In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-type q-Euler polynomials and their interpolation functions. In this paper, we give another definition
of the multiple Hurwitz type q-zeta function. This function interpolates Nörlund-type q-Euler polynomials at negative integers. We also give some identities related to these polynomials
and functions. Furthermore, we give some remarks about approximations of Bernoulli and
Euler polynomials.
1. Introduction, Definitions and Notations
The classical Euler numbers and polynomials have been studied by many mathematicians, which are defined as follows, respectively,
2et+1=∑n=0∞Entnn!,|t|<π,2etxet+1=∑n=0∞En(x)tnn!,|t|<π,
cf.[1–51]. Observe that En(0)=En.
These numbers and polynomials are interpolates by the Euler zeta function and Hurwitz-type-zeta functions, respectively,
ζE(s)=∑n=1∞(-1)nns,s∈ℂ,ζE(s,x)=∑n=0∞(-1)n(n+x)s,s∈ℂ.
Let [x]=[x:q]=1-qx/1-q. Observe that limq→1[x]=x, cf. [3, 47].
Various kinds of the q-analogue of the Euler numbers and polynomials, recently, have been studied by many mathematicians. In this paper, we use Kim's [13, 21] and Simsek's [43] methods. By using p-adic q-Volkenborn integral [12], Kim [13, 26] constructed many kinds of generating functions of the q-Euler numbers and polynomials and their interpolation functions. He also gave many applications of these numbers and functions. Simsek [40, 43] studied on the generating functions of the Euler numbers and Bernoulli numbers. By using these generating functions, Simsek constructed q-Dedekind-type sums and q-Hardy-type sums as well.
Recently, Cangul et al. gave higher-order q-Genocchi numbers and their interpolation functions. Applying p-adic q-fermionic integral on p-adic integers, they also gave Witt's formula of these numbers.
In [21], by using multivariate fermionic p-adic integral on ℤp, Kim constructed generating function of the Nörlund-type q-Euler polynomials of higher order. Main motivation of this paper is to construct interpolation function of the Nörlund-type q-Euler polynomials. Therefore, we firstly give generating function of the Nörlund-type q-Euler polynomials.
Kim [21] defined Nörlund type q-extension Euler polynomials of higher order. He gave many applications and interesting identities. We give some of them in what follows.
Let q∈ℂ with |q|<1:
Fq(t,x)=2∑m=0∞(-1)me[m+x]t=∑n=0∞En,q(x)tnn!.
Observe that Fq(t)=Fq(t,0). Hence, we have En,q(0)=En,q. If q→1 into (1.5), then we easily obtain (1.2).
Higher-order q-Euler polynomials of the Nörlund type are defined by Kim [21]. He gave generating functions related to Euler numbers of higher-order. In this paper, we use generating functions in [21]. Especially, we can use the following generating function, which are proved by Kim [21, Theorem 2.3, page 5].
Theorem 1.1 ([21, Theorem 2.3, page 5]).
For r∈N, and n≥0, one has
Fq(r)(t,x)=2r∑m=0∞(-1)m(m+r-1m)e[m+x]t=∑n=0∞En,q(r)(x)tnn!.
It is noted that if r=1, then (1.6) reduces to (1.5).
Remark 1.2.
In (1.6); we easily see that
limq→1Fq(r)(t,x)=2r∑m=0∞(-1)m(m+r-1m)e(m+x)t=2rext∑m=0∞(-1)m(m+r-1m)emt=2rext(1+et)r=F(r)(t,x).
From the above, we obtain generating function of the Nörlund Euler numbers of higher order. That is
F(r)(t,x)=2rext(1+et)r=∑n=0∞En(r)(x)tnn!.
Thus, we have
limq→1En,q(r)(x)=En(r)(x).
cf.[21].
In Section 2, we study on modified generating functions of higher-order Nörlund-type q-Euler polynomials and numbers. We obtain some relations related to these numbers and polynomials.
In Section 3, we give interpolation functions of the higher order Nörlund-type q-Euler polynomials.
