)(n2k) 2k∑l=0(2kl)(−1)2k−l⎛⎝[2]q + wq1+h G(h,α)n1+n2−l+1,q−1 ,w−1n1 + n2 − l + 1⎞⎠=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩⎛⎝[2]q + wq1+h G(h,α)n1+n2−l+1,q−1 ,w−1n1 + n2 − l + 1⎞⎠ , if k = 0,wq1+h(n1k)(n2k)∑2kl=0(2kl)(−1)2k−lG(h,α)n1+n2−l+1,q−1 ,w−1n1 + n2 − l + 1 , if k > 0,(2:13)Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 6 of 9And we can easily have that∫Zpqx(h−1)φω(x)B(α)k,n1(x, q)B(α)k,n2 (x, q)dμ−q(x)=∫Zpqx(h−1)φω(x)(n1k)[x]kqα [1 − x]n1−kq−α(n2k)[x]kqα [1 − x]n2−kq−α dμ−q(x)=(n1k)(n2k)∫Zpqx(h−1)φw(x)[x]2kqα [1 − x]n1+n2−kq−α dμ−q(x)=(n1k)(n2k)∫Zpqx(h−1)φw(x)[x]2kqα (1 − [x]qα)n1+n2−2kdμ−q(x)=(n1k)(n2k)∫Zpqx(h−1)φw(x)[x]2kqαn1+n2−2k∑l=0(n1 + n2 − 2kl)(−1)l[x]lqαdμ−q(x)=(n1k)(n2k) n1+n2−2k∑l=0(−1)l(n1 + n2 − 2kl)∫Zpqx(h−1)φw(x)[x]2k+1qα dμ−q(x)=(n1k)(n2k) n1+n2−2k∑l=0(−1)l(n1 + n2 − 2kl)G(h,α)2k+l+1,q,w2k + l + 1,where n1,n2, k ∈ Z+.(2:14)Therefore, by (2.14) and Theorem 8, we obtain the theorem below.Theorem 9. Let n1,n2, k ∈ Z+ with n1 + n2 >2k + 1. Then we have2k∑l=0(2kl)(−1)2k−l⎛⎝[2]q + wq1+h G(h,α)n1+n2−l+1,q−1 ,w−1n1 + n2 − l + 1⎞⎠=n1+n2−2k∑l=0(−1)l(n1 + n2 − 2kl)G(h,α)2k+l+1,q,w2k + l + 1.For n1,n2, . . . ,ns, k ∈ Z+,n1 + n2 + · · · + ns > sk + 1, then by the symmetry of q-Bern-stein polynomials with weight a, we see that∫Zpqx(h−1)φw(x)s∏i=1B(α)k,ni(x, q)dμ−q(x)=s∏i=1(nik)∫Zpqx(h−1)φw(x)[x]skqα [1 − x]n1+n2+···+ns−skq−α dμ−q(x)=s∏i=1(nik)∫Zpqx(h−1)φw(x)(1 − [1 − x]q−α )sk[1 − x]n1+n2+···+ns−skq−α dμ−q(x)=s∏i=1(nik) sk∑l=0(skl)(−1)sk−l∫Zpqx(h−1)φw(x)[1 − x]n1+n2+···+ns−lq−α dμ−q(x)=s∏i=1(nik) sk∑l=0(skl)(−1)sk−l⎛⎝[2]q + wq1+h G(h,α)n1+n2+···+ns−l+1,q−1 ,w−1n1 + n2 + · · · + ns − l + 1⎞⎠ .Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 7 of 9Therefore, we have the theorem below.Theorem 10. For n1,n2,n3, . . . ,ns, k ∈ Z+ with n1 + n2 + . . . + ns > sk + 1, we have∫Zpqx(h−1)φw(x)s∏i=1B(α)k,ni(x, q)dμ−q(x)=s∏i=1(nik) sk∑l=0(skl)(−1)sk−l⎛⎝[2]q + wq1+h G(h,α)n1+n2+···+ns−l+1,q−1,w−1n1 + n2 + · · · + ns − l + 1⎞⎠ .In the same manner as in (2.11), we can get the following relation:∫Zpqx(h−1)φw(x)s∏i=1B(α)k,ni(x, q)dμ−q(x)=s∏i=1(nik)∫Zpqx(h−1)φw(x)[x]skqα (1 − [x]qα)n1+n2+···+ns−skdμ−q(x)=s∏i=1(nik)∫Zpqx(h−1)φw(x)[x]skqαn1+n2+···+ns−sk∑l=0(−1)l(n1 + n2 + · · · + ns − skl)(−1)l[x]lqαdμ−q(x)=s∏i=1(nik) n1+n2+···+ns−sk∑l=0(−1)l(n1 + n2 + · · · + ns − skl)∫Zpqx(h−1)φw(x)[x]sk+lqα dμ−q(x)=s∏i=1(nik) n1+n2+···+ns−sk∑l=0(−1)l(n1 + n2 + · · · + ns − skl)G(h,α)sk+l+1,q,wsk + l + 1,where n1,n2, . . . ,ns, k ∈ Z+ with n1 + n2 + . . . + ns > sk + 1.By Theorem 11 and (2.9), we have the following corollary.Corollary 11. Let m ∈ N. For n1,n2, . . . ,ns, k ∈ Z+ with n1 + . . . + ns > sk + 1, wehavesk∑l=0(skl)(−1)sk−l⎛⎝[2]q + wq1+h G(h,α)n1+n2+···+ns−l+1,q−1,w−1n1 + n2 + · · · + ns − l + 1⎞⎠=n1+n2+···+ns−sk∑l=0(−1)l(n1 + n2 + · · · + ns − skl)G(h,α)sk+l+1,q,wsk + l + 1,Authors’ contributionsAll authors contributed equally to the manuscript and read and approved the final manuscript.Competing interestsThe authors declare that they have no competing interests.Received: 16 December 2011 Accepted: 23 March 2012 Published: 23 March 2012References1. Jung, NS, Ryoo, CS: On the twisted (h; q)-Genocchi numbers and polynomials associated with weight α. Proc. JangjeonMath. Soc. . 