Applied Mathematics Letters 21 (2008) 934–939
www.elsevier.com/locate/aml
A new extension of q-Euler numbers and polynomials related to
their interpolation functions
Hacer Ozdena,∗, Yilmaz Simsekb
a University of Uludag, Faculty of Arts and Science, Department of Mathematics, Bursa, Turkey
b University of Akdeniz, Faculty of Arts and Science, Department of Mathematics, Antalya, Turkey
Received 9 March 2007; received in revised form 30 July 2007; accepted 18 October 2007
Abstract
In this work, by using a p-adic q-Volkenborn integral, we construct a new approach to generating functions of the (h, q)-Euler
numbers and polynomials attached to a Dirichlet character χ . By applying the Mellin transformation and a derivative operator to
these functions, we define (h, q)-extensions of zeta functions and l-functions, which interpolate (h, q)-extensions of Euler numbers
at negative integers.
©c 2007 Elsevier Ltd. All rights reserved.
Keywords: p-adic Volkenborn integral; Twisted q-Euler numbers and polynomials; Zeta and l-functions
1. Introduction, definitions and notation
Let p be a fixed odd prime number. Throughout this work, Zp, Qp, C and Cp respectively denote the ring of p-
adic rational integers, the field of p-adic rational numbers, the complex numbers field and the completion of algebraic
closure of Qp. Let vp be the normalized exponential valuation of C with |p| p−vp(p) 1p p = = p . When one talks of
q-extension, q is considered in many ways, e.g. as an indeterminate, a complex number q ∈ C, or a p-adic number
1
q ∈ Cp. If q ∈ C we assume that |q| < 1. If q ∈ Cp, we assume that |1− q|p < p
− p−1 , so that qx = exp(x log q)
for |x |q 6 1; cf. [3,2,5–7,4,11,14,16,1]. We use the following notation:
1 x x− q 1− (−q)
[x]q = , [x]−q = ,1− q 1+ q
where limq→( 1 [)x]q = x ; cf. [5]. ( )
Let U D Zp be the set of uniformly differentiable functions on Zp. For f ∈ U D Zp , Kim [3] originally defined
the p-adic invar∫iant q-integral on Zp as follows:
1 p∑N−1
Iq( f ) = f (x)dµq(x) = lim [ ] f (x)qx ,
Z Np N→∞ p q x=0
∗ Corresponding author.
E-mail addresses: hozden@uludag.edu.tr (H. Ozden), ysimsek@akdeniz.edu.tr (Y. Simsek).
0893-9659/$ - see front matter©c 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2007.10.005
H. Ozden, Y. Simsek / Applied Mathematics Letters 21 (2008) 934–939 935
where N is a natural number and p is an odd prime number. The q-deformed p-adic invariant integral on Zp, in the
fermionic sense, is defined by ∫
I−q( f ) = lim Iq( f ) = f (x)dµ−q(x), cf. [3,5,6,4].
q→−q Zp
Recently, twisted (h, q)-Bernoulli and Euler numbers and polynomials were studied by several authors (see [10,2,15,
16,9,8,13,1]).
By definition of µ−q(x), we see that
I−1( f1)+ I−1( f ) = 2 f (0), cf. [5], (1.1)
where f1(x) = f (x + 1).
In this study, we define new (h, q)-extension of Euler numbers and polynomials. By using a derivative operator on
these functions, we derive (h, q)-extensions of zeta functions and l-functions, which interpolate (h, q)-extensions of
Euler numbers at negative integers.
2. A new approach to q-Euler numbers
In this section, we define (h, q)-extension of Euler numbers and polynomials. Substituting f (x) qhx= et x , with
h ∈ Z, into (1.1) we have
2 ∑∞h tnFq (t) = I−1(qhxet x ) = h t = E (h)n,q , |h log q + t | < π, (2.1)q e + 1 n!
n=0
where E (h)n,q is called the (h, q)-extension of Euler numbers. lim
(h)
q→1 En,q = En , where En is the classical Euler
numbers. That is
2 ∑∞ tn
t = En cf. [8,4,12,17].e + 1 n!
n=0
(h, q)-extensions of Euler polynomials, E (h)n,q(x), are defined by the following generating function:
t x ∑∞ n
Fh
2e t
q (t, x) = F
h(t)et x (h)q = h t = En,q(x) . (2.2)q e + 1 n!
n=0
By∫usin∑g the Maclaurin series of e
t x in (2.1), we have
∞ tn xn ∑∞ tn
qhx dµ−1(x) = E (h)n,q .
