Computers and Mathematics with Applications 56 (2008) 898–908
www.elsevier.com/locate/camwa
Interpolation function of the (h, q)-extension of
twisted Euler numbers
Hacer Ozdena, Yilmaz Simsekb,∗
aUniversity of Uludag, Faculty of Arts and Science, Department of Mathematics, Bursa, Turkey
bUniversity of Akdeniz, Faculty of Arts and Science, Department of Mathematics, Antalya, Turkey
Received 24 May 2007; received in revised form 31 July 2007; accepted 7 January 2008
Abstract
In [H. Ozden, Y. Simsek, I.N. Cangul, Generating functions of the (h, q)-extension of Euler polynomials and numbers, Acta
Math. Hungarica, in press (doi:10.1007/510474-008-7139-1)], by using p-adic q-invariant integral on Zp in the fermionic sense,
Ozden et al. constructed generating functions of the (h, q)-extension of Euler polynomials and numbers. They defined (h, q)-Euler
zeta functions and (h, q)-Euler l-functions. They also raised the following problem: “Find a p-adic twisted interpolation function
of the generalized twisted (h, q)-Euler numbers, E(h)n,χ,w(q)”. The aim of this paper is to give a partial answer to this problem.
Therefore, we constructed t∑wisted (h, q)-partial zeta function and twisted p-adic (h, q)-Euler l-functions:∞
(h) χ(m)(−1)mξmqhmlE p q (s, χ) = 2, ,ξ, s ,
m m=1
(m,p)=1
which interpolate (h, q)-extension of Euler numbers, at negative integers:
l(h) (h) n (h)( pE p q −n, χ) = En,ξ,χ (q)− p χ, ,ξ, n n(p)En,ξ p,χ (q ).n
By using this interpolation function and twisted (h, q)-partial zeta function, we proved distribution relations of the (h, q)-extension
of generalized Euler polynomials. Consequently we find a partial answer to the above question.
©c 2008 Elsevier Ltd. All rights reserved.
Keywords: p-adic q-deformed fermionic integral; Twisted q-Euler numbers and polynomials; Zeta and l-functions; Twisted p-adic (h, q)-l-
functions
1. Introduction, definitions and notations
p-adic numbers were invented by Kurt Hensel around the end of the nineteenth century. In spite of their being
already one hundred years old, these numbers are still today enveloped in an aura of mystery within the scientific
community. Although they have penetrated several mathematical fields, Number Theory, Algebraic Geometry,
Algebraic Topology, Analysis, Mathematical Physics, String Theory, Field Theory, Stochastic Differential Equations
∗ Corresponding author.
E-mail addresses: hozden@uludag.edu.tr (H. Ozden), ysimsek@akdeniz.edu.tr, simsekyil63@yahoo.com (Y. Simsek).
0898-1221/$ - see front matter©c 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2008.01.020
H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908 899
on real Banach Spaces and Manifolds and other parts of the natural sciences, they have yet to reveal their full potential
in (for example) physics. While solving mathematical and physical problems and while constructing and investigating
measures on manifolds, the p-adic numbers are used. There is an unexpected connection of the p-adic Analysis with
q-Analysis and Quantum Groups, Quantum Top, and thus with Noncommutative Geometry, and q-Analysis is a sort
of q-deformation of the ordinary analysis. Spherical functions on Quantum Groups are q-special functions cf. [2–25].
Kubota and Leopoldt proved the existence of meromorphic functions, L p(s, χ), which is defined over the p-adic
number field. This function interpolates the nth generalized Bernoulli numbers associate with the primitive Dirichlet
character χ , and χn = χw−n , with the Teichmüller character cf. [3–5,8,11,12,16,19,21,23,25–33]. Young [24] defined
p-adic integral representation for the two-variable p-adic L-functions, introduced by Fox [4]. For powers of the
Teichmüller character, he used the integral representation to extend the L-function to the large domain, in which it is
a meromorphic function in the first variable and an analytical element in the second. These integral representations
imply systems of congruences for the generalized Bernoulli polynomials. In [12], Kim constructed the two-variable p-
adic q-L-function, which interpolates the generalized q-Bernoulli polynomials. This function is the q-extension of the
two-variable p-adic L-function. He gave a p-adic integral representation for this two-variable p-adic q-L-function.
