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). 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Some results for the q-Bernoulli, q-Euler numbers and polynomials Daeyeoul Kim 1 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, Z p , Q p , and C p denote the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of Q p , 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 C p . The Bernoulli polynomials, denoted by B n (x), are defined as (1.1) B n (x) = n  k=0  n k  B k x n−k , n ≥ 0, where B k are the Bernoulli numbers given by the coefficients in the power series (1.2) t e t − 1 = ∞  k=0 B k t k k! . From the above definition, we see B k ’s are all rational numbers. Since t e t −1 − 1 + t 2 is an even function (i.e., invariant under x → −x), we see that B k = 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 (1.3) B k = lim N→∞ 1 p N p N −1  a=0 a k (see [15,16]). Notice that, from the definition B k ∈ Q, and these integrals are independent of the prime p which used to compute them. The examples of (1.3) are: (1.4) lim N→∞ 1 p N p N −1  a=0 a = lim N→∞ 1 p N p N (p N − 1) 2 = − 1 2 = B 1 , lim N→∞ 1 p N p N −1  a=0 a 2 = lim N→∞ 1 p N p N (p N − 1)(2p N − 1) 6 = 1 6 = B 2 . Euler numbers E k , k ≥ 0 are integers given by (cf. [17–19]) (1.5) E 0 = 1, E k = − k−1  i=0 2|k−i  k i  E i for k = 1, 2, . . . . The Euler polynomial E k (x) is defined by (see [20, p. 25]): (1.6) E k (x) = k  i=0  k i  E i 2 i  x − 1 2  k−i , 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 2 k gives the Euler numbers (1.7) E k = 2 k E k  1 2  , where E 0 = 1, E 2 = −1, E 4 = 5, E 6 = −61, . . . . Therefore, E k = E k (0), in fact ([19, p. 374 (2.1)]) (1.8) E k (0) = 2 k + 1 (1 − 2 k+1 )B k+1 , where B k 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 Z p . 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 Z p 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 UD(Z p , C p ) denote the space of all uniformly (or strictly) differentiable C p - valued functions on Z p . The p-adic q-integral of a function f ∈ UD(Z p ) on Z p is defined by (2.1) I q (f) = lim N→∞ 1 [p N ] q p N −1  a=0 f(a)q a =  Z p f(z)dµ q (z), where [x] q = (1 − q x )/(1 − q), and the limit taken in the p-adic sense. Note that (2.2) lim q →1 [x] q = x for x ∈ Z p , where q tends to 1 in the region 0 < |q − 1| p < 1 (cf. [22,5,12]). The bosonic p-adic integral on Z p is considered as the limit q → 1, i.e., (2.3) I 1 (f) = lim N→∞ 1 p N p N −1  a=0 f(a) =  Z p f(z)dµ 1 (z). From (2.1), we have the fermionic p-adic integral on Z p as follows: (2.4) I −1 (f) = lim q→−1 I q (f) = lim N→∞ p N −1  a=0 f(a)(−1) a =  Z p f(z)dµ −1 (z). In particular, setting f(z) = [z] k q in (2.3) and f(z) =  z + 1 2  k q in (2.4), respectively, we get the following formulas for the p-adic q-Bernoulli and p-adic q-Euler numbers, respectively, if q ∈ C p with 0 < |q − 1| p < 1 as follows (2.5) B k (q) =  Z p [z] k q dµ 1 (z) = lim N→∞ 1 p N p N −1  a=0 [a] k q , (2.6) E k (q) = 2 k  Z p  z + 1 2  k q dµ −1 (z) = 2 k lim N→∞ p N −1  a=0  a + 1 2  k q (−1) a . 