Cheon Seoung Ryoo

CS Ryoo

E-J Moon

Hui Young Lee

IN Cangul

Jung Yoog Kang

LC Jang

Nam Soon Jung

T Kim

V Kurt

Advances in Difference Equations

English

Crossref

RESEARCH Open AccessSome identities on the higher-order-twistedq-Euler numbers and polynomials with weight aHui Young Lee*, Nam Soon Jung, Jung Yoog Kang and Cheon Seoung Ryoo* Correspondence: normaliz@hnu.krDepartment of Mathematics,Hannam University, Daejeon 306-791, KoreaAbstractIn this article, we introduce some properties of higher-order-twisted q-Euler numbersand polynomials with weight a, and we observe some properties of higher-order-twisted q-Euler numbers and polynomials with weight a for several cases. Inparticular, by using the the fermionic p-adic q-integral on ℤp, we give a new conceptof twisted q-Euler numbers and polynomials with weight a.2000 Mathematics Subject Classification: 11B68; 11S40; 11S80.Keywords: Euler numbers and polynomials, q-Euler numbers and polynomials,higher-order-twisted q-Euler numbers and polynomials with weight α1. IntroductionLet p be a fixed odd prime. Throughout this article ℤp, Qp, ℂ, and ℂp, will, respectively,denote the ring of p-adic rational integers, the field of p-adic rational numbers, thecomplex number field, and the completion of algebraic closure of Qp. Let N be the setof natural numbers and ℤ+ = N∪{0}. Let νp be the normalized exponential valuation ofℂp with∣∣p∣∣p = p−νp(p) = p−1 (see [1-14]). When one speaks of q-extension, q can beregarded as an indeterminate, a complex number q Î ℂ, or p-adic number q Î ℂp; itis always clear from context. If q Î ℂ, we assume |q| < 1. If q Î ℂp, then we assume |1 - q|p < 1 (see [1-14]).In this article, we use the notation of q-number as follows (see [1-14]):[x]q =1 − qx1 − q .Note that limq®1[x]q = x for any x with |x|p ≤ 1 in the p-adic case.Let C(ℤp) be the space of continuous functions on ℤp. For f Î C(ℤp), Kim defined thefermionic p-adic q-integral on ℤp as follows (see [6,7]):I−q(f ) =∫Zpf (x)dμ−q(x) = limN→∞1[pN]−qpN−1∑x=0f (x)(−q)x,= limN→∞[2]q1 + qpNpN−1∑x=0f (x)(−q)x.(1)Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21© 2012 Lee 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.From (1), we note thatqI−q(f1) + I−q(f ) = [2]qf (0),where f1(x) = f(x + 1).It is well known that the ordinary Euler polynomials are defined by2et + 1ext = eE(x)t =∞∑n=0En(x)tnn!,with the usual convention of replacing En(x) by En(x).In the special case, x = 0, En(0) = En are called the nth Euler numbers (see [1-14]).By (2), we get the following recurrence relation as follows:E0 = 1, and (E + 1)n + E ={2, if n = 0,0, if n > 0.