He also derived q-extension of the generalized formula of Diamond and Ferrero and Greenberg formula for the two
variable p-adic L-function in terms of the p-adic gamma and log gamma functions.
Twisted q-Bernoulli and Euler numbers and polynomials are very important not only in Mathematics and Statistics,
but also Mathematical Physics. Recently, these numbers and polynomials have been studied by several authors cf.
[9,19,21,30,31,34–38]. In [10,39], by using q-Volkenborn integration, Kim constructed the new (h, q)-extension of
the Bernoulli numbers and polynomials. He defined (h, q)-extension of the zeta functions which interpolate new
(h, q)-extension of the Bernoulli numbers and polynomials. In [21,40], the second author defined (h, q)-extension
of Bernoulli numbers and polynomials. He also constructed their interpolation functions at negative integers. Ozden
et al. [41,42] studied on (h, q)-extension of Euler numbers and polynomials. In [29], Kim and Rim constructed the
q-twisted Euler numbers and polynomials, by using p-adic q-invariant integral on Zp in the fermionic sense. They
also defined interpolation functions of them at negative integers, explicitly.
In this paper, we define twisted (h, q)-partial zeta function, which interpolate (h, q)-extension of Euler polynomials
at negative integers. We find the relation between twisted (h, q)-partial zeta function and the twisted (h, q)-zeta func-
tion. By using this function, we constructed twisted p-adic (h, q)-Euler l-functions, which interpolate (h, q)-extension
of Euler numbers, at negative integers. This result gave us a partial answer of the problem Ozden et al. [1], which is
given by: “Find a p-adic twisted interpolation function of the generalized twisted h q -Euler numbers, E (h)( , ) n,χ,w(q).”
Ozden et al. [1], by using p-adic q-Volkenborn integration, they constructed generating functions of the new
twisted (h, q)-Euler numbers and generalized twisted (h, q)-Euler numbers and polynomials. By applying the Mellin
transformation and derivative operator to these generating functions, they defined integral representation of the new
twisted (h, q)-zeta functions and twisted (h, q)-lE -functions which interpolate twisted (h, q)-Euler numbers and
generalized twisted (h, q)-Euler numbers at nonpositive integers.
Throughout this paper Z, Zp, Qp and Cp will denote the ring of rational integers, the ring of p-adic integers, the
field of p-adic rational numbers and the completion of the algebraic closure of Qp, respectively. Let Z Z++ = ∪ {0}.
Let vp be the normalized exponential valuation of Cp with −v (p) −1| p | pp = p = p . When one talks of q-extension, q
is variously considered as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ Cp, then we
normally assume | q − 1 |p < 1, so that qx = exp(x log q) for | x |p ≤ 1. If q ∈ C, then we assume that | q |< 1. Let1− qx , if q 6= 1[x]q = 1− qx, if q = 1.
For
f ∈ UD(Zp,Cp) = { f | f : Zp → Cp is uniformly differentiable function},
the p-adic invar∫iant q-integral on Zp was defined by Kim [10,28] as follows:p∑N1 −1
Iq( f ) = f (x)dµq(x) lim qx= N f (x),Zp N→∞ [p ]q x=0
900 H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908
where and p is a odd prime number. Recently, many applications of this integral have been given. For detail see cf.
[10,11,21–23,32,34,39–49].
The q-deformed p-adic invar∫iant integral on Zp, in the fermionic sense is defined by [28]:p∑N−1
I−1( f ) = lim Iq( f ) = f (x)dµ−1(x) = lim (−1)x f (x). (1.1)
q→−1 Zp N→∞ x=0
By using p-adic q-integral on Zp, Kim[44] proved the integral equations related to the p-adic q-integral.