Zp n! n!n=0 n=0
n
By comparing coefficients of tn on either side of the above equation, we obtain the Witt formula, which is given by!
the following theorem.
Theor∫em 1 (Witt Formula). For h ∈ Z, q ∈ Cp with |1− q|p < 1,
qhx xndµ−1(x) (h)= En,q , (2.3)
Zp
and ∫
qhy(x + y)ndµ (h)−1(y) = En,q(x).
Zp
936 H. Ozden, Y. Simsek / Applied Mathematics Letters 21 (2008) 934–939
Fro∑m (2.2), we have∞ tn ∑∞ n ∑∞ n
E (h) xn
t
E (h)
t
n,q =n n n,q
(x) .
! ! n!
n=0 n=0 n=0
By the∑Ca∑uchy product, we see that∞ n tk tn−k ∑∞ n
E (h)
t
k,q x
n−k E (h)= n,q(x) .k! (n − k)! n!
n=0 k=0 n=0
n
By comparing coefficients of tn , we arrive at the following theorem:!
Theorem 2. Let n∑∈ Z+ = Z ∪ {0}. Then we haven ( )
(h) nE (x) xn−k (h)n,q = Ek k,q
. (2.4)
k=0
Let d be a fixed in(teger. For a)ny positive integer N , we set ⋃ ( )
X X lim Z/dpN Z , X Z , X∗ N= d = 1 = p = a + dp Zp ,
←−
N { ( )} 0<a<dp(a,p)=1
a + dpN Zp = x ∈ X : x ≡ a mod dpN ,
where∫a ∈ Z with 0 6 a < ∫dpN (cf. [3]). It is known that
f (x)dµ−1(x) = f (x)dµ−1(x), cf. [3].
Zp X
From t∫his we note that ∑d−1 ∫ ( )a x k ( )ht
(x k ht k a
+
+ t) q dµ−1(t) = d (−1) qha t + qd dµ−1(t), (2.5)
X a=0 Zp d
where d is an odd positive integer. From (2.2) and (2.5), we obtain the following theorem.
Theorem 3 (Distrib∑ution Relation). Fo(r d an o)dd positive integer, k ∈ Z+, we haved−1 x + a
E (h) k (h)k,q(x) = d (−1)
aqha E .
k,qd d
a=0
By (1.1), Kim [5] defined the following integral equation:
∑n−1
I−1( fn)+ (−1)n−1 I−1( f ) = 2 ( 1)n−1−l− f (l), (2.6)
l=0
where n ∈ N, fn(x) = f (x + n).
Let d be an odd positive integer and χ be the Dirichlet character with conductor d; substituting f (x) qhx= χ(x)et x ,
for h ∈ Z, into (2.6), we obtain
d∑−1
2 ( 1)aχ(a)eta− qha ∑∞ n
Fh t a=0( , χ) (h)
t π
q = hd td = En,χ,q , |t + h log q| < , (2.7)q e + 1 n! d
n=0
where E (h)n,χ,q denote (h, q)-extensions of generalized Euler numbers.
H. Ozden, Y. Simsek / Applied Mathematics Letters 21 (2008) 934–939 937
Fro∫m (2.7), we see that ∑d−1 ∫ ( )hx ( )n
χ(x)qhx xndµ (x) dn χ(a)qha a d
a
−1 = (−1) q + x dµ−1(x). (2.8)
X a=0 Zp d
By Theorem 1 and (2.8), we obtain the following theorem.
Theorem 4. Let d ∑be an odd positive intege(r an)d χ be Dirichlet’s character with conductor d. Then we haved−1
E (h) dn a qha 1 a E (h)
a
n,χ,q = χ( ) (− ) .n,qd d
a=0
From (2.6), we also note that
d∑−1
2 ( 1)aχ(a)et (a+x)− qha
∞ n
Fh t x a=0
∑
( , , χ) (h)
t
q = qhdetd
= En,χ,q(x) , (2.9)
+ 1 n!
n=0
where h ∈ Z, E (h)n,χ,q(x) are called generalized (h, q)-extensions of Euler polynomials attached to χ and Fhq (t, x, χ) =
Fhq (t, χ)e
t x .
By ∫(2.9), we easily see that
(x n+ y) χ(y)qhydµ (h)−1(y) = En,χ,q(x). (2.10)
X
By using (2.10), we arrive at the following theorem.