In Section 4, we obtain some relations related to he higher order Nörlund-type q-Euler polynomials.
In Section 5, we give remarks and observations on an Approximation theory related to Bernoulli and Euler polynomials.
2. Modified Generating Functions of Higher-Order Nörlund-Type q-Euler Polynomials and Numbers
In this section we define generating function of modified higher order Nörlund type q-Euler polynomials and numbers, which are denoted by En,q(r)(x), and En,q(r) respectively. We give relations between these numbers and polynomials.
We modify (1.6) as follows:
ℱq(r)(t,x)=Fq(r)(q-xt,x),
where Fq(r)(t,x) is defined in (1.6). From the above we find that
ℱq(r)(t,x)=∑n=0∞q-nxEn,q(r)(x)tnn!.
After some elementary calculations, we obtain
ℱq(r)(t,x)=exp([x]q-xt)𝔣q(r)(t),
where
𝔣q(r)(t)=(2r∑m=0∞(-1)m(m+r-1m)e[m]t)=∑n=0∞En,q(r)tnn!.
From the above we have
ℱq(r)(t,x)=∑n=0∞εn,q(r)(x)tnn!,
where
εn,q(r)(x)=q-nxEn,q(r)(x).
By using Cauchy product in (2.3), we arrive at the following theorem.
Theorem 2.1.
For r∈ℕ, and n≥0, one has
εn,q(r)(x)=∑j=0n(nj)qjx[x]n-kEj,q(r).
By using (2.7), we easily obtain the following result.
Corollary 2.2.
For r∈ℕ, and n≥0, one has
εn,q(r)(x)=∑m=0∞∑j=0n∑l=0n-j(nj,l,n-j-l)(n-j+m-1m)(-1)lqm+x(l+j)Ej,q(r).
We now give some identity related to Nörlund type Euler polynomials and numbers of higher-order.
Substituting x=0 into (1.10), we find that
En(r)=2r∑n1,n2,…,nr=0∞∑j1,…,jr=0j1+⋯+jr=n(nj1,…,jr)(-1)n1+n2+⋯+nr∏k=0rnkjk.
By (1.10) and (2.9), we arrive at the following theorem.
Theorem 2.3.
For r∈ℕ, and n≥0, one has
En(r)=∑j=0n(nj)(-x)n-jEj(r)(x).
By using (1.10) and [35, Theorem 3.6, page 7] , we easily arrive at the following result.
Corollary 2.4.
For r, v∈ℕ, and n≥0, one has
(E(r)(x)+E(v)(y))n=∑j=0n(nj)xn-jEj(r+v)(y),
where (E(r)(x))n is replaced by En(r)(x).
3. Interpolation Function of Higher-Order Nörlund-Type q-Euler Polynomials
Recently, higher-order Bernoulli polynomials and Euler polynomials have studied by many mathematicians. Especially, in this paper, we study on higher-order Euler polynomials which are constructed by Kim (see, e.g., [14, 17, 21, 24, 27, 28, 33]) and see also the references cited in each of these earlier works.
In [20], by using the fermionic p-adic invariant integral on ℤp, the set of p-adic integers, Kim gave a new construction of q-Genocchi numbers, Euler numbers of higher order. By using q-Genoucchi, Euler numbers of higher order, he investigated the interesting relationship between w-q-Euler polynomials and w-q-Genocchi polynomials. He also defined the multiple w-q-zeta functions which interpolate q-Genoucchi, Euler numbers of higher order.
By using similar method of the papers given by Kim [20, 21], in this section, applying derivative operator dk/dtk|t=0 and Mellin Transformation to the generating functions of the higher-order Nörlund-type q-Euler polynomials, we give interpolation function of these polynomials.
By applying operator dk/dtk|t=0 to (1.6), we obtain the following theorem.
Theorem 3.1.
Let r,k∈ℤ+ and x∈ℝ with 0<x≤1. Then one has
Ek,q(r)(x)=2r∑m=0∞(-1)m(m+r-1m)[m+x]k.