Let p be a fixed prime. For a fixed positive integer f with (p, f ) = 1, we set
X = X f = limZ/ f pNZ,
←−
N
X1 = Z⋃p,
X∗ = a + f pZp
0<a< f p
(a,p)=1
and { }
a f pN+ Zp = x ∈ X | x ≡ a(mod f pN ) ,
where∫a ∈ Z satisfies the co∫ndition 0 ≤ a < f pN . For f ∈ UD(Zp,Cp),
f (x)dµ−1(x) = f (x)dµ−1(x), cf. [14,27–29,39,41,49].
Zp X
From (1.1), Kim [28] gave the following interesting identity:
I−1( f1)+ I−1( f ) = 2 f (0), (1.2)
where f1(x) = f (x + 1). ∑n−1
I ( f ) ( 1)n−1 n−1−x−1 n + − I−1( f ) = 2 (−1) f (x), (1.3)
x=0
where fn(x) = f (x + n), n +∈ Z .
In [1], Ozden et al. constructed generating functions of the twisted (h, q)-Euler numbers and polynomials. By
using these generating functions and the q-deformed p-adic invariant integral on Zp in the fermionic sense, they
defined twisted (h, q)-Euler numbers, polynomials and twisted generalized (h, q)-Euler numbers and polynomials
with attached to Dirichlet character. Moreover, they proved twisted version of Witt’s formula for the (h, q)-Euler
numbers and polynomials.
Let ⋃
Tp = C pn = lim C pn ,
n→∞
n≥1{ n }
where Tpn = ξ ∈ C pp | ξ = 1 is the cyclic group of order pn . For ξ ∈ Tp, we denote by φξ : Zp → Cp the
locally constant function x ξ x→ (see [9,21,29,36]).
We give application of the q-deformed p-adic invariant integral on Zp in the fermionic sense related to [1]. By
substituting f (h)ξ,q(x, t) ξ xqhxet x= into (1.2), we define twisted (h, q)-extension of Euler numbers, En,ξ (q) by means
of the following generating function: ∑∞ n
F (h)(t) I (ξ xqhxet x
2 (h) t
ξ,q = −1 ) = h t = En,ξ (q) cf. [1]. (1.4)ξq e + 1 n!
n=0
We note that if ξ = 1, th∑en E (h) q E (h)n,ξ ( ) = n (q) and F (h) (h) 2ξ,q (t) = Fq (t) = h t cf. [42]. If q → 1, thenq e +1
F (h)
n
q (t) F(t)
2 ∞ t
→ = et 1 = n 1 En n , where En is usual Euler numbers, cf. [23,27,37].+ = !
H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908 901
Twisted (h)(h, q)-extension of Euler polynomials E (z, q) are defined by means of the generating function
∑ n,ξ∞ tn
F (h)ξ,q (t, z) = F
(h) t z (h)
ξ,q (t)e = En,ξ (z, q) cf. [1].n!
n=0
We note that E (h)n,ξ (0
(h)
, q) = En,ξ (q). If 1, then E
(h) (h) (h) (h)
ξ = n,ξ (z, q) → En (z, q) and Fξ,q (t, z) → Fq (t, z) =
F (h)(t)et zq cf. [42].
Twisted versions of Witt’s formula for E (h) (h)n,ξ (z, q) and En,ξ (q) are given by the following theorem:
Theorem 1 ([1])∫. For h ∈ Z, ξ ∈ Tp and q ∈ Cp with | q − 1 |p < 1, we obtain
E (h)n,ξ (q) = ξ
xqhx xndµ−1(x),
Zp
and ∫
E (h)n,ξ (z, q) = ξ
xqhx (x n+ z) dµ−1(x). (1.5)
Zp
Remark 1. If q → 1, ξ = 1, then F (h)ξ,q(t, z) → F(t, z) 2 et z= et 1 and En,ξ (z, q) → En(z), the usual Euler+
polynomials, cf. [23,42].
