CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector 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 | | = −vp(p) = 1 closure of Qp. Let vp be the normalized exponential valuation of Cp with p p 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 p−1 x q ∈ Cp. If q ∈ C we assume that |q| < 1. If q ∈ Cp, we assume that |1 − q|p < p , so that q = 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 − qx 1 − (−q)x [x] = , [x]− = , q 1 − q q 1 + q where limq→1 [x]q = x; cf. [5]. Let UD Zp be the set of uniformly differentiable functions on Zp. For f ∈ UD Zp , Kim [3] originally defined the p-adic invariant q-integral on Zp as follows: Z pN −1 1 X x Iq ( f ) = f (x)dµq (x) = lim f (x)q , N→∞ N Zp p q x=0 ∗ Corresponding author. E-mail addresses: [email protected] (H. Ozden), [email protected] (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 Z 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 etx , with h ∈ Z, into (1.1) we have ∞ n h hx tx 2 X (h) t F (t) = I− (q e ) = = E , |h log q + t| < π, (2.1) q 1 qhet + 1 n,q n! n=0 (h) (h) where En,q is called the (h, q)-extension of Euler numbers. limq→1 En,q = En, where En is the classical Euler numbers. That is ∞ 2 X tn = En cf. [8,4,12,17]. et + 1 n! n=0 (h) (h, q)-extensions of Euler polynomials, En,q (x), are defined by the following generating function: ∞ 2etx X tn Fh(t, x) = Fh(t)etx = = E(h) (x) . (2.2) q q qhet + 1 n,q n! n=0 By using the Maclaurin series of etx in (2.1), we have Z ∞ n n ∞ n X hx t x X (h) t q dµ− (x) = E . n! 1 n,q n! Zp n=0 n=0 tn By comparing coefficients of n! on either side of the above equation, we obtain the Witt formula, which is given by the following theorem. Theorem 1 (Witt Formula). For h ∈ Z, q ∈ Cp with |1 − q|p < 1, Z hx n (h) q x dµ−1(x) = En,q , (2.3) Zp and Z hy n (h) q (x + y) dµ−1(y) = En,q (x). Zp 936 H. Ozden, Y. Simsek / Applied Mathematics Letters 21 (2008) 934–939 From (2.2), we have ∞ ∞ ∞ X tn X tn X tn E(h) xn = E(h) (x) . n,q n! n! n,q n! n=0 n=0 n=0 By the Cauchy product, we see that ∞ n k n−k ∞ n X X h t t X t E( ) xn−k = E(h) (x) . k,q k! (n − k)! n,q n! n=0 k=0 n=0 tn By comparing coefficients of n! , we arrive at the following theorem: Theorem 2. Let n ∈ Z+ = Z ∪ {0}. Then we have n X n h E(h) (x) = xn−k E( ). (2.4) n,q k k,q k=0 Let d be a fixed integer. For any positive integer N, we set = = N = ∗ = [ + N X Xd lim←− Z/dp Z , X1 Zp, X a dp Zp , N 0<a<dp (a,p)=1 N n N o a + dp Zp = x ∈ X : x ≡ a mod dp , N where a ∈ Z with 0 6 a < dp (cf. [3]). It is known that Z Z f (x)dµ−1(x) = f (x)dµ−1(x), cf. [3]. Zp X From this we note that − Z d 1 Z k ht k ht k X a ha a + x d (x + t) q dµ− (t) = d (−1) q t + q dµ− (t), (2.5) 1 d 1 X a=0 Zp where d is an odd positive integer. From (2.2) and (2.5), we obtain the following theorem. Theorem 3 (Distribution Relation). For d an odd positive integer, k ∈ Z+, we have d−1 h X h x + a E( )(x) = dk (−1)aqha E( ) . k,q k,qd d a=0 By (1.1), Kim [5] defined the following integral equation: n−1 n−1 X n−1−l I−1( fn) + (−1) I−1( f ) = 2 (−1) 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)etx , for h ∈ Z, into (2.6), we obtain d−1 P − a ta ha 2 ( 1) χ(a)e q ∞ n a=0 X t π Fh(t, χ) = = E(h) , |t + h log q| < , (2.7) q qhd etd + 1 n,χ,q n! d n=0 (h) where En,χ,q denote (h, q)-extensions of generalized Euler numbers. H. Ozden, Y. Simsek / Applied Mathematics Letters 21 (2008) 934–939 937 From (2.7), we see that − Z d 1 Z hx n hx n n X ha a d a χ(x)q x dµ− (x) = d χ(a)q (−1) q + x dµ− (x). (2.8) 1 d 1 X a=0 Zp By Theorem 1 and (2.8), we obtain the following theorem. Theorem 4. Let d be an odd positive integer and χ be Dirichlet’s character with conductor d. Then we have d−1 X h a E(h) = dn χ(a)qha(−1)a E( ) . n,χ,q n,qd d a=0 From (2.6), we also note that d−1 P − a t(a+x) ha 2 ( 1) χ(a)e q ∞ n a=0 X t Fh(t, x, χ) = = E(h) (x) , (2.9) q qhd etd + 1 n,χ,q n! n=0 (h) h where h ∈ Z, En,χ,q (x) are called generalized (h, q)-extensions of Euler polynomials attached to χ and Fq (t, x, χ) = h tx Fq (t, χ)e . By (2.9), we easily see that Z n hy (h) (x + y) χ(y)q dµ−1(y) = En,χ,q (x). (2.10) X By using (2.10), we arrive at the following theorem. Theorem 5. Let d be an odd integer. Then we have d−1 X h a + x E(h) (x) = dn (−1)aχ(a)qha E( ) . n,χ,q n,qd d a=0 3. A new approach to the (h, q)-Euler zeta function In this section, we assume that q ∈ C with |q| < 1. By using a geometric series in (2.2), we obtain ∞ ∞ X X tn 2ext qhnetn(−1)n = E(h) (x) . n,q n! n=0 n=0 k By applying the derivative operator d | to the above equation, we have dtk t=0 ∞ (h) = X − n hn + k Ek,q (x) 2 ( 1) q (x n) . (3.1) n=0 By (3.1), we define new extensions of Hurwitz type (h, q)-Euler zeta functions as follows: Definition 1.