Let us define interpolation function of higher-order Nörlund-type q-Euler numbers as follows. Definition 3.2.
Let q,s∈ℂ with |q|<1, and 0<x≤1. Then we define
ζq(r)(s,x)=2r∑n=0∞(-1)n(n+r-1n)[n+x]s.
Remark 3.3.
It holds that
limq→1ζq(r)(s,x)=2r∑n=0∞(-1)n(n+r-1n)[n+x]s.
For detail about the above function (see [5–38, 44, 47]). By applying dk/dtk|t=0 derivative operator to (1.10), we easily see that
ζ(r)(s,x)=2r∑n1,…,nr=0∞(-1)n1+⋯+nr(∑j=1rnj+x)s,
where s∈ℂ.
The functions in (3.3) and (3.4) interpolate same numbers at negative integers. That is, these functions interpolate higher-order Nörlund-type Euler numbers at negative integers. So, by (3.3), we modify (3.4) in sense of q-analogue.
In [3–51], many authors extensively have studied on similar type of (3.4).
In (3.3) and (3.4), setting r=1, we have
ζ(1)(s,x)=2∑n=0∞(-1)n(n+x)s=ζE(s,x)
where ζE(s,x) denotes Hurwitz type Euler zeta function, which interpolates classical Euler polynomials at negative integers.Theorem 3.4.
Let n∈ℤ+. Then one has
ζq(r)(-n,x)=En,q(r)(x).
Proof.
Substituting s=-k,k∈ℤ+ into (3.2). Then we have
ζq(r)(-k,x)=2r∑n=0∞(-1)n(n+r-1n)[n+x]k.
Setting (3.1) into the above, and after some elementary calculations, we easily arrive at the desired result.
By applying the Mellin transformation to (2.5), we find that
1Γ(s)∫0∞ts-1ℱq(r)(-t,x)dt=2r∑m=0∞(-1)m(m+r-1m)qnxs[m+x]s.
From the above we define the following function, which interpolate En,q(r)(x) at negative integers.
Definition 3.5.
Let q,s∈ℂ with |q|<1, and 0<x≤1. Then we define
𝒵q(r)(s,x)=2r∑m=0∞∑j=0∞(-1)m(r+m-1m)(s+j-1j)qx(ns+j)+mj.
Theorem.
Let n∈ℤ+. Then one has
𝒵q(r)(-n,x)=εn,q(r)(x).
By Theorems 5, 6, a relation between the functions ζq(r)(-n,x) and ζq(r)(-n,x) is given by the following corollary.
Corollary 3.7.
𝒵q(r)(-n,x)=q-nxζq(r)(-n,x).
Remark 3.8.
Recently many authors have studied on the Riemann zeta function, Hurwitz zeta function, Lerch zeta function, Dirichlet series for the polylogarithm function and Dirichlet's eta function and the other functions. The Lerch trancendentΦ(z,s,a) is the analytic continuation of the series
Φ(z,s,a)=1as+z(a+1)s+z(a+2)s+⋯=∑n=0∞zn(n+a)s,
which converges for (a∈ℂ∖‚ℤ0-, s∈ℂ when |z|<1; ℜ(s)>1 when |z|=1), where as usual
ℤ0-=ℤ-∪{0},ℤ-={-1,-2,-3,…}.