Theorem 2 ([1]). L∑et n(∈)Z+. Then we haven
(h) nEn,ξ (z, q) = z
n−kE (h)
k k,ξ
(q).
k=0
The generalized twisted (h, q)-extension of Euler polynomials, E (h)n,χ,ξ (z, q) are defined by means of the following
generating function: let χ be a Dirichlet character with conductor f (=odd)
∑f−1
(h) 2( a a−1) χ(a)ξ qhae(z+a)t ∑∞ tnFχ,ξ,q(t z (h), ) = f f h f t = En,χ,ξ (z, q) cf. [1].ξ q e + 1 n!
a=0 n=0
Note that, by substituting z = 0 in the above, E (h)n,χ,ξ (0, q E
(h)
) = n,χ,ξ (q) is the nth twisted (h, q)-extension of Euler
number. If (h) (h)ξ = 1, then En,χ,ξ (q)→ En,χ (q) cf. [42].
E (h)n,χ,ξ (z, q) is given explicitly in the following theorem [1]:
Theorem 3. Let χ be∑a D(iric)hlet character. We haven n
E (h) z q zn−kE (h)n,χ,ξ ( , ) = (q).k k,χ,ξ
k=0
Observe that if ξ = 1, then F (h) (h)χ,ξ,q(t, z) reduces to Fχ,q(t, z):
∑f−1 2( 1)aχ(a)qhae(z+a)t
F (h)
−
χ,q(t, z) = qhe f t
, cf. [42].
+ 1
a=0 ∑ a (a+z)t
If q → 1 in the above, then F t z F f−1 2(−1) χ(a)eχ,q( , )→ χ (t, z) zt= Fχ (t)e = a 0 f t and En,χ (z, q)→ E= e 1 n,χ (z)+
are the usual generalized Euler polynomials cf. [23,42].
Here, we assume that q ∈ C with | q |< 1 and s ∈ C.
902 H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908
Ozden et al. [1] defined twisted (h, q)-extensions of Hurwitz type Euler zeta functions as follows:
Definition 1. Let s ∈ C, z ∈ R+. We define
∑∞
(h) (−1)nξnqnh
ζE,ξ,q(s, z) = 2 s . (1.6)(n + z)
n=0
(h)
ζE,ξ,q(s, z) is an analytical function in the whole complex s-plane.
Remark 2. Observe∑that if z = 1, then twisted (h, q)-extensions of Euler zeta function is defined by∞ n n hn
(h) (−1) ξ q
ζE,ξ,q(s) = 2 s , cf. [42].n
n=1
By (1.6), for ξ = 1, we easily see th∑at∞ n nh
(h) (h) (−1) q
ζE,1,q(s, z) = ζE,q(s, z) = 2 , cf. [42].(n + z)s
n=0
If ξ = 1 and q → 1∑in (1.6), then we obtain∞ ( 1)n−
ζE (s, z) = 2 ,
(n + z)s
n=0
cf. [14,28,29,42]. For q-zeta and q-Hurwitz zeta functions similar results were obtained cf. (see for detail [23,36–39,
49]).
The value of (h)ζE,ξ,q(s, z) at negative integers is given explicitly as follows:
Theorem 4 ([1]). Let n ∈ Z+. Then we have
(h) n z E (h)ζE,ξ,q(− , ) = n,ξ (z, q).
Twisted (h, q)-extensions of l-function is defined by
Definition 2 ([1]). L∑et χ be a Dirichlet character. For s ∈ C, we have∞ n
(h) 2(−1) qnhχ(n)ξnlE,ξ,q(s, χ) = .ns
n=1
l(h)E,ξ,q(s, χ) is an analytical function in the whole complex s-plane.
Note that if ξ =∑1, then, we have in the above∞ n nh
l(h)
2(−1) q χ(n)
E,q(s, χ) = s , cf. [42],n
n=1
and ∑∞
(h) 2( n−1) χ(n)lim l
q 1 E,q
(s, χ) = lE (s, χ) =
→ ns
cf. [14,28,29,42].
n=1
Relation between (h) (h)ζE,ξ,q(s, z) and lE,ξ,q(s, χ) is given by the following theorem:
H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908 903
Theorem 5 ([1]). Let s ∈ C. Let χ be a Dirichlet char(acter w)ith conductor f (=odd). We have∑f
l(h)
1 (h) a
E,ξ,q(s, χ) ( 1)
aqhaξa= s − χ(a)ζ s, . (1.7)f E,ξ f ,q f f
a=1
The value of twisted (h, q)-lE -function at negative integers is given explicitly by the following theorem:
Theorem 6 ([1]). Let n ∈ Z+. Then we have
l(h)E,ξ,q(−n, χ) = E
(h)
n,χ,ξ (q).