(2)Recently, (h, q)-Euler numbers are defined byE(h)0,q =21 + qh, and qh(qE(h)q + 1)n + E(h)q ={2, if n = 0,0, if n > 0,with the usual convention about replacing(E(h)q)nby E(h)n,q (see [1-16]).Note that limq→1E(h)n,q = En.Let Tp = ∪N≥1CpN = limN→∞CpN, where CpN = {w|wpN = 1} is the cyclic group of orderpN. For w Î Tp, we denote by jw : ℤp ® ℂp the locally constant function x ↦ wx.For a Î N and w Î Tp, the twisted q-Euler numbers with weight a are also defined byE˜(α)0,q,w =[2]qwq + 1, and wq(qα E˜(α)q,w + 1)n+ E˜(α)n,q,w ={[2]q, if n = 0,0, if n > 0,with the usual convention about replacing(E˜(α)q,w)nby E˜(α)n,q,w (see [2,5]).The main purpose of this article is to present a systemic study of some families of higher-order-twisted q-Euler numbers and polynomials with weight a. In Section 2, we investigatehigher-order-twisted q-Euler numbers and polynomials with weight a and establish interest-ing properties. In Sections 3, 4, and 5, we observe some properties for special cases.2. Higher-order-twisted q-Euler numbers and polynomials with weight aFor h Î ℤ, a, k Î N, w Î Tp and n Î ℤ+, let us consider the expansion of higher-order-twisted q-Euler polynomials with weight a as follows:E˜(α)n,q,w(h, k|x)=∫Zp· · ·∫Zpw︸ ︷︷ ︸k−times∑ki=1xi[k∑i=1xi + x]nqαqx1(h−1)+···+xk(h−k)dμ−q(x1) · · · dμ−q(xk). (3)From (1) and (3), we note thatE˜(α)n,q,w(h, k|x) =[2]kq(1 − qα)nn∑l=0(nl)(−1)l qαlx(1 + wqαl+h) · · · (1 + wqαl+h−k+1) . (4)Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 2 of 10In the special case, x = 0 E˜(α)n,q,w(h, k|0) = E˜(α)n,q,w(h, k) are called the higher-order-twisted q-Euler numbers with weight a.By (3), we getE˜(α)n,q,w(h, k) = (qα − 1)E˜(α)n+1,q,w(h − α, k) + E˜(α)n,q,w(h − α, k). (5)From (5) and mathematical induction, we get the following theorem.Theorem 1. For a, k Î N and n Î ℤ+, we haven−1∑i=0(−1)i−1(qα − 1)n−1−iE˜(α)n−i,q,w(h, k)= (qα − 1)n−1E˜(α)n,q,w(h − α, k) + (−1)nE(α)1,q,w(h − α, k).For complex number q Î ℂp, m Î ℤ+, we get the following;qα(x1+···+xk+1)m =(1 − (1 − qα(x1+···+xk+1)))m=m∑l=0(ml)(−1)l(1 − qα(x1+···+xk+1))l=m∑l=0(ml)(−1)l(1 − qα)l (1 − qα(x1+···+xk+1))l(1 − qα)l=m∑l=0(ml)(−1)l(1 − qα)l[x1 + x2 + · · · + xk+1]lqα .From (3), (4), and above property, we haveE˜(α)0,q,w(mα, k + 1)=∫Zp· · ·∫Zpw∑k+1j=1 xjq∑k+1j=1 (mα−j)xjdμ−q(x1) · · · dμ−q(xk+1)=m∑l=0(ml)(qα − 1)l∫Zp· · ·∫Zpw∑k+1j=1 xj⎡⎣ k+1∑j=1xj⎤⎦lqαq−∑k+1j=1 jxjdμ−q(x1) · · · dμ−q(xk+1)=m∑l=0(ml)(qα − 1)lE˜(α)l,q,w(0, k + 1)=[2]k+1q(1 + wqαm)(1 + wqαm−1) . . . (1 + wqαm−k).(6)From (3), we can derive the following equation.i∑j=0(ij)(qα − 1)j∫Zp· · ·∫Zpw∑ks=1 xs[k∑s=1xs]n−i+jqαq∑ks=1 (h−α−s)xs dμ−q(x1) · · · dμ−q(xk)=∫Zp· · ·∫Zpw∑ks=1 xs[k∑s=1xs]n−iqαq∑ks=1 (h−s)xsqα(∑ks=1 xs)(i−1)dμ−q(x1) · · · dμ−q(xk)=i−1∑j=0(qα − 1)j(i − 1j)E˜(α)n−i+j,q,w(h, k).(7)Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 3 of 10By (3), (4), (5), and (6), we see thati∑j=0(qα − 1)j(ij)E˜(α)n−1+j,q,w(h − α, k) =i−1∑j=0(qα − 1)j(i − 1j)E˜(α)n−i+j,q,w(h, k).Therefore, we obtain the following theorem.Theorem 2. For a, k Î N and n, i Î ℤ+, we havei∑j=0(ij)(qα − 1)jE˜(α)n−i+j,q,w(h − α, k) =i−1∑j=0(qα − 1)j(i − 1j)E˜(α)n−i+j,q,w(h, k).By simple calculation, we easily see thatm∑j=0(mj)(qα − 1)jE˜(α)j,q,w(0, k) =[2]kq(1 + wqαm)(1 + wqαm−1) · · · (1 + wqαm−k+1) .3. Polynomials E˜(α)n,q,w(0, k|x)We now consider the polynomials E˜(α)n,q,w(0, k|x) (in qx) byE˜(α)n,q,w(0, k|x)=∫Zp· · ·∫Zp︸ ︷︷ ︸k−timeswx1+···+xk[x +k∑i=1xi]nqαq−∑kj=1 jxjdμ−q(x1) · · · dμ−q(xk). (8)By (8) and (4), we get(qα − 1)nE˜(α)n,q,w(0, k|x) = [2]kqn∑l=0(nl)qαlx(−1)n−1 1(1 + wqαl) · · · (1 + wqαl−k+1) . (9)From (8) and (9), we can derive the following equation.∫Zp· · ·∫Zpwx1+···+xkq∑kj=1 (αn−j)xj+αnxdμ−q(x1) · · · dμ−q(xk)=n∑j=0(nj)[α]jq(q − 1)jE˜(α)j,q,w(0, k|x),and ∫Zp· · ·∫Zpwx1+···+xkq∑kj=1 (αn−j)xj+αnxdμ−q(x1) · · · dμ−q(xk)=[2]kqqαnx(1 + wqαn) · · · (1 + wqαn−k+1) .(10)Therefore, by (9) and (10), we obtain the following theorem.Theorem 3. For a Î N and n, k Î ℤ+, we haveE˜(α)n,q,w(0, k|x) =[2]kq[α]nq(1 − q)nn∑l=0(nl)(−1)lqαlx 1(−wqαl−k+1 : q)k,Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 4 of 10andn∑l=0(nl)[α]lq(q − 1)lE˜(α)l,q,w(0, k|x) =qαnx[2]kq(−wqαn−k+1 : q)k,where (a : q)0 = 1 and (a : q)k = (1 - a)(1 - aq) ... (1 - aqk-1).Let d Î N with d ≡ 1 (mod 2). Then we have∫Zp· · ·∫Zpwx1+···+xk⎡⎣x + k∑j=1xj⎤⎦nqαq−∑kj=1 jxjdμ−q(x1) · · · dμ−q(xk)=[d]nqα[d]k−qd−1∑a1,··· ,ak=0wa1+···+akq−∑kj=2 (j−1)aj(−1)∑kj=1 aj×∫Zp· · ·∫Zpwd(x1+···+xk)⎡⎣x +∑kj=1 ajd+k∑j=1xj⎤⎦nqαdq−d∑kj=1 