However, Φ denotes the familiar Hurwitz-Lerch Zeta function (cf. e. g., [8], [49, page 121 et seq.] ). Some special cases of the function Φ(z,s,a) are given by the following relations (e.g., and details see [8], [49, page 121 et seq.] ):
the Riemann zeta function
Φ(1,s,1)=ζ(s)=∑n=1∞1ns,ℜ(s)>1,
the Hurwitz zeta function
Φ(1,s,a)=ζ(s,a)=∑n=0∞1(n+a)s,ℜ(s)>1,
the Dirichlet's eta function
Φ(-1,s,1)=ζ*(s)=∑n=1∞(-1)n-1ns,
the Dirichlet beta function
Φ(-1,s,1/2)2s=β(s)=∑n=0∞(-1)n(2n+1)s,
the Legendre chi function
zΦ(z2,s,1/2)2s=χs(z)=∑n=0∞z2n+1(2n+1)s,(|z|≤1;ℜ(s)>1),
the polylogarithm
zΦ(z,n,1)=Lim(z)=∑n=0∞zknm,
the Lerch zeta function (sometimes called the Hurwitz-Lerch zeta function)
L(λ,α,s)=Φ(e2πiλ,s,α),
which is a special function and generalizes the Hurwitz zeta function and polylogarithm cf. [6, 8, 20, 46, 49] and see also the references cited in each of these earlier works. Consequently, the functions 𝒵q(r)(-n,x) and ζq(r)(-n,x) are related to the Hurwitz-Lerch zeta function and the other special functions, which are defined above:
2Φ(-1,s,x)=ζ(1)(s,x)=ζE(s,x).
4. Some Relations Related to Higher-Order Nörlund q-Euler Polynomials
In this section, by using generating function of higher-order Nörlund q-Euler polynomials, which is defined by Kim [20, 21], we obtain the following identities.
Theorem 4.1.
Let q∈ℂ with |q|<1. Let r be a positive integer. Then one has
Ek,q(r)(x)=2r∑j=0k∑a=0j(-1)a(ka,j-a,k-j)qja[x]k-j(1-q)j(1+qk-j)r-1.
Proof.
By using (3.1), we have
Ek,q(r)(x)=2r∑m=0∞(-1)m(m+r-1m)([m]+qm[x])k=2r∑m=0∞(-1)m(m+r-1m)∑j=0k(kj)[m]jqm(k-j)[x]k-j=2r∑m=0∞(-1)m(m+r-1m)∑j=0k(kj)(1-qm)j(1-q)jqm(k-j)·[x]k-j=2r∑m=0∞(-1)m(m+r-1m)∑j=0k∑a=0j(kj)(ja)(-1)aqaj+m(k-j)·[x]k-j(1-q)j=2r∑j=0k∑a=0j(kj)(ja)(-1)aqja·[x]k-j(1-q)j∑m=0∞(-1)m(m+r-1m)qm(k-j)=2r∑j=0k∑a=0j(kj)(ja)(-1)aqja·[x]k-j(1-q)j(1+qk-j)r-1=2r∑j=0k∑a=0j(-1)a(ka,j-a,k-j)qja[x]k-j(1-q)j(1+qk-j)r-1.
Thus, we complete the proof.
Theorem 4.2.
Let q∈ℂ with |q|<1. Let r be a positive integer. Then one has
En,q(r)(x)=∑j=0n(-1)j(nj)qjx(1-qj)-r(1-q)-n.
Proof.
By using (1.6)
Fq(r)(t,x)=2r∑m=0∞(m+r-1m)(-1)me[m+x]t=2r∑m=0∞∑n=0∞(m+r-1m)(-1)m(1-qm+x1-q)ntnn!=2r∑m=0∞∑n=0∞(m+r-1m)(-1)m(1-q)nn!(∑j=0n(nj)(-1)j·qjx+jm)tn=2r∑n=0∞∑j=0∞(nj)(-1)j.qjx(1-q)n·tnn!∑m=0∞(m+r-1m)(-1)m·qjm
Thus we have;
∑n=0∞En,q(r)(x)tnn!=2r∑n=0∞(∑j=0n(nj)(-1)jqjx(1-qj)-r(t1-q)n1n!).
By comparing the coefficients tn/n! both sides in the above, we arrive at the desired result.
Theorem 4.3.
Let r,y∈ℤ+. Then one has
∑j=0k(kj)Ej,q(r)(x)Ek-j,q(y)(x)=2r+y∑n=0∞∑j=0n(-1)n(j+r-1j)(n-j+y-1n-j)([x+y]+[n-j+x])k.
Proof.