2. Partial twisted (h, q)-zeta function
In this section, we construct a twisted partial (h, q)-Euler zeta function. We assume that q ∈ C with | q |< 1. Let
R, be the field of real numbers and let ξ be the r -th root of unity.
Let s ∈ C and a, F ∈ Z with F is an odd integer and 0 < a < F . Then twisted partial (h, q)-Euler zeta function
is as follows: ∑ ( 1)mξmqhm
H (h)
−
E,ξ,q(s, a|F) = 2 s . (2.1)
m≡a(modF m)
m>0
We give relationship between H (h) (h)E,ξ,q(s, a|F) and ζE,ξ,q(s, z) as follows:
Substituting m = a + nF∑with F is odd into (2.1), we havem m hm
H (h)
(−1) ξ q
E,ξ,q(s, a|F) = 2 s
∑m≡a(modF
m
)
m>0
∞
( 1)a+nFξa+nFqh(a+nF)−
= 2
(a nF)s
n=0
( a a ha−1) ξ q ∑
+
∞
(−(1)nξnF)qnF= 2
F s
n n
a s
+
=0 F
(−1)aξaqha ( )(h) a
= ζ
F s E,ξ F ,qF
s, . (2.2)
F
By substituting s = −n, n ∈ Z+ in the above and using Theorem 4, after some elementary calculations, we arrive
at the following theorem:
Theorem 7. Let F be odd and s ∈ C. Then we(havea a ha )
(h) (−1) ξ qH (h)
a
E,ξ,q(s, a|F) = ζ s, .F s E,ξ F ,qF F
Let n ∈ Z+, then we have ( )
H (h)
a
E,ξ,q(−n, a
a ha a n (h) F
|F) = (−1) q ξ F E F , q .n,ξ F
Let χ be a Dirichlet character with conducto(r )fχ(and) fχ |F . By using Theorem 2 in Theorem 7, we get∑n n−k
H (h) a
n a (h)
E,ξ,q(−n, a|F) = (−1) ξ
aqhaFn E (qF ). (2.3)
k F n,ξ F
k=0
904 H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908
From Theorems 5 and 7, we have
∑F
l(h) (h)E,ξ,q(s, χ) = χ(a)HE,ξ,q(s, a|F). (2.4)
a=1
For s = −n with n ∈ Z+
∑F
l(h) (h)E,ξ,q(−n, χ) = χ(a)HE,ξ,q(−n, a|F). (2.5)
a=1
By using Theorem 7 in the above, we have
∑F ( )
l(h) a ha a n (h)
a F
E,ξ,q(−n, χ) = (−1) χ(a)q ξ F E F , q . (2.6)n,ξ F
a=1
By using Theorem 6 in the above, we arrive at the following theorem:
Theorem 8. Let χ∑be a Dirichlet character with co(nductor)F. Then we haveF
(h) aEn,ξ,χ (q
(h)
) = (−1)a χ(a)qhaξaFnE , qF .
n,ξ F F
a=1
Remark 3. By applying the q-deformed p-adic invariant integral on Zp, in the fermionic sense, Ozden et al. [1]
proved Theorem 8 by the different method. This theorem gives us the distribution relation of the generalized (h, q)
twisted Euler numbers. Distribution relations of Bernoulli and Euler numbers are very important in Number Theory,
Statistics and Measure Theory.