Theorem 5. Let d be a∑n odd integer. Then we h(aved−1 )
E (h) n
a + x
n,χ,q(x) = d (−1
a a qha E (h)) χ( )
n,qd
.
d
a=0
3. A new approach to the (h, q)-Euler zeta function
In this s∑ection, we assume th∑at q ∈ C with |q| < 1. By using a geometric series in (2.2), we obtain∞ ∞ n
2ext qhnetn( 1)n E (h)
t
− = n,q(x) .n!
n=0 n=0
k
By applying the derivative operator d k |t=0 to the above equation, we have
∑ dt∞
E (h)k,q(x) = 2 (−1)
nqhn(x n)k+ . (3.1)
n=0
By (3.1), we define new extensions of Hurwitz type (h, q)-Euler zeta functions as follows:
Definition 1. For h ∈∑Z, s ∈ C and 0 < x ≤ 1, we define∞ n hn
(h) (−1) q
ζE,q(s, x) = 2 (n x)s
. (3.2)
+
n=0
(h)
ζE,q(s, x) is an analytic function on the whole complex s-plane. If x = 1, then we define the (h, q)-Euler zeta
function as follows:
938 H. Ozden, Y. Simsek / Applied Mathematics Letters 21 (2008) 934–939
∑∞ n hn
(h) (−1) q
ζE,q(s) = 2 .ns
n=1
For s = −k, k ∈ Z+ in (3.2) and using (3.1), we arrive at the following theorem.
Theorem 6. For k ∈ Z+, we have
(h)
ζE,q(−k x
(h)
, ) = Ek,q(x). (3.3)
Remark 1. ∫By applying the Mellin transformation to the generating function of (h, q)-Euler polynomials, for s ∈ C,
1 ∞
Fh t x (h)q (− , )t
s−1dt = ζE,q(s, x).Γ (s) 0
By substituting s = −n, n ∈ Z+ and using the Cauchy residue theorem, we obtain another proof of Theorem 6.
By us∑ing (2.7) we have with χ(a∑+ d) = χ(a), where d is an odd positive integer,∞ ∞ n
2 ( 1)mχ(m)etmqhm E (h)
t
− = n,χ,q . (3.4)n!
m=0 n=0
k
By applying the derivative operator d k |t=0 to the above equation, we have
∑ dt∞
E (h) 2 ( 1)mqhmχ(m)mkk,χ,q = − . (3.5)
m=0
By using (3.5), we define new extensions of (h, q)-Euler l-functions as follows:
Definition 2. Let s ∈∑C. We define∞
(h) (
m hm
−1) q χ(m)
lE,q(s, χ) = 2 . (3.6)ms
m=1
l(h)E,q(s, x) is an analytic function on the whole complex s-plane. From (3.5) and (3.6), we arrive at the following
theorem.
Theorem 7. For k ∈ Z+, we have
l(h) (h)E,q(−k, χ) = Ek,χ,q . (3.7)
Remark 2. ∫
1 ∞
Fh ( t)t s−1dt l(h)
Γ (s) q,χ
− = E,q(s, χ).
0
By using the Cauchy residue theorem we obtain another proof of Theorem 7.
By substituting m ∑= a∑+ dn, a = 1, . . . , d , d is odd, n = 0, 1, 2, . . ., into (3.6), we haved ∞ ( 1)a+dmqdhm+ha(h) − χ(dm + a)lE,q(s, χ) = 2 (a + dm)s
a=1∑m=0d ∑∞ m dhm
d−s ( 1)aχ(a)qha
2((−1) q= − a )s
∑a m +=1 m=0 dd (
s a a
)
= d− (−1) χ(a qha (h)) ζ s, .
E,qd d
a=1
H. Ozden, Y. Simsek / Applied Mathematics Letters 21 (2008) 934–939 939
By substituting s = −n,∑n ∈ Z+, into the above e(quation), we haved
(h) n a ha (h) alE,q(−n, χ) = d (−1) χ(a)q ζ d −n,E,q d
∑a=1d ( )
dn a ha (h)
a
= (−1) χ(a)q E . (3.8)
n,qd d
a=1
By using (2.4), (3.7) and (3.8), we obtained the following theorem.
Theorem 8 (Distribution Relations for the Generalized (h, q)-Extension of Euler Numbers). Let d be an odd integer.
Then we have ∑d ∑n ( )n
E (h) ( 1)aχ(a)qha n−k k (h)n,χ,q = − a d E .k k,qd
a=1 k=0
Acknowledgement
The second author is supported by the research fund of Akdeniz University.
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