By using (1.6), we have
∑n=0∞En,q(r)(x)tnn!∑n=0∞En,q(y)(x)tnn!=2r+y∑n=0∞(-1)n(n+r-1n)e[n+x]t∑n=0∞(-1)n(n+y-1n)e[n+x]t.
By using Cauchy product into the above, we obtain
∑n=0∞(∑j=0nEj,q(r)(x)En-j,q(y)(x)1j!(n-j)!)tn=2r+y∑n=0∞(∑j=0n(j+r-1j)(n-j+y-1n-j)(-1)ne[j+x]te[n-j+x]t),
From the above, we have
∑m=0∞(∑j=0mEj,q(r)(x)Em-j,q(y)(x)1j!(m-j)!)tm=∑m=0∞(2r+y∑n=0∞∑j=0n(j+r-1j)(n-j+y-1n-j)([j+x]+[n-j+x])m)tm
By comparing the coefficients of both sides of tn in the above we arrive at the desired result.
Remark 4.4.
In (4.1) setting y=1, we have
∑j=0m(mj)Ej,q(r)(x)Em-j,q(x)=2r+1∑n=0∞(-1)n∑j=0n(j+r-1j)([j+x]+[n-j+x])m.
The above relations give us (3.1) related to (4.1).
5. Further Remarks and Observations on Approximation
Apostol [1, page 481] gave Weierstrass theorem as follows.
Theorem 5.1.
Let f be real valued and continuous on a closed interval [a,b]. Then, given any ε>0, there exists a polynomial p (which may be depend on ε) such that
|f(x)-p(x)|<ε,
for every x∈[a,b].
According to Apostol [1]; the above theorem is described by saying that every continuous function can be “uniformly approximated” by a polynomial.
We now give, more useful, and more interesting result concerning the approximation by polynomials which is related to the Bernstein polynomials (cf. [1, 2, 4, 34, 39]).
Definition 5.2.
([2]) Let f be a function with domain I=[0,1] and range R. The nth Bernstein polynomial for f is defined to be
Bn(x)=Bn(f;x)=∑k=0nf(kn)(nk)xk(1-x)n-k.
These Bernstein polynomials are not only used with probability (the Binomial Distribution) but also used in the approximation theory.
Let f be continuous on I with values in ℝ. Then the sequence of Bernstein polynomials for f, defined in (5.2) converges uniformly on I to f (cf.[2]).
In [4], Costabile and Dell'Accio collected classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only. Their problem is considered from the point of view of the existence of explicit formulas, interpolation to boundary data, bounds for the remainder and convergence of the polynomial series. Applications to some problems of numerical analysis are pointed out, such as nonlinear equations, numerical differentiation and integration formulas, specially associated differential boundary value problems. Some polynomial expansions for smooth enough functions defined in rectangles or in triangles of ℝ2 as well as in cuboids or in tetrahedrons in ℝ3 and their applications are also discussed. They also used Bernoulli and Euler polynomials for the polynomial approximation cf. see for detail [4].
Lopez and Temme [34] studied on uniform approximations of the Bernoulli and Euler polynomials for large values of the order in terms of hyperbolic functions. They obtained convergent expansions for
Bn(nz+12),En(nz+12)
in powers of 1/n, and coefficients are rational functions of z and hyperbolic functions of argument 1/2z, here Bn(x) and En(x) denote Bernoulli and Euler polynomials, respectively. Their expansions are uniformly valid for |z±i/2π|>1/2π and |z±i/π|>1/π, respectively. For a real argument, the accuracy of these approximations is restricted to the monotonic region cf. see for detail [34].
Recently, many authors studied on very different type of the approximation theory. Consequently, by using the above motivations, we conclude this section by the following questions:
Bernoulli functions and Euler functions are related to trigonometric polynomials cf. [46]. Approximation by q-analogue of these functions may be possible.
whether or not define better uniform approximations for the Nörlund q-Euler polynomials higher order;
is it possible to define uniform expansions of the Nörlund q-Euler polynomials higher order?