By using (2.3) and (2∑.6), we haveF ∑n ( )( )
l(h
n a n−k) (h)
E,ξ,q(−n, χ) = ( 1)
a
− χ(a)ξaqhaFn E (qF ).
k F k,ξ F
a=1 k=0
From above, we have
∑F ∑n ( )( )k
l(h) ( n, χ) ( 1)aχ(a)ξa ha n
n F (h) F
E,ξ,q − = − q a E (q ). (2.7)k a k,ξ F
a=1 k=0
By using (2.3), we have ∑n ( )( )k
H (h) a a ha n
n F (h) F
E,ξ,q(−n, a|F) = (−1) ξ q a E (q ).k a k,ξ F
k=0
By using the above equation, we modify twisted partial (h, q)-zeta function as follows:
Definition 3. Let s ∈ C. Then we define∑∞ ( )( )k
H (h)
−s F
E,ξ,q(s, a|F
(h)
) = ( 1)aξaqhaa−s F− E
k a k,ξ F
(q ). (2.8)
k=0
By using (2.5) and (2.8), we obtain
∑F ∑∞ ( )( )k
l(h)
−s F (h)
E,ξ,q(s, χ) ( 1)
aχ(a)ξaqhaa−s= − E F (q
F ), (2.9)
k a k,ξ
a=1 k=0
where s ∈ C.
H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908 905
3. Twisted p-adic (h, q)-Euler lE-function
Here, we use some notations which are due to Washington [8] and Kim [13]. The integer p∗ is defined by p∗ = p
if p > 2 and p∗ = 4 if p = 2 cf. [8, p. 51]. Let w denote the Teichmüller character, having conductor fw = p∗. For
an arbitrary character χ with conductor f = fχ . We define χn = χw−n , where n ∈ Z, in the sense of the product of
characters. In this section, if q ∈ Cp, then we assume | 1− q |p < 1. Let
a
〈a〉 = w−1(a)a = .
w(a)
Note that 〈a〉 ≡ 1(mod fw) cf. [8, p. 51].
We are now ready to construct twisted p-adic (h, q)-lE -function, which interpolate (h, q)-extension of Euler
numbers, at negative integers. So we obtain a partial answer to the question of Ozden et al. [1]: “Find a p-adic
twisted interpolation function of the generalized twisted (h, q)-Euler numbers, E (h)n,χ,w(q)”.
In Section 2, we define analytical functions on whole complex s-plane. In this section, we construct p-adic
analogues of these functions, our method is similar to that of [8,13]. Let χ be Dirichlet character with conductor
d (=odd) and F be a p∑ositive integer multiple of p∑an(d d. )W(e defi)ne p-adic analogues of (2.9) as follows:F ∞ k
l(h) (s, χ) ( 1)aχ(a)ξaqha a −s
−s F (h) F
E,p,ξ,q = − 〈 〉 Ek a k,ξ F
(q ),
a=1 k=0
and we also define p-adic analogues of (2.8)∑as f(ollows:∞ )( )k
H (h) (s, a F) ( 1)aξaqha a −s
−s F
E (h) (qFE,p,ξ,q | = − 〈 〉 F ), (3.1)k a k,ξ
k=0
where p - a and p∗|F .
Rema∑rk 4(. w
−)s((a)〈a∞ ) s〉 and
−s F k
E (h) (qF ),
k a k,ξ F
k=0
1
where F is multiple of p and fχ , are analytical in D s C s < p∗ p−= { ∈ p || | p−1p } cf. [7,20,8]. Therefore,
l(h) (h)E,p,ξ,q(s, χ) and HE,p,ξ,q(s, a|F) are analytical on D.
By (2.8), we have ∑n ( )( )k
H (h) a
n F (h)
E,p,ξ,q(−n, a|F) = (−1) ξ
aqha〈a n〉 E (qF )
k
k=0 ∑n (a)(
k),ξ
F
n a −k
1 a aqhaan −n a E (h)( ) ξ w ( ) (qF= − )
k F k,ξ F
k=0
w−n= (a)H (h)E,ξ,q(−n, a|F). (3.2)
Thus we get
H (h) −n (h)E,p,ξ,q(−n, a|F) = w (a)HE,ξ,q(−n, a|F).
