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Advances in Difference Equations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Some results for the q-Bernoulli, q-Euler numbers and polynomials Advances in Difference Equations 2011, 2011:68 doi:10.1186/1687-1847-2011-68 Daeyeoul Kim (daeyeoul@nims.re.kr) Min-Soo Kim (minsookim@kaist.ac.kr) ISSN 1687-1847 Article type Research Submission date 2 September 2011 Acceptance date 23 December 2011 Publication date 23 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/68 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Kim and Kim ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Some results for the q -Bernoulli, q -Euler numbers and polynomials Daeyeoul Kim1 and Min-Soo Kim∗2 1 National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu Daejeon 305-340, South Korea 2 Department of Mathematics, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea ∗ Corresponding author: minsookim@kaist.ac.kr Email address: DK: daeyeoul@nims.re.kr Abstract The q -analogues of many well known formulas are derived by us- ing several results of q -Bernoulli, q -Euler numbers and polynomials. The q - analogues of ζ -type functions are given by using generating functions of q - Bernoulli, q -Euler numbers and polynomials. Finally, their values at non- positive integers are also been computed. 2010 Mathematics Subject Classiﬁcation: 11B68; 11S40; 11S80. Keywords: Bosonic p-adic integrals; Fermionic p-adic integrals; q -Bernoulli polynomials; q -Euler polynomials; generating functions; q -analogues of ζ -type functions; q -analogues of the Dirichlet’s L-functions. 1. Introduction Carlitz [1,2] introduced q -analogues of the Bernoulli numbers and polynomials. From that time on these and other related subjects have been studied by various authors (see, e.g., [3–10]). Many recent studies on q -analogue of the Bernoulli, Euler numbers, and polynomials can be found in Choi et al. [11], Kamano [3], Kim [5,6,12], Luo [7], Satoh [9], Simsek [13,14] and Tsumura [10]. For a ﬁxed prime p, Zp , Qp , and Cp denote the ring of p-adic integers, the ﬁeld of p-adic numbers, and the completion of the algebraic closure of Qp , respectively. Let | · |p be the p-adic norm on Q with |p|p = p−1 . For convenience, | · |p will also be used to denote the extended valuation on Cp . The Bernoulli polynomials, denoted by Bn (x), are deﬁned as n n Bk xn−k , (1.1) Bn (x) = n ≥ 0, k k=0 where Bk are the Bernoulli numbers given by the coeﬃcients in the power series ∞ tk t (1.2) = Bk . et − 1 k! k=0 t t From the above deﬁnition, we see Bk ’s are all rational numbers. Since et −1 − 1 + 2 is an even function (i.e., invariant under x → −x), we see that Bk = 0 for any odd integer k not smaller than 3. It is well known that the Bernoulli numbers can also 1
2 be expressed as follows pN −1 1 ak (1.3) Bk = lim N N →∞ p a=0 (see [15,16]). Notice that, from the deﬁnition Bk ∈ Q, and these integrals are independent of the prime p which used to compute them. The examples of (1.3) are: pN −1 1 pN (pN − 1) 1 1 lim a = lim = − = B1 , N →∞ pN N →∞ pN 2 2 a=0 (1.4) pN −1 1 pN (pN − 1)(2pN − 1) 1 1 a2 = lim lim = = B2 . N →∞ pN N N →∞ p 6 6 a=0 Euler numbers Ek , k ≥ 0 are integers given by (cf. [17–19]) k−1 k (1.5) E0 = 1, Ek = − Ei for k = 1, 2, . . . . i i=0 2|k−i The Euler polynomial Ek (x) is deﬁned by (see [20, p. 25]): k k −i k Ei 1 (1.6) Ek (x) = x− , i 2i 2 i=0 which holds for all nonnegative integers k and all real x, and which was obtained by Raabe [21] in 1851. Setting x = 1/2 and normalizing by 2k gives the Euler numbers 1 Ek = 2k Ek (1.7) , 2 where E0 = 1, E2 = −1, E4 = 5, E6 = −61, . . . . Therefore, Ek = Ek (0), in fact ([19, p. 374 (2.1)]) 2 (1 − 2k+1 )Bk+1 , (1.8) Ek (0) = k+1 where Bk are Bernoulli numbers. The Euler numbers and polynomials (so-named by Scherk in 1825) appear in Euler’s famous book, Institutiones Calculi Diﬀerentialis (1755, pp. 487–491 and p. 522). In this article, we derive q -analogues of many well known formulas by using sev- eral results of q -Bernoulli, q -Euler numbers, and polynomials. By using generating functions of q -Bernoulli, q -Euler numbers, and polynomials, we also present the q -analogues of ζ -type functions. Finally, we compute their values at non-positive integers. This article is organized as follows. In Section 2, we recall deﬁnitions and some properties for the q -Bernoulli, Euler numbers, and polynomials related to the bosonic and the fermionic p-adic integral on Zp . In Section 3, we obtain the generating functions of the q -Bernoulli, q -Euler num- bers, and polynomials. We shall provide some basic formulas for the q -Bernoulli and q -Euler polynomials which will be used to prove the main results of this article.