Acknowledgment
Y. Simsek is supported by the research fund of Akdeniz University.
ApostolT. M.1957Reading, Mass, USAAddison-Wesleyxii+553MR0087718ZBL0077.05501BartleR. G.19762ndNew York, NY, USAJohn Wiley & Sonsxv+480MR0393369ZBL0309.26003CenkciM.CanM.Some results on q-analogue of the Lerch zeta function2006122213223MR2213080ZBL1098.11016CostabileF. A.costabil@unical.itDell'AccioF.fdellacc@unical.itPolynomial approximation of CM functions by means of boundary values and applications: a survey20072101-211613510.1016/j.cam.2006.10.059CangulI. N.KurtV.OzdenH.SimsekY.On higher order w-q Genocchi numbers2009191CangulI. N.cangul@uludag.edu.trOzdenH.hozden@uludag.edu.trSimsekY.ysimsek@akdeniz.edu.trA new approach to q-Genocchi numbers and their interpolation functionsNonlinear Analysis: Theory, Methods & Applications. In press10.1016/j.na.2008.11.040CangulI. N.cangul@uludag.edu.trOzdenH.hozden@uludag.edu.trSimsekY.simsekyil63@yahoo.comGenerating functions of the (h,q) extension of twisted Euler polynomials and numbers20081203281299MR242952510.1007/s10474-008-7139-1GuilleraJ.SondowJ.Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent2008163247270MR2429900JangL.-C.KimS.-D.ParkD. W.RoY. S.A note on Euler number and polynomials2006200653460210.1155/JIA/2006/34602ZBL1091.11007MR2221227JangL.KimT.q-Genocchi numbers and polynomials associated with fermionic p-adic invariant integrals on ℤp20082008823218710.1155/2008/232187MR2407281ZBL1149.11010KimT.JangL.-C.RyooC.-S.Note on q-extensions of Euler numbers and polynomials of higher order20082008937129510.1155/2008/371295MR2379521KimT.q-Volkenborn integration200293288299MR1965383ZBL1092.11045KimT.Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials20031019198ZBL1072.11090MR2013106KimT.tkim@kongju.ac.krOn Euler-Barnes multiple zeta functions2003103261267MR2012900ZBL1038.11058KimT.A note on the q-multiple zeta function200482111113MR2042852ZBL1081.11061KimT.tkim@kongju.ac.krAnalytic continuation of multiple q-zeta fand their values at negative integers20041117176MR2134649ZBL1115.11068KimT.tkim@kongju.ac.krq-generalized Euler numbers and polynomials2006133293298MR226283110.1134/S1061920806030058KimT.tkim@knu.ac.krOn the analogs of Euler numbers and polynomials associated with p-adic q-integral on ℤp at q=−120073312779792MR2313680ZBL1120.1101010.1016/j.jmaa.2006.09.027KimT.tkim@kongju.ac.krOn the q-extension of Euler and Genocchi numbers2007326214581465MR228099610.1016/j.jmaa.2006.03.037ZBL1112.11012KimT.New approach to q-Genocch, Euler numbers and polynomials and their interpolation functions2009182105112KimT.Some identities on the q-Euler polynomials of higher order and
q-Stirling numbers by the fermionic p-adic integral on
ℤpto appear in Russian Journal of Mathematical PhysicsKimT.tkim@knu.ac.krq-Euler numbers and polynomials associated with p-adic q-integrals20071411527MR228783110.2991/jnmp.2007.14.1.3ZBL1158.11009KimT.A note on p-adic q-integral on ℤp associated with q-Euler numbers2007152133137ZBL1132.11369MR2356172KimT.q-extension of the Euler formula and trigonometric functions2007143275278MR234177510.1134/S1061920807030041KimT.The modified q-Euler numbers and polynomials2008162161170MR2404632KimT.tkim@knu.ac.krEuler numbers and polynomials associated with zeta functions200820081158158210.1155/2008/581582ZBL1145.11019MR2407279KimT.tkkim@kw.ac.krOn the multiple q-Genocchi and Euler numbers2008154481486MR247085010.1134/S1061920808040055KimT.tkim@knu.ac.krq-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients20081515157MR2390694KimT.ChoJ.