jxjdμ−qd(x1) · · · dμ−qd(xk)(11)Thus, by (11), we obtain the following theorem.Theorem 4. For d Î N with d ≡ 1 (mod 2), we haveE˜(α)n,q,w(0, k|x)=[d]nqα[d]k−qd−1∑a1,··· ,ak=0(−w)a1+···+akq−∑kj=2 (j−1)aj E˜(α)n,qd,wd(0, k|x + a1 + · · · + akd).Moreover,E˜(α)n,q,w(0, k|dx)=[d]nqα[d]k−qd−1∑a1,··· ,ak=0(−w)a1+···+akq−∑kj=2 (j−1)aj E˜(α)n,qd,wd(0, k|x + a1 + · · · + akd).By (8), we getE˜(α)n,q,w(0, k|x =n∑l=0(nl)[x]n−lqα qαlxE˜(α)l,q,w(0, k)=([x]qα + qαxE˜(α)q,w(0, k))n,where E˜(α)n,q,w(0, k|0) = E˜(α)n,q,w(0, k).Thus, we note thatE˜(α)n,q,w(0, k|x + y =n∑l=0(nl)[y]n−lqα qαlyE˜(α)l,q,w(0, k|x)=([y]qα + qαyE˜(α)q,w(0, k|x))n.4. Polynomials E˜(α)n,q,w(h, 1|x)Let us define polynomials E˜(α)n,q,w(h, 1|x) as follows:E˜(α)n,q,w(h, 1|x) =∫Zpwx1 [x + x1]nqαqx1(h−1)dμ−q(x1). (12)Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 5 of 10From (12), we haveE˜(α)n,q,w(h, 1|x) =[2]q(1 − qα)nn∑l=0(nl)(−1)lqαlx 1(1 + wqαl+h).By the calculation of the fermionic p-adic q-integral on ℤp, we see thatqαx∫Zpwx1 [x + x1]nqαqx1(h−1)dμ−q(x1)= (qα − 1)∫Zpwx1 [x + x1]n+1qα qx1(h−α−1)dμ−q(x1) +∫Zpwx1 [x + x1]nqαqx1(h−α−1)dμ−q(x1).(13)Thus, by (13), we obtain the following theorem.Theorem 5. For a Î N and h Î ℤ, we haveqαxE˜(α)n,q,w(h, 1|x) = (qα − 1)E˜(α)n+1,q,w(h − α, 1|x) + E˜(α)n,q,w(h − α, 1|x).It is easy to show thatE˜(α)n,q,w(h, 1|x) =∫Zpwx1 [x + x1]nqαqx1(h−1)dμ−q(x1)=n∑l=0(nl)[x]n−1qα qαlx∫Zpwx1 [x1]lqαqx1(h−1)dμ−q(x1)=n∑l=0(nl)[x]n−1qα qαlxE˜(α)l,q,w(h, 1)=(qαxE˜(α)q,w(h, 1) + [x]qα)n, for n ≥ 1,(14)with the usual convention about replacing (E˜(α)q,w(h, 1))n by E˜(α)n,q,w(h, 1).From qI-q(f1) + I-q(f) = [2]qf(0), we havewqh∫Zpwx1 [x + x1 + 1]nqαqx1(h−1)dμ−q(x1) +∫Zpwx1 [x + x1]nqαqx1(h−1)dμ−q(x1)= [2]q[x]nqα .(15)By (13) and (15), we getwqhE˜(α)n,q,w(h, 1|x + 1) + E˜(α)n,q,w(h, 1|x) = [2]q[x]nqα . (16)For x = 0 in (16), we havewqhE˜(α)n,q,w(h, 1|1) + E˜(α)n,q,w(h, 1) ={[2]q, if n = 0,0, if n > 0.(17)Therefore, by (14) and (17), we obtain the following theorem.Theorem 6. For h Î ℤ and n Î ℤ+, we havewqh(qαE˜(α)q,w(h, 1) + 1)nE˜(α)n,q,w(h, 1) ={[2]q, if n = 0,0, if n > 0,with the usual convention about replacing (E˜(α)q,w(h, 1))n by E˜(α)n,q,w(h, 1).Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 6 of 10From the fermionic p-adic q-integral on ℤp, we easily getE˜(α)0,q,w(h, 1) =∫Zpwx1qx1(h−1)dμ−q(x1) =[2]q[2]wqh.By (12), we see thatE˜(α)n,q−1,w−1(h, 1|1 − x) =∫Zpw−1[1 − x + x1]nq−αq−x1(h−1)dμ−q−1 (x1)= (−1)nwqαn+h−1 [2]q(1 − qα)nn∑l=0(nl)(−1)lqαlx 11 + wqαl+h= (−1)nwqαn+h−1E˜(α)n,q,w(h, 1|x)(18)Therefore, by (18), we obtain the following theorem.Theorem 7. For a Î N, h Î ℤ and n Î ℤ+, we haveE˜(α)n,q−1,w−1 (h, 1|1 − x) = (−1)nwqαn+h−1E˜(α)n,q,w(h, 1|x).In particular, for x = 1, we getE˜(α)n,q,w(h, 1)= (−1)nwqαn+h−1E˜(α)n,q,w(h, 1|1)= (−1)n+1qαn−1E˜(α)n,q,w(h, 1) if n ≥ 1.Let d Î N with d ≡ 1 (mod 2). Then we have∫Zpwx1qx1(h−1)[x + x1]nqαdμ−q(x1)=[d]nqα[d]−qd−1∑a=0waqha(−1)a∫Zpwdx1[x + ad+ x1]nqαdqx1(h−1)ddμ−qd(x1).(19)Therefore, by (19), we obtain the following theorem.Theorem 8 (Multiplication formula). For d Î N with d ≡ 1 (mod 2), we haveE˜(α)n,q,w(h, 1|x) =[d]nqα[d]−qd−1∑a=0waqha(−1)qE˜(α)n,qd,wd(h, 1|x + ad).5. Polynomials E˜(α)n,q,w(h, k|x) and k = hIn (3), we know thatE˜(α)n,q,w(h, k|x)=∫Zp· · ·∫Zpwx1+···+xk[x1 + · · · + xk + x]nqαq(h−1)x1+···+(h−k)xkdμ−q(x1) · · · dμ−q(xk).Thus, we get(qα − 1)nE˜(α)n,q,w(h, k|x) = [2]kqn∑l=0(nl)(−1)n−l qαlx(1 + wqαl+h) · · · (1 + wqαl+h−k+1) ,Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 7 of 10andwqh∫Zp· · ·∫Zpwx1+···+xk [x + 1 +k∑i=1xi]nqαq∑ki=1 (h−i)xidμ−q(x1) · · · dμ−q(xk)= −∫Zp· · ·∫Zpwx1+···+xk [x +k∑i=1xi]nqαq∑ki=1 (h−i)xidμ−q(x1) · · · dμ−q(xk)+ [2]q∫Zp· · ·∫Zpwx2+···+xk [x +k∑i=2xi]nqαq∑ki=2 (h−i)xidμ−q(x2) · · · dμ−q(xk).(20)Therefore, by (3) and (20), we obtain the following theorem.Theorem 9. For h Î ℤ, a Î N and n Î ℤ+, we havewqhE˜(α)n,q,w(h, k|x + 1) + E˜(α)n,q,w(h, k|x) = [2]qE˜(α)n,q,w(h − 1, k − 1|x).Note thatqαx∫Zp· · ·∫Zpwx1+···+wk [x +k∑i=1xi]nqαq∑ki=1 (h−i)xidμ−q(x1) · · · dμ−q(xk)= (qα − 1)∫Zp· · ·∫Zpwx1+···+xk [x +k∑i=1xi]n+1qα q∑ki=1 (h−α−i)xi dμ−q(x1) · · · dμ−q(xk)+∫Zp· · ·∫Zpwx1+···+xk [x +k∑i=1xi]nqαq∑ki=1 (h−α−i)xidμ−q(x1) · · · dμ−q(xk)= (qα − 1)E˜(α)n+1,q,w(h − α, k|x) + E˜(α)n,q,w(h − α, k|x).(21)Therefore, by (21), we obtain the following theorem.Theorem 10. For n Î ℤ+, we haveqαxE˜(α)n,q,w(h, k|x) = (qα − 1)E˜(α)n+1,q,w(h − α, k|x) + E˜(α)n,q,w(h − α, k|x).Let d Î N with d ≡ 1 (mod 2). Then we get∫Zp· · ·∫Zpwx1+···+xk [x +k∑j=1xj]nqαq∑kj=1 (h−j)xjdμ−q(x1) · · · dμ−q(xk)=[d]nqα[d]k−qd−1∑a1,··· ,ak=0wa1+···+akqh∑kj=1 aj−∑kj=2 (j−1)aj(−1)∑kj=1 aj×∫Zp· · ·∫Zpwd(x1+···+xk)⎡⎣x +∑kj=1 ajd+k∑j=1xj⎤⎦nqαdqd∑kj=1 (h−j)xjdμ−qd(x1) · · · dμ−qd(xk).