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]) B k (q) → B k =  Z p z k dµ 1 (z), E k (q) → E k =  Z p (2z + 1) k dµ −1 (z). 4 For |q − 1| p < 1 and z ∈ Z p , we have (2.7) q iz = ∞  n=0 (q i − 1) n  z n  and |q i − 1| p ≤ |q − 1| p < 1, 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 p N and x ∈ p N Z p (see [16]). By (2.3) and (2.7), we obtain (2.8) I 1 (q iz ) = 1 q i − 1 lim N→∞ (q i ) p N − 1 p N = 1 q i − 1 lim N→∞ 1 p N  ∞  m=0  p N m  (q i − 1) m − 1  = 1 q i − 1 lim N→∞ 1 p N ∞  m=1  p N m  (q i − 1) m = 1 q i − 1 lim N→∞ ∞  m=1 1 m  p N − 1 m − 1  (q i − 1) m = 1 q i − 1 ∞  m=1 1 m  −1 m − 1  (q i − 1) m = 1 q i − 1 ∞  m=1 (−1) m−1 (q i − 1) m m = i log q q i − 1 since the series log(1 + x) =  ∞ m=1 (−1) m−1 x m /m converges at |x| p < 1. Similarly, by (2.4), we obtain (see [4, p. 4, (2.10)]) (2.9) I −1 (q iz ) = lim N→∞ p N −1  a=0 (q i ) a (−1) a = 2 q i + 1 . From (2.5), (2.6), (2.8) and (2.9), we obtain the following explicit formulas of B k (q) and E k (q): (2.10) B k (q) = log q (1 − q) k k  i=0  k i  (−1) i i q i − 1 , (2.11) E k (q) = 2 k+1 (1 − q) k k  i=0  k i  (−1) i q 1 2 i 1 q i + 1 , 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 Z p : (2.12) B k (x, q) =  Z p [x + z] k q dµ 1 (z) and E k (x, q) =  Z p [x + z] k q dµ −1 (z), 5 where q ∈ C p with 0 < |q − 1| p < 1 and x ∈ Z p , 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 (2.13) B k (x, q) = k  i=0  k i  [x] k−i q q ix B i (q) = (q x B(q) + [x] q ) k and (2.14) E k (x, q) = k  i=0  k i  E i (q) 2 i  x − 1 2  k−i q q i(x− 1 2 ) =  q x− 1 2 2 E(q) +  x − 1 2  q  k , where the symbol B k (q) and E k (q) are interpreted to mean that (B(q)) k and (E(q)) k must be replaced by B k (q) and E k (q) when we expanded the one on the right, respectively, since [x + y] k q = ([x] q + q x [y] q ) k and (2.15) [x + z] k q =  1 2  k q  [2x − 1] q 1 2 + q x− 1 2  1 2  −1 q  z + 1 2  q  k =  1 2  k q k  i=0  k i  [2x − 1] k−i q q (x− 1 2 )i  1 2  −i q  z + 1 2  i q (cf. [4,5]). The above formulas can be found in [7] which are the q-analogues of the corresponding classical formulas in [17, (1.2)] and [23], etc. Obviously, put x = 1 2 in (2.14). Then (2.16) E k (q) = 2 k E k  1 2 , q  = E k (0, q) and lim q →1 E k (q) = E k , where E k are Euler numbers (see (1.5) above). Lemma 2.2 (Addition theorem). B k (x + y, q) = k  i=0  k i  q iy B i (x, q)[y] k−i q (k ≥ 0), E k (x + y, q) = k  i=0  k i  q iy E i (x, q)[y] k−i q (k ≥ 0). 6 Proof. Applying the relationship [x + y − 1 2 ] q = [y] q + q y [x − 1 2 ] q to (2.14) for x → x + y, we have E k (x + y, q) =  q x+y − 1 2 2 E(q) +  x + y − 1 2  q  k =  q y  q x− 1 2 2 E(q) +  x − 1 2  q  + [y] q  k = k  i=0  k i  q iy  q x− 1 2 2 E(q) +  x − 1 2  q  i [y] k−i q = k  i=0  k i  q iy E i (x, q)[y] k−i q . Similarly, the first identity follows.  Remark 2.3. From (2.12), we obtain the not completely trivial identities lim q →1 B k (x + y, q) = k  i=0  k i  B i (x)y k−i = (B(x) + y) k , lim q →1 E k (x + y, q) = k  i=0  k i  E i (x)y k−i = (E(x) + y) k , where q ∈ C p tends to 1 in |q − 1| p < 1. Here B i (x) and E i (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  i=0  k i  q i [n] i q B i (x, q n ) = [n] k q B k  x + 1 n , q n  , k  i=0  k i  q i [n] i q E i (x, q n ) = [n] k q E k  x + 1 n , q n  . 