3 In Section 4, we construct the q -analogue of the Riemann’s ζ -functions, the Hurwitz ζ -functions, and the Dirichlet’s L-functions. We prove that the value of their functions at non-positive integers can be represented by the q -Bernoulli, q - Euler numbers, and polynomials. 2. q -Bernoulli, q -Euler numbers and polynomials related to the Bosonic and the Fermionic p-adic integral on Zp In this section, we provide some basic formulas for p-adic q -Bernoulli, p-adic q -Euler numbers and polynomials which will be used to prove the main results of this article. Let U D(Zp , Cp ) denote the space of all uniformly (or strictly) diﬀerentiable Cp - valued functions on Zp . The p-adic q -integral of a function f ∈ U D(Zp ) on Zp is deﬁned by pN −1 1 f (a)q a = (2.1) Iq (f ) = lim f (z )dµq (z ), [pN ]q N →∞ Zp a=0 where [x]q = (1 − q x )/(1 − q ), and the limit taken in the p-adic sense. Note that (2.2) lim [x]q = x q →1 for x ∈ Zp , where q tends to 1 in the region 0 < |q − 1|p < 1 (cf. [22,5,12]). The bosonic p-adic integral on Zp is considered as the limit q → 1, i.e., pN −1 1 (2.3) I1 (f ) = lim N f (a) = f (z )dµ1 (z ). N →∞ p Zp a=0 From (2.1), we have the fermionic p-adic integral on Zp as follows: pN −1 f (a)(−1)a = (2.4) I−1 (f ) = lim Iq (f ) = lim f (z )dµ−1 (z ). q →−1 N →∞ Zp a=0 k In particular, setting f (z ) = [z ]k in (2.3) and f (z ) = z + 1 q in (2.4), respectively, q 2 we get the following formulas for the p-adic q -Bernoulli and p-adic q -Euler numbers, respectively, if q ∈ Cp with 0 < |q − 1|p < 1 as follows pN −1 1 [z ]k dµ1 (z ) [a]k , (2.5) Bk (q ) = = lim N q q N →∞ p Zp a=0 pN −1 k k 1 1 k k (−1)a . (2.6) Ek (q ) = 2 z+ dµ−1 (z ) = 2 lim a+ 2 2 N →∞ Zp q q a=0 Remark 2.1. The q -Bernoulli numbers (2.5) are ﬁrst deﬁned by Kamano [3]. In (2.5) and (2.6), take q → 1. Form (2.2), it is easy to that (see [17, Theorem 2.5]) z k dµ1 (z ), (2z + 1)k dµ−1 (z ). Bk (q ) → Bk = Ek (q ) → Ek = Zp Zp
4 For |q − 1|p < 1 and z ∈ Zp , we have ∞ z q iz = (q i − 1)n and |q i − 1|p ≤ |q − 1|p < 1, (2.7) n n=0 where i ∈ Z. We easily see that if |q − 1|p < 1, then q x = 1 for x = 0 if and only if q is a root of unity of order pN and x ∈ pN Zp (see [16]). By (2.3) and (2.7), we obtain N (q i )p − 1 1 iz I1 (q ) = i lim pN q − 1 N →∞ ∞ pN 1 1 (q i − 1)m − 1 = lim q i − 1 N →∞ pN m m=0 ∞ N 1 1 p (q i − 1)m = lim q i − 1 N →∞ pN m m=1 ∞ 1 pN − 1 1 (q i − 1)m (2.8) = lim q i − 1 N →∞ m=1 m m − 1 ∞ 1 1 −1 (q i − 1)m = i−1 q m m−1 m=1 ∞ (q i − 1)m 1 (−1)m−1 = i−1 q m m=1 i log q = qi − 1 ∞ since the series log(1 + x) = m=1 (−1)m−1 xm /m converges at |x|p < 1. Similarly, by (2.4), we obtain (see [4, p. 4, (2.10)]) pN −1 2 iz (q i )a (−1)a = (2.9) I−1 (q ) = lim . qi + 1 N →∞ a=0 From (2.5), (2.6), (2.8) and (2.9), we obtain the following explicit formulas of Bk (q ) and Ek (q ): k log q k i (−1)i i (2.10) Bk (q ) = , (1 − q )k i q −1 i=0 k 2k+1 k 1 1 (−1)i q 2 i i (2.11) Ek (q ) = , (1 − q )k i q +1 i=0 where k ≥ 0 and log is the p-adic logarithm. Note that in (2.10), the term with i = 0 is understood to be 1/log q (the limiting value of the summand in the limit i → 0). We now move on to the p-adic q -Bernoulli and p-adic q -Euler polynomials. The p-adic q -Bernoulli and p-adic q -Euler polynomials in q x are deﬁned by means of the bosonic and the fermionic p-adic integral on Zp : [x + z ]k dµ1 (z ) [x + z ]k dµ−1 (z ), (2.12) Bk (x, q ) = and Ek (x, q ) = q q Zp Zp
5 where q ∈ Cp with 0 < |q − 1|p < 1 and x ∈ Zp , respectively. We will rewrite the above equations in a slightly diﬀerent way. By (2.5), (2.6), and (2.12), after some elementary calculations, we get k k [x]k−i q ix Bi (q ) = (q x B (q ) + [x]q )k (2.13) Bk (x, q ) = q i i=0 and (2.14) k k k −i 1 q x− 2 k Ei (q ) 1 1 i(x− 1 ) Ek (x, q ) = x− q = E (q ) + x − , 2 i i 2 2 2 2 q q i=0 where the symbol Bk (q ) and Ek (q ) are interpreted to mean that (B (q ))k and (E (q ))k must be replaced by Bk (q ) and Ek (q ) when we expanded the one on the right, respectively, since [x + y ]k = ([x]q + q x [y ]q )k and q k k −1 1 1 1 x− 1 z ]k [x + = [2x − 1] +q z+ 2 1 q 2 2 2 q2 q q q (2.15) k k −i i 1 k 1 1 1 k−i [2x − 1]q q (x− 2 )i = z+ 2 i 2 2 q i=0 q q (cf. [4,5]). The above formulas can be found in [7] which are the q -analogues of the 1 corresponding classical formulas in [17, (1.2)] and [23], etc. Obviously, put x = 2 in (2.14). Then 1 Ek (q ) = 2k Ek (2.16) ,q = Ek (0, q ) and lim Ek (q ) = Ek , 2 q →1 where Ek are Euler numbers (see (1.5) above). Lemma 2.2 (Addition theorem). k k iy q Bi (x, q )[y ]k−i Bk (x + y, q ) = (k ≥ 0), q i i=0 k k iy q Ei (x, q )[y ]k−i Ek (x + y, q ) = (k ≥ 0). q i i=0