(22)Therefore, by (22), we obtain the following theorem.Theorem 11. For d Î N with d ≡ 1 (mod 2), we haveE˜(α)n,q,w(h, k|dx)=[d]nqα[d]k−qd−1∑a1,··· ,ak=0wa1+···+akqh∑kj=1 aj−∑kj=2 (j−1)aj(−1)∑kj=1 aj E˜(α)n,qd ,wd(h, k|x + a1 + · · · + akd).Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 8 of 10Let E˜(α)n,q,w(k, k|x) = E˜(α)n,q,w(k|x). Then we get(qα − 1)nE˜(α)n,q,w(k|x) =n∑l=0(nl)(−1)n−lqαlx [2]kq(1 + wqαl+k) · · · (1 + wqαl+1) ,and∫Zp· · ·∫Zpw−(x1+···+xk)[k − x +k∑i=1xi]nq−αq−∑ki=1 (k−i)xidμ−q−1 (x1) · · · dμ−q−1 (xk)=q⎛⎝ k2⎞⎠(1 − q−α)n [2]kqn∑l=0(nl)(−1)lqαlx 1(1 + wqαl+1) · · · (1 + wqαl+k)= (−1)nqnαq⎛⎝ k2⎞⎠ [2]kq(1 − qα)nn∑l=0(nl)(−1)lqαlx(1 + wqαl+1) · · · (1 + wqαl+k)= (−1)nqαn+⎛⎝ k2⎞⎠E˜(α)n,q,w(k|x).(23)Therefore, by (23), we obtain the following theorem.Theorem 12. For n Î ℤ+, we haveE˜(α)n,q−1,w−1 (k|k − x) = (−1)nwkqαn+⎛⎝ k2⎞⎠E˜(α)n,q,w(k|x).Let x = k in Theorem 12. Then we see thatE˜(α)n,q−1,w−1(k|0) = (−1)nwkqαn+⎛⎝ k2⎞⎠E˜(α)n,q,w(k|k).(24)From (15), we note thatwqkE˜(α)n,q,w(k|x + 1) + E˜(α)n,q,w(k|x) = [2]qE˜(α)n,q,w(k − 1|x). (25)It is easy to show that(qα − 1)nE˜(α)n,q,w(k|0) =n∑l=0(nl)(−1)l+n [2]kq(1 + wqαl+1) · · · (1 + wqαl+k) .By simple calculation, we getn∑l=0(nl)(qα − 1)l∫Zp· · ·∫Zpw∑ki=1 xk [k∑i=1xk]lqαq∑kl=i (k−i)xidμ−q(x1) · · · dμ−q(xk)=[2]kq(1 + wqαn+k)(1 + wqαn+k−1) · · · (1 + wqαn+1) .(26)From (26), we note thatn∑l=0(nl)(qα − 1)lE˜(α)l,q,w(k|0) =[2]kq(1 + wqαn+k)(1 + wqαn+k−1) · · · (1 + wqαn+1) ,Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 9 of 10andE˜(α)n,q,w(k|x) =∫Zp· · ·∫Zpw∑ki=1 xk [x +k∑i=1xk]nqαq∑ki=1 (k−i)xidμ−q(x1) · · · dμ−q(xk)=n∑l=0(nl)[x]n−lqα qαlxE˜(α)l,q,w(k|0)=(qxαE˜(α)q,w(k|0) + [x]qα)n,n ∈ Z+,with the usual convention about replacing (E˜(α)q,w(k|0))n by E˜(α)n,q,w(k|0).Put x = 0 in (25), we getwqkE˜(α)n,q,w(k|1) + E˜(α)n,q,w(k|0) = [2]qE˜(α)n,q (k − 1|0).Thus, we havewqk(qα E˜(α)q,w(k|0) + 1)n + E˜(α)n,q,w(k|0) = [2]qE˜(α)n,q,w(k − 1|0),with the usual convention about replacing (E˜(α)q,w(k|0))n by E˜(α)n,q,w(k|0).AcknowledgementsThe authors express their gratitude to the referee for his/her valuable comments.