15, 1–9 (2012). doi:10.1155/2011/1234832. Rim, SH, Jin, JH, Moon, EJ, Lee, SJ: Some identities on the q-Genocchi polynomials of higher-order and q-Stirlingnumbers by the fermionic p-adic integral on Zp. International Journal of Mathematics and Mathematical Sciences 2010(2010). Art. ID 8602803. Ryoo, CS: Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials.Proc Jangjeon Math Soc. 14, 239–248 (2011)4. Ryoo, CS: Some relations between twisted q-Euler numbers and Bernstein polynomials. Adv Stud Contemp Math. 21,217–223 (2011)5. Simesk, Y, Kurt, V, Kim, D: New approach to the complete sum of products of the twisted (h, q )-Bernoulli numbers andpolynomials. J Nonlinear Math Phys. 14, 44–56 (2007). doi:10.2991/jnmp.2007.14.1.5Jung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 8 of 96. 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)7. Kim, T, Jang, LC, Yi, H: A note on the modified q-Bernstein polynomials. Discrete Dynamics in Nature and Society 2010,12 (2010). Article ID 7064838. Kim, T: Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adicintegral on Zp. Russian J Math Phys. 16, 484–491 (2009). doi:10.1134/S10619208090400379. Kim, T: Note on the Euler numbers and polynomials. Adv Stud Contemp Math. 17, 131–136 (2008)10. Kim, T: q-Volkenborn integration. Russ J Math Phys. 9, 288–299 (2002)11. Kim, T, Choi, J, Kim, YH, Ryoo, CS: On the fermionic p-adic integral representation of Bernstein polynomials associatedwith Euler numbers and polynomials. J Inequal Appl 2010, 12 (2010). Art ID 86424712. Simsek, Y, Acikgoz, M: A new generating function of q-Bernstein-type polynomials and their interpolation function.Abstract and Applied Analysis 2010, 12 (2010). Article ID 769095doi:10.1186/1029-242X-2012-67Cite this article as: Jung et al.: The twisted (h, q)-Genocchi numbers and polynomials with weight a and q-bernstein polynomials with weight a. Journal of Inequalities and Applications 2012 2012:67.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.comJung et al. Journal of Inequalities and Applications 2012, 2012:67http://www.journalofinequalitiesandapplications.com/content/2012/1/67Page 9 of 9

The twisted (h, q)-Genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α

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The twisted (h, q)-Genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α

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