Definition 4. Let s ∈ D. Let χ be Dirichlet character with conductor d (=odd) and F be a positive integer multiple
of p and d . Then we defi∑neF
l(h)E,p,ξ,q(s, χ) = χ(a H
(h)
) E,p,ξ,q(s, a|F). (3.3)
a=1
(a,p)=1
906 H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908
By substituting (3.2) into (3.3), Definition 4 is modified to the following definition:
Definition 5. Let s ∈ D. T∑hen we define∞
(h) χ(m)(−1)mξmqhmlE,p,ξ,q(s, χ) = 2 s .
m m=1
(m,p)=1
By using Definition 5, we easily obtain
∑∞ χ(m)( 1)m mqhm ∑∞(h) − ξ χ(mp)(−1)mξmpqhpmlE,p,ξ,q(s, χ) = 2 − 2 . (3.4)ms psms
m=1 m=1
By using Definition 2, we obtain the following corollary:
Corollary 1. Let s ∈ D. Then we have
l(h) s l(h) −s (h)E,p,ξ,q( , χ) = E,ξ,q(s, χ)− p χ(p)lE,ξ p,q p (s, χ).
The function, l(h)E,p,ξ,q(s, χ) is analytical on D. This function interpolates twisted generalized (h, q)-Euler numbers
at negative integers, which is given as follows:
By substituting s = −n, n ∈ Z+, into (3.3), we have∑F
l(h) (h)E,p,ξ,q(−n, χ) = χ(a)HE,p,ξ,q(−n, a|F).
a=1
(a,p)=1
By (3.2) and (3.4), we get
∑F
l(h)E,p,ξ,q(−n, χ) = χn(a H
(h)
) E,ξ,q(−n, a|F)
a=1
(a,p)=1
∑F ∑Fp
(h) (h)
= χn(a)HE,ξ,q(−n, a|F)− χn(ap)HE,ξ,q(−n, ap|F)
∑a=1 a=1 FF ∑p
= χn(a H
(h) (h)
) E,ξ,q(−n, a|F)− χn(p) χn(a)HE,ξ,q(−n, ap|F)
a=1 a=1
By using Theorem 7 and (3.1) in the above, we have
∑F (
(h) a a ha n (h) a
)
l FE,p,ξ,q(−n, χ) = χn(a)(−1) ξ q F E , qn,ξ F F
a=1 ∑F ( )p ( )n
pnχ (p) χ (a)( 1)paqhpa pa
F
E (h)
a p F
− n n − ξ F , (q ) pp n,(ξ p p Fa=1 ) p
E (h)= n,ξ,χ (q) p
n
− χn(p)E
(h) p
n n,ξ p,χ
(q ).
n
Thus we arrive at the following theorem:
Theorem 9. Let n ∈ Z+. Then we have
l(h)E,p,ξ,q(−n, χ) = E
(h) n
n,ξ,χ (q)− p χn(p E
(h)
) p
n n,ξ p,χ
(q ).
n
H. Ozden, Y. Simsek / Computers and Mathematics with Applications 56 (2008) 898–908 907
Observe that ∑∞ χ(m)( m m−1) ξ
lim l(h)
q 1 E,p,ξ,q
(s, χ) = lE,p,ξ (s, χ) = 2 ,
→
m 1 m
s
=
(m,p)=1
which is called twisted p-adic lE -function. This function interpolates twisted generalized Euler numbers at negative
integers.
In [∫42], Ozden and Simsek gave the following Witt’s formula for the numbers E (h)n,χ,q(x)
ynχ(y)qhydµ−1(y) E (h)= n,χ (q).
X
By using the above equ∫ation, we obtain
l(h) (s, χ) y −sE,p,ξ,q = 〈 〉 χ(y)ξ
yqhydµ−1(y).
X∗
Conclusion 1. Thus we derive p-adic (h, q)-twisted interpolation function which interpolates E (h)n,ξ,χ numbers. This
gives a part of answer to the question in [1]. For q → 1, Definition 5 reduces to the ordinary twisted p-adic lE -
function, which interpolates twisted generalized Euler numbers at negative integers by using Theorem 9. p adic (h, q)
twisted interpolation function, l(h)E,p,ξ,q(s, χ), for q → 1, coincides with that of [18]. These functions are used not
only in Mathematics and Mathematical analysis but also used in Physics and its applications.
Acknowledgements
We thank the referees for their valuable comments on this present form. The second author is supported by the
research fund of Akdeniz University.
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