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: 19 September 2011 Accepted: 29 February 2012 Published: 29 February 2012References1. Ryoo, CS: On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity. ProcJangjeon Math Soc. 13, 255–263 (2010)2. Ryoo, CS: A note on the weighted q-Euler numbers and polynomials. Adv Stud Contemp Math. 21, 47–54 (2011)3. Moon, E-J, Rim, S-H, Jin, J-H, Lee, S-J: On the symmetric properties of higher-order twisted q-Euler numbers andpolynomials. Adv Diff Equ 2010, 8 (2010). (Art ID 765259)4. Cangul, IN, Kurt, V, Ozden, H, Simsek, Y: On the higher-order w-q-Genocchi numbers. Adv Stud Contemp Math. 19, 39–57(2009)5. Jang, LC: A note on Nörlund-type twisted q-Euler polynomials and numbers of higher order associated with fermionicinvariant q-integrals. J Inequal Appl 2010, 12 (2010). (Art ID 417452)6. Kim, T: q-Volkenborn integration. Russ J Math Phys. 9, 288–299 (2002)7. Kim, T: A note on q-Volkenborn integration. Proc Jangjeon Math Soc. 8, 13–17 (2005)8. Kim, T, Choi, J, Kim, YH: On extended Carlitz’s type q-Euler numbers and polynomials. Adv Stud Contemp Math. 20,499–505 (2010)9. Kim, T: The modified q-Euler numbers and polynomials. Adv Stud Contemp Math. 16, 161–170 (2008)10. Kim, T: q-Euler numbers and polynomials associated with p-adic q-integrals. J Nonlinear Math Phys. 14, 15–27 (2007).doi:10.2991/jnmp.2007.14.1.311. Kim, T: Note on the Euler q-zeta functions. J Number Theory. 129, 1798–1804 (2009). doi:10.1016/j.jnt.2008.10.00712. Kim, T: Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adicintegral on ℤp. Russ J Math Phys. 16, 484–491 (2009). doi:10.1134/S106192080904003713. Kim, T: Barnes type multiple q-zeta function and q-Euler polynomials. J Phys A: Math Theor 43, 11 (2010). (Art ID 255201)14. 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 864247)15. Kim, T, Choi, J, Kim, YH, Ryoo, CS: A note on the weighted p-adic q-Euler measure on ℤp. Adv Stud Contemp Math. 21,35–40 (2011)16. Kurt, V: A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials. Appl Math Sci. 3, 53–56 (2009)doi:10.1186/1687-1847-2012-21Cite this article as: Lee et al.: Some identities on the higher-order-twisted q-Euler numbers and polynomials withweight a. Advances in Difference Equations 2012 2012:21.Lee et al. Advances in Difference Equations 2012, 2012:21http://www.advancesindifferenceequations.com/content/2012/1/21Page 10 of 10

Some identities on the higher-order-twisted q-Euler numbers and polynomials with weight α

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Some identities on the higher-order-twisted q-Euler numbers and polynomials with weight α

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