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  i=0  k i  n i B i (x) = n k B k  x + 1 n  , k  i=0  k i  n i E i (x) = n k E k  x + 1 n  , respectively. We can now obtain the multiplication formulas by using p-adic integrals. 7 From (2.3), we see that (2.17) B k (nx, q) =  Z p [nx + z] k q dµ 1 (z) = lim N→∞ 1 np N np N −1  a=0 [nx + a] k q = 1 n lim N→∞ 1 p N n−1  i=0 p N −1  a=0 [nx + na + i] k q = [n] k q n n−1  i=0  Z p  x + i n + z  k q n dµ 1 (z) is equivalent to (2.18) B k (x, q) = [n] k q n n−1  i=0 B k  x + i n , q n  . If we put x = 0 in (2.18) and use (2.13), we find easily that (2.19) B k (q) = [n] k q n n−1  i=0 B k  i n , q n  = [n] k q n n−1  i=0 k  j=0  k j  i n  k−j q n q ij B j (q n ) = 1 n k  j=0 [n] j q  k j  B j (q n ) n−1  i=0 q ij [i] k−j q . Obviously, Equation (2.19) is the q-analogue of B k = 1 n(1 − n k ) k−1  j=0 n j  k j  B j n−1  i=1 i k−j , which is true for any positive integer k and any positive integer n > 1 (see [24, (2)]). From (2.4), we see that (2.20) E k (nx, q) =  Z p [nx + z] k q dµ −1 (z) = lim N→∞ n−1  i=0 p N −1  a=0 [nx + na + i] k q (−1) na+i = [n] k q n−1  i=0 (−1) i  Z p  x + i n + z  k q n dµ (−1) n (z). By (2.12) and (2.20), we find easily that (2.21) E k (x, q) = [n] k q n−1  i=0 (−1) i E k  x + i n , q n  if n odd. 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 B k (x, q) = [n] k q n n−1  i=0 B k  x + i n , q n  , E k (x, q) = [n] k q n−1  i=0 (−1) i E k  x + i n , q n  if n odd. 3. Construction generating functions of q-Bernoulli, q-Euler numbers, and polynomials In the complex case, we shall explicitly determine the generating function F q (t) of q-Bernoulli numbers and the generating function G q (t) of q-Euler numbers: (3.1) F q (t) = ∞  k=0 B k (q) t k k! = e B(q)t and G q (t) = ∞  k=0 E k (q) t k k! = e E(q)t , where the symbol B k (q) and E k (q) are interpreted to mean that (B(q)) k and (E(q)) k must be replaced by B k (q) and E k (q) when we expanded the one on the right, respectively. Lemma 3.1. F q (t) = e t 1−q + t log q 1 − q ∞  m=0 q m e [m] q t , G q (t) = 2 ∞  m=0 (−1) m e 2[m+ 1 2 ] q t . Proof. Combining (2.10) and (3.1), F q (t) may be written as F q (t) = ∞  k=0 log q (1 − q) k k  i=0  k i  (−1) i i q i − 1 t k k! = 1 + log q ∞  k=1 1 (1 − q) k t k k!  1 log q + k  i=1  k i  (−1) i i q i − 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  k − 1 i − 1  = i  k i  (k ≥ i ≥ 1). 9 Thus, we find that F q (t) = 1 + log q ∞  k=1 1 (1 − q) k t k k!  1 log q + k k  i=1  k − 1 i − 1  (−1) i 1 q i − 1  = ∞  k=0 1 (1 − q) k t k k! + log q ∞  k=1 k (1 − q) k t k k! ∞  m=0 q m k−1  i=0  k − 1 i  (−1) i q mi = e t 1−q + log q 1 − q ∞  k=1 k (1 − q) k−1 t k k! ∞  m=0 q m (1 − q m ) k−1 = e t 1−q + t log q 1 − q ∞  m=0 q m ∞  k=0  1 − q m 1 − q  k t k k! . Next, by (2.11) and (3.1), we obtain the result G q (t) = ∞  k=0 2 k+1 (1 − q) k k  i=0  k i  (−1) i q 1 2 i 1 q i + 1 t k k! = 2 ∞  k=0 2 k ∞  m=0 (−1) m  1 − q m+ 1 2 1 − q  k t k k! = 2 ∞  m=0 (−1) m ∞  k=0  m + 1 2  k q (2t) k k! = 2 ∞  m=0 (−1) m e 2[m+ 1 2 ] q t . 