-S.A note on multiple Dirichlet's q-L-function20051115760MR2151284ZBL1078.11013KimT.RimS.-H.On Changhee-Barnes' q-Euler numbers and polynomials2004928186MR2090111ZBL1065.11010KimT.KimY.-H.HwangK.-W.Note on the generalization of the higher order q-Genocchi numbers and q-Euler numberspreprint, http://arxiv.org/abs/0901.1697KimT.KimM.-S.JangL.RimS.-H.New q-Euler numbers and polynomials associated with p-adic q-integrals2007152243252MR2356180KimT.tkim64@hanmail.netChoiJ. Y.abyss1225@hanmail.netSugJ. Y.jysug@knu.ac.krExtended q-Euler numbers and polynomials associated with fermionic P-adic q-integral on ℤp2007142160163MR2318827ZLB1132.3333110.1134/S1061920807020045LópezJ. L.jllopez@posta.unizar.esTemmeN. M.nicot@cwi.nlUniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions19991033241258MR171263310.1111/1467-9590.00126ZBL1006.41003OzdenH.CangulI. N.SimsekY.Remarks on sum of products of (h,q)-twisted Euler polynomials and numbers20082008881612910.1155/2008/816129MR2379516ZBL1140.11314OzdenH.CangulI. N.SimsekY.Multivariate interpolation functions of higher-order q-Euler numbers and their applications200820081639085710.1155/2008/390857MR2393118ZBL1140.11313OzdenH.hozden@uludag.edu.trSimsekY.ysimsek@akdeniz.edu.trA new extension of q-Euler numbers and polynomials related to their interpolation functions2008219934939MR2436527ZBL1152.1100910.1016/j.aml.2007.10.005RimS.-H.shrim@knu.ac.krKimT.tkim@kongju.ac.krA note on q-Euler numbers associated with the basic q-zeta function2007204366369MR230336210.1016/j.aml.2006.04.019OstrovskaS.On the q-Bernstein polynomials2005112193204MR2169894ZBL1116.41013SimsekY.q-analogue of twisted l-series and q-twisted Euler numbers20051102267278ZBL1114.11019MR212260910.1016/j.jnt.2004.07.003SimsekY.yilmazsimsek@hotmail.comq-Dedekind type sums related to q-zeta function and basic L-series20063181333351MR221089210.1016/j.jmaa.2005.06.007ZBL1149.11054SimsekY.Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h,q)-zeta function and L-function20063242790804ZBL1139.11051MR226508110.1016/j.jmaa.2005.12.057SimsekY.Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions2008162251278MR2404639SimsekY.ysimsek@akdeniz.edu.trq-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functionsNonlinear Analysis: Theory, Methods & Applications. In press10.1016/j.na.2008.11.014SimsekY.Complete sums of products of (h,q)-extension of Euler numbers and polynomialsto appear in Journal of Difference Equations and Applicationshttp://arxiv.org/abs/0707.284910.1080/10236190902813967SimsekY.Special functions related to dedekind type DC-sums and their applicationspreprint, http://arxiv.org/abs/0902.0380SrivastavaH. M.harimsri@math.uvic.caKimT.tkim@mail.kongju.ac.krSimsekY.ysimsek@mersin.edu.trq-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series2005122241268MR2199005SrivastavaH. M.harimsri@math.uvic.caRemarks on some relationships between the Bernoulli and Euler polynomials2004174375380MR204574010.1016/S0893-9659(04)90077-8ZBL1070.33012SrivastavaH. M.ChoiJ.2001Dordrecht, The NetherlandsKluwer Academic Publishersx+388MR1849375ZBL1014.33001SitaramachandraraoR.Dedekind and Hardy sums197848325340ZhaoJ.Multiple q-zeta functions and multiple q-polylogarithms2007142189221MR2341851