6 Proof. Applying the relationship [x + y − 1 ]q = [y ]q + q y [x − 1 ]q to (2.14) for 2 2 x → x + y, we have k 1 q x+y− 2 1 Ek (x + y, q ) = E (q ) + x + y − 2 2 q k x− 1 q 1 2 qy = E (q ) + x − + [ y ]q 2 2 q i k x− 1 k iy q 1 2 [y ]k−i = q E (q ) + x − q i 2 2 q i=0 k k iy q Ei (x, q )[y ]k−i . = q i i=0 Similarly, the ﬁrst identity follows. Remark 2.3. From (2.12), we obtain the not completely trivial identities k k Bi (x)y k−i = (B (x) + y )k , lim Bk (x + y, q ) = i q →1 i=0 k k Ei (x)y k−i = (E (x) + y )k , lim Ek (x + y, q ) = i q →1 i=0 where q ∈ Cp tends to 1 in |q − 1|p < 1. Here Bi (x) and Ei (x) denote the classical Bernoulli and Euler polynomials, see [17,15] and see also the references cited in each of these earlier works. Lemma 2.4. Let n be any positive integer. Then k kii 1 q [n]q Bi (x, q n ) = [n]k Bk x + , q n , q i n i=0 k kii 1 q [n]q Ei (x, q n ) = [n]k Ek x + , q n . q i n i=0 Proof. Use Lemma 2.2, the proof can be obtained by the similar way to [7, Lemma 2.3]. We note here that similar expressions to those of Lemma 2.4 are given by Luo [7, Lemma 2.3]. Obviously, Lemma 2.4 are the q -analogues of k k ki 1 ki 1 n Bi (x) = nk Bk x + n Ei (x) = nk Ek x + , , i n i n i=0 i=0 respectively. We can now obtain the multiplication formulas by using p-adic integrals.
7 From (2.3), we see that [nx + z ]k dµ1 (z ) Bk (nx, q ) = q Zp npN −1 1 [nx + a]k = lim q N →∞ npN a=0 (2.17) n−1 pN −1 1 1 [nx + na + i]k = lim q n N →∞ pN i=0 a=0 n−1 k [n]k i q = x+ +z dµ1 (z ) n n Zp qn i=0 is equivalent to n−1 [n]k x+i n q (2.18) Bk (x, q ) = Bk ,q . n n i=0 If we put x = 0 in (2.18) and use (2.13), we ﬁnd easily that n−1 [ n] k in q Bk (q ) = Bk ,q n n i=0 [n]k n−1 k k −j k i q q ij Bj (q n ) = (2.19) n j n qn i=0 j =0 k n−1 1 k [ n] j Bj (q n ) q ij [i]k−j . = q q n j j =0 i=0 Obviously, Equation (2.19) is the q -analogue of k−1 n−1 1 k j ik−j , Bk = n Bj n(1 − nk ) j j =0 i=1 which is true for any positive integer k and any positive integer n > 1 (see [24, (2)]). From (2.4), we see that [nx + z ]k dµ−1 (z ) Ek (nx, q ) = q Zp n−1 pN −1 [nx + na + i]k (−1)na+i = lim (2.20) q N →∞ i=0 a=0 n−1 k i = [n]k (−1)i x+ +z dµ(−1)n (z ). q n Zp qn i=0 By (2.12) and (2.20), we ﬁnd easily that n−1 x+i n Ek (x, q ) = [n]k (−1)i Ek (2.21) ,q if n odd. q n i=0
8 From (2.18) and (2.21), we can obtain Proposition 2.5 below. Proposition 2.5 (Multiplication formulas). Let n be any positive integer. Then n−1 [ n] k x+i n q Bk (x, q ) = Bk ,q , n n i=0 n−1 x+i n Ek (x, q ) = [n]k (−1)i Ek ,q if n odd. q n i=0 3. Construction generating functions of q -Bernoulli, q -Euler numbers, and polynomials In the complex case, we shall explicitly determine the generating function Fq (t) of q -Bernoulli numbers and the generating function Gq (t) of q -Euler numbers: ∞ ∞ tk tk = eB (q)t = eE (q)t , (3.1) Fq (t) = Bk (q ) and Gq (t) = Ek (q ) k! k! k=0 k=0 where the symbol Bk (q ) and Ek (q ) are interpreted to mean that (B (q ))k and (E (q ))k must be replaced by Bk (q ) and Ek (q ) when we expanded the one on the right, respectively. Lemma 3.1. ∞ t log q t q m e[m]q t , Fq (t) = e 1−q + 1−q m=0 ∞ 1 (−1)m e2[m+ 2 ]q t . Gq (t) = 2 m=0 Proof. Combining (2.10) and (3.1), Fq (t) may be written as ∞ k i tk log q k (−1)i i Fq (t) = (1 − q )k i q − 1 k! i=0 k=0 ∞ k tk 1 1 k i (−1)i i = 1 + log q + . (1 − q )k k ! log q i=1 i q −1 k=1 Here, the term with i = 0 is understood to be 1/log q (the limiting value of the summand in the limit i → 0). Speciﬁcally, by making use of the following well- known binomial identity k−1 k k =i (k ≥ i ≥ 1). i−1 i
9 Thus, we ﬁnd that ∞ k tk 1 1 k−1 1 (−1)i i Fq (t) = 1 + log q +k (1 − q )k k ! log q i−1 q −1 i=1 k=1 ∞ ∞ ∞ k−1 tk tk 1 k k−1 qm (−1)i q mi = + log q (1 − q )k k ! (1 − q )k k ! m=0 i i=0 k=0 k=1 ∞ ∞ tk log q k t q m (1 − q m )k−1 =e + 1−q k −1 k ! 1−q (1 − q ) m=0 k=1 ∞ ∞ k 1 − qm tk t log q t m =e + q . 1−q 1−q 1−q k! m=0 k=0 Next, by (2.11) and (3.1), we obtain the result ∞ k 2k+1 1 tk k 1 (−1)i q 2 i i Gq (t) = (1 − q )k i q + 1 k! i=0 k=0 k ∞ ∞ 1 1 − q m+ 2 tk k m =2 2 (−1) 1−q k! m=0 k=0 ∞ ∞ k (2t)k 1 (−1)m =2 m+ 2 k! q m=0 k=0 ∞ 1 (−1)m e2[m+ 2 ]q t . =2 m=0 This completes the proof. Remark 3.2. The remarkable point is that the series on the right-hand side of Lemma 3.1 is uniformly convergent in the wider sense. From (2.13)and (2.14), we deﬁne the q -Bernoulli and q -Euler polynomials by ∞ ∞ tk tk (q x B (q ) + [x]q )k (3.2) Fq (t, x) = Bk (x, q ) = , k! k! k=0 k=0 k ∞ ∞ tk tk E (q ) 1 x− 1 (3.3) Gq (t, x) = Ek (x, q ) = q + x− . 2 k! 2 2 k! q k=0 k=0 Hence, we have Lemma 3.3. ∞ t log q t Fq (t, x) = e[x]q t Fq (q x t) = e 1−q + q m+x e[m+x]q t . 1−q m=0
10 Proof. From (3.1) and (3.2), we note that ∞ tk (q x B (q ) + [x]q )k Fq (t, x) = k! k=0 x = e(q B (q )+[x]q )t x = eB (q)q t e[x]q t = e[x]q t Fq (q x t). The second identity leads at once to Lemma 3.1. Hence, the lemma follows. Lemma 3.4. ∞ 1 q x− 2 [x − 1 ]q t (−1)m e[m+x]q t . Gq (t, x) = e Gq t =2 2 2 m=0 Proof. By similar method of Lemma 3.3, we prove this lemma by (3.1), (3.3), and Lemma 3.1. Corollary 3.5 (Diﬀerence equations). q x log q (k + 1)[x]k Bk+1 (x + 1, q ) − Bk+1 (x, q ) = (k ≥ 0), q q−1 Ek (x + 1, q ) + Ek (x, q ) = 2[x]k (k ≥ 0). q Proof. By applying (3.2) and Lemma 3.3, we obtain (3.4) ∞ tk Fq (t, x) = Bk (x, q ) k! k=0 ∞ ∞ tk+1 1 log q q m+x [m + x]k =1+ + (k + 1) . q (1 − q )k+1 1−q (k + 1)! m=0 k=0 By comparing the coeﬃcients of both sides of (3.4), we have B0 (x, q ) = 1 and ∞ 1 log q q m+x [m + x]k−1 (3.5) Bk (x, q ) = +k (k ≥ 1). q k (1 − q ) 1−q m=0 Hence, q x log q k−1 Bk (x + 1, q ) − Bk (x, q ) = k [x]q (k ≥ 1). q−1 Similarly we prove the second part by (3.3) and Lemma 3.4. This proof is complete. From Lemma 2.2 and Corollary 3.5, we obtain for any integer k ≥ 0, k+1 1 q−1 k+1 i [x]k = q Bi (x, q ) − Bk+1 (x, q ) , q k + 1 q x log q i i=0 k 1 ki [x]k = q Ei (x, q ) + Ek (x, q ) q 2 i i=0
11 which are the q -analogues of the following familiar expansions (see, e.g., [7, p. 9]): k k 1 k+1 1 k xk = and xk = Bi (x) Ei (x) + Ek (x) , k+1 i 2 i i=0 i=0 respectively. Corollary 3.6 (Diﬀerence equations). Let k ≥ 0 and n ≥ 1. Then 1n 1−n n Bk+1 x + ,q − Bk+1 x + ,q n n nq n(x−1)+1 log q k + 1 (1 + q [nx − n]q )k , = [n]k+1 q−1 q 1n 1−n n 2 (1 + q [nx − n]q )k . Ek x + ,q + Ek x + ,q = [n]k n n q Proof. Use Lemma 2.4 and Corollary 3.5, the proof can be obtained by the similar way to [7, Lemma 2.4]. Letting n = 1, Corollary 3.6 reduces to Corollary 3.5. Clearly, the above dif- ference formulas in Corollary 3.6 become the following diﬀerence formulas when q→1: k−1 1 1−n 1−n (3.6) Bk x + − Bk x + =k x+ (k ≥ 1, n ≥ 1), n n n k 1 1−n 1−n (3.7) Ek x + + Ek x + =2 x+ (k ≥ 0, n ≥ 1), n n n respectively (see [7, (2.22), (2.23)]). If we now let n = 1 in (3.6) and (3.7), we get the ordinary diﬀerence formulas Bk+1 (x + 1) − Bk+1 (x) = (k + 1)xk−1 and Ek (x + 1) + Ek (x) = 2xk for k ≥ 0. In Corollary 3.5, let x = 0. We arrive at the following proposition. Proposition 3.7. logp q if k = 1 (qB (q ) + 1)k − Bk (q ) = q −1 B0 (q ) = 1, 0 if k > 1, k k E (q ) 1 E (q ) 1 1 1 −2 E0 (q ) = 1, q +− + q + =0 if k ≥ 1. 2 2 2 2 2 q q Proof. The ﬁrst identity follows from (2.13). To see the second identity, setting x = 0 and x = 1 in (2.14) we have k k 1 1 q− 2 1 q2 1 Ek (0, q ) = E (q ) + − and Ek (1, q ) = E (q ) + . 2 2 2 2 q q This proof is complete.
12 Remark 3.8. (1). We note here that quite similar expressions to the ﬁrst identity of Proposition 3.7 are given by Kamano [3, Proposition 2.4], Rim et al. [8, Theorem 2.7] and Tsumura [10, (1)]. (2). Letting q → 1 in Proposition 3.7, the ﬁrst identity is the corresponding classical formulas in [8, (1.2)]: 1 if k = 1 (B + 1)k − Bk = B0 = 1, 0 if k > 1 and the second identity is the corresponding classical formulas in [25, (1.1)]: (E + 1)k + (E − 1)k = 0 E0 = 1, if k ≥ 1. 4. q -analogues of Riemann’s ζ -functions, the Hurwitz ζ -functions and the Didichlet’s L-functions Now, by evaluating the k th derivative of both sides of Lemma 3.1 at t = 0, we obtain the following ∞ k k d 1 k log q q m [m]k−1 , (4.1) Bk (q ) = Fq (t) = − q dt 1−q q−1 t=0 m=0 ∞ k k d 1 = 2k+1 (−1)m m + (4.2) Ek (q ) = Gq (t) dt 2 t=0 q m=0 for k ≥ 0. Deﬁnition 4.1 (q -analogues of the Riemann’s ζ -functions). For s ∈ C, deﬁne ∞ qm 1 1 log q ζ q ( s) = + , s−1 q − 1 m=1 [m]s s−1 1 q 1−q ∞ (−1)m 2 ζq,E (s) = s . 1s 2 m+ 2 q m=0 Note that ζq (s) is a meromorphic function on C with only one simple pole at s = 1 and ζq,E (s) is a analytic function on C. Also, we have ∞ ∞ (−1)m 1 (4.3) lim ζq (s) = = ζ (s) and lim ζq,E (s) = 2 = ζE (s). ms (m + 1)s q →1 q →1 m=1 m=0 (In [26, p. 1070], our ζE (s) is denote φ(s).) The values of ζq (s) and ζq,E (s) at non-positive integers are obtained by the following proposition. Proposition 4.2. For k ≥ 1, we have Bk (q ) ζq (1 − k ) = − and ζq,E (1 − k ) = Ek−1 (q ). k Proof. It is clear by (4.1) and (4.2). We can investigate the generating functions Fq (t, x) and Gq (t, x) by using a method similar to the method used to treat the q -analogues of Riemann’s ζ -functions in Deﬁnition 4.1.
13 Deﬁnition 4.3 (q -analogues of the Hurwitz ζ -functions). For s ∈ C and 0 < x ≤ 1, deﬁne ∞ q m+x 1 1 log q ζq (s, x) = + , s−1 q − 1 m=0 [m + x]s s−1 1 q 1−q ∞ (−1)m ζq,E (s, x) = 2 . [m + x]s q m=0 Note that ζq (s, x) is a meromorphic function on C with only one simple pole at s = 1 and ζq,E (s, x) is a analytic function on C. The values of ζq (s, x) and ζq,E (s, x) at non-positive integers are obtained by the following proposition. Proposition 4.4. For k ≥ 1, we have Bk (x, q ) ζq (1 − k, x) = − and ζq,E (1 − k, x) = Ek−1 (x, q ). k Proof. From Lemma 3.3 and Deﬁnition 4.3, we have k d Fq (t, x) = −kζq (1 − k, x) dt t=0 for k ≥ 1. We obtain the desired result by (3.2). Similarly the second form follows by Lemma 3.4 and (3.3). Proposition 4.5. Let d be any positive integer. Then d−1 1 x+i Fq (t, x) = Fq d [d]q t, , d d i=0 d−1 x+i (−1)i Gqd Gq (t, x) = [d]q t, if d odd. d i=0 Proof. Substituting m = nd + i with n = 0, 1, . . . and i = 0, . . . , d − 1 into Lemma 3.3, we have ∞ t log q t q m+x e[m+x]q t Fq (t, x) = e + 1−q 1−q m=0 d−1 ∞ [d]q t log q d 1 [d]q t q nd+x+i e[nd+x+i]q t = e 1−qd + 1 − qd d n=0 i=0 d− 1 ∞ [d]q t log q d 1 [d]q t [n+ x+i ]qd [d]q t x+i (q d )n+ = e 1−qd + e , d d 1 − qd d n=0 i=0 where we use [n +(x + i)/d]qd [d]q = [nd + x + i]q . So we have the ﬁrst form. Similarly the second form follows by Lemma 3.4. From (3.2), (3.3), Propositions 4.4 and 4.5, we obtain the following: Corollary 4.6. Let d and k be any positive integer. Then d−1 [d]k x+i q ζq (1 − k, x) = ζq d 1 − k, , d d i=0
14 d−1 x+i ζq,E (−k, x) = [d]k (−1)i ζqd ,E −k, if d odd. q d i=0 Let χ be a primitive Dirichlet character of conductor f ∈ N. We deﬁne the generating function Fq,χ (x, t) and Gq,χ (x, t) of the generalized q -Bernoulli and q - Euler polynomials as follows: ∞ tk Fq,χ (t, x) = Bk,χ (x, q ) k! k=0 (4.4) f 1 a+x = χ(a)Fqf [f ]q t, f f a=1 and ∞ tk Gq,χ (t, x) = Ek,χ (x, q ) k! k=0 (4.5) f a+x (−1)a χ(a)Gqf = [f ]q t, if f odd, f a=1 where Bk,χ (x, q ) and Ek,χ (x, q ) are the generalized q -Bernoulli and q -Euler poly- nomials, respectively. Clearly (4.4) and (4.5) are equal to ∞ t log q χ(m)q m+x e[m+x]q t , Fq,χ (t, x) = (4.6) 1−q m=0 ∞ (−1)m χ(m)e[m+x]q t Gq,χ (t, x) = 2 if f odd, (4.7) k=0 respectively. As q → 1 in (4.6) and (4.7), we have Fq,χ (t, x) → Fχ (t, x) and Gq,χ (t, x) → Gχ (t, x), where Fχ (t, x) and Gχ (t, x) are the usual generating func- tion of generalized Bernoulli and Euler numbers, respectively, which are deﬁned as follows [13]: f ∞ χ(a)te(a+x)t tk (4.8) Fχ (t, x) = = Bk,χ (x) , ef t − 1 k! a=1 k=0 f ∞ (−1)a χ(a)e(a+x)t tk (4.9) Gχ (t, x) = 2 = Gk,χ (x) if f odd. ef t + 1 k! a=1 k=0 From (3.2), (3.3), (4.4) and (4.5), we can easily see that f [f ]k a+x f q (4.10) Bk,χ (x, q ) = χ(a)Bk ,q , f f a=1 f a+x f [ f ]k (−1)a χ(a)Ek (4.11) Ek,χ (x, q ) = ,q if f odd. q f a=1 By using the deﬁnitions of ζq (s, x) and ζq,E (s, x), we can deﬁne the q -analogues of Dirichlet’s L-function.
15 Deﬁnition 4.7 (q -analogues of the Dirichlet’s L-functions). For s ∈ C and 0 < x ≤ 1, ∞ χ(m)q m+x log q Lq (s, x, χ) = , q − 1 m=0 [m + x]s q ∞ (−1)m χ(m) q (s, x, χ) =2 . [m + x]s q m=0 Similarly, we can compute the values of Lq (s, x, χ) at non-positive integers. Theorem 4.8. For k ≥ 1, we have Bk,χ (x, q ) Lq (1 − k, x, χ) = − and q (1 − k, x, χ) = Ek−1,χ (x, q ). k Proof. Using Lemma 3.3 and (4.4), we obtain f ∞ ∞ tk [f ]q t log q f 1 [f ]q t [n+ x+a ]qf [f ]q t x+a (q f )n+ Bk,χ (x, q ) = χ(a) e 1−qf + e f f 1 − qf k! f a=1 n=0 k=0 ∞ t log q χ(m)q m+x e[m+x]q t , = 1−q m=0 f where we use [n + (a + x)/f ]qf [f ]q = [nf + a + x]q and χ(a) = 0. Therefore, a=1 we obtain ∞ k tk d Bk,χ (x, q ) = Bk,χ (x, q ) dt k! t=0 k=0 ∞ k log q χ(m)q m+x [m + x]k−1 . = q 1−q m=0 Hence for k ≥ 1 ∞ Bk,χ (x, q ) log q χ(m)q m+x [m + x]k−1 − = q k q − 1 m=0 = Lq (1 − k, x, χ). Similarly the second identity follows. This completes the proof. Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors have equal contributions to each part of this paper. All the authors read and approved the ﬁnal manuscript. Acknowledgment This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2011-0001184).
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