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 (3.2) F q (t, x) = ∞  k=0 B k (x, q) t k k! = ∞  k=0 (q x B(q) + [x] q ) k t k k! , (3.3) G q (t, x) = ∞  k=0 E k (x, q) t k k! = ∞  k=0  q x− 1 2 E(q) 2 +  x − 1 2  q  k t k k! . Hence, we have Lemma 3.3. F q (t, x) = e [x] q t F q (q x t) = e t 1−q + t log q 1 − q ∞  m=0 q m+x e [m+x] q t . [...]... [10] Tsumura, H: A note on q-analogue of the Dirichlet series and q-Bernoulli numbers J Number Theory 39, 251–256 (1991) [11] Choi, J, Anderson, PJ, Srivastava, HM: Carlitz’s q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions Appl Math Comput 215(3), 1185–1208 (2009) [12] Kim, T: On the analogs of Euler numbers and polynomials associated with p-adic q-integral... functions J Number Theory 129(7), 1798–1804 (2009) [7] Luo, Q-M: Some results for the q-Bernoulli and q-Euler polynomials J Math Anal Appl 363(1), 7–18 (2010) [8] Rim, S-H, Bayad, A, Moon, E-J, Jin, J-H, Lee, S-J: A new costruction on the q-Bernoulli polynomials J Inequal Appl Adv Diff Equ 2011, 34 (2011) [9] Satoh, J: q-analogue of Riemann’s ζ-function and q-Euler numbers J Number Theory 31, 346–62... → 1 in Proposition 3.7, the first identity is the corresponding classical formulas in [8, (1.2)]: 1 0 (B + 1)k − Bk = B0 = 1, if k = 1 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 kth derivative of both... q-Bernoulli numbers and their denominators Int J Number Theory 4(6), 911–925 (2008) [4] Kim, M-S, Kim, T, Ryoo, C-S: On Carlitz’s type q-Euler numbers associated with the fermionic p-adic integral on Zp J Inequal Appl 2010, Article ID 358986, 13 (2010) [5] Kim, T: On a q-analogue of the p-adic log gamma functions and related integrals J Number Theory 76, 320–329 (1999) [6] Kim, T: Note on the Euler... = 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) and ζq,E (1 − k, x) = Ek−1 (x, q) ζq (1 − k, x) = − k Proof From Lemma 3.3 and Definition 4.3, we have k d dt Fq (t, x) = −kζq (1 − k, x) t=0 for k ≥ 1 We obtain the desired result by (3.2) Similarly the. .. 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) 16 References [1] Carlitz, L: q-Bernoulli numbers and polynomials Duke Math J 15, 987–1000 (1948) [2] Carlitz, L: q-Bernoulli and Eulerian numbers Trans Am Math Soc 76, 332–350... [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 ζq (1 − k, x) = [d]k q d d−1 ζq d i=0 1 − k, x+i d , 14 d−1 ζq,E (−k, x) = [d]k q (−1)i ζqd ,E −k, i=0 x+i d if d odd 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 qEuler polynomials as follows: ∞ Fq,χ (t, x) = Bk,χ (x, q) k=0 (4.4) 1 f = tk k! f χ(a)Fqf [f ]q t, a=1 a+x f and ∞ Gq,χ (t, x) = Ek,χ (x, q) k=0 (4.5) tk k! f (−1)a χ(a)Gqf = [f ]q t, a=1 a+x f if f odd, where Bk,χ (x, q) and Ek,χ (x, q) are the generalized q-Bernoulli and q-Euler polynomials, respectively Clearly (4.4) and (4.5) are... + x]q and we obtain ∞ k d tk Bk,χ (x, q) = Bk,χ (x, q) dt k! = k log q 1−q k=0 ∞ f a=1 χ(a) = 0 Therefore, t=0 χ(m)q m+x [m + x]k−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... (1851) [22] Kim, T: On explicit formulas of p-adic q