A New Extension of Q-Euler Numbers and Polynomials Related to Their Interpolation Functions

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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

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. For h ∈ Z, s ∈ C and 0 < x ≤ 1, we deﬁne ∞ n hn h X (−1) q ζ ( ) (s, x) = 2 . (3.2) E,q (n + x)s 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 X (−1) q ζ ( ) (s) = 2 . E,q 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) − = (h) ζE,q ( k, x) 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 Z ∞ h − s−1 = (h) Fq ( t, x)t dt 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 using (2.7) we have with χ(a + d) = χ(a), where d is an odd positive integer, ∞ ∞ X X tn 2 (−1)mχ(m)etmqhm = E(h) . (3.4) n,χ,q n! m=0 n=0

k By applying the derivative operator d | to the above equation, we have dtk t=0 ∞ (h) = X − m hm k Ek,χ,q 2 ( 1) q χ(m)m . (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 ∞ m hm h X (−1) q χ(m) l( ) (s, χ) = 2 . (3.6) E,q ms m=1

(h) lE,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 (h) − = (h) lE,q ( k, χ) Ek,χ,q . (3.7)

Remark 2. 1 Z ∞ h − s−1 = (h) Fq,χ ( t)t dt lE,q (s, χ). Γ (s) 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 have

d ∞ a+dm dhm+ha h X X (−1) q χ(dm + a) l( ) (s, χ) = 2 E,q (a + dm)s a=1 m=0 d ∞ X X 2(−1)mqdhm = d−s (−1)aχ(a)qha + a s a=1 m=0 m d d X h a = d−s (−1)aχ(a)qhaζ ( ) 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 equation, we have d h X h a l( ) (−n, χ) = dn (−1)aχ(a)qhaζ ( ) −n, E,q E,qd d a=1 d X h a = dn (−1)aχ(a)qha 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 X X n h E(h) = (−1)aχ(a)qhaan−kdk E( ) . n,χ,q k k,qd a=1 k=0

Acknowledgement

The second author is supported by the research fund of Akdeniz University.

References

[1] M. Cenkci, M. Can, V. Kurt, p-adic interpolation functions and Kummer-type congruences for q-twisted and q-generalized twisted Euler numbers, Adv. Stud. Contemp. Math. 9 (2) (2004) 203–216. [2] L.C. Jang, S.-D. Kim, D.-W. Park, Y.S. Ro, A note on Euler number and polynomials, J. Inequal. Appl. (2006) 5 pp. Art. ID 34602. [3] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002) 288–299. [4] T. Kim, On Euler–Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (3) (2003) 261–267. [5] T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Z p at q = −1, J. Math. Anal. Appl. 331 (2007) 779–792. [6] T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458–1465. [7] T. Kim, A new approach to q-zeta function, J. Comput. Anal. Appl. 9 (2007) 395–400. [8] T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys. 14 (1) (2007) 15–27. [9] T. Kim, The modiﬁed q-Euler numbers and polynomials. arxivmath.NT/0702523. [10] T. Kim, A note on p-adic invariant integral in the rings of p-adic integers, Adv. Stud. Contemp. Math. 13 (1) (2006) 95–99. [11] T. Kim, An invariant p-adic q-integral on Zp, Appl. Math. Lett., in press (doi:10.1016/j.aml.2006.11.011). [12] H. Ozden, Y. Simsek, S.-H. Rim, I.N. Cangul, A note on p-adic q-Euler Measure, Adv. Stud. Contemp. Math. 14 (2) (2007) 233–239. [13] S-H. Rim, T. Kim, A note on q-Euler numbers associated with the basic q-zeta function, Appl. Math. Lett. 20 (4) (2007) 366–369. [14] M. Schork, Ward’s “calculus of sequences”, q-calculus and the limit q to −1, Adv. Stud. Contemp. Math. 13 (2) (2006) 131–141. [15] Y. Simsek, q-analogue of twisted l-series and q-twisted Euler numbers, J. Number Theory 110 (2) (2005) 267–278. [16] Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl. 324 (2006) 790–804. [17] H.M. Srivastava, T. Kim, Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russ. J. Math. Phys. 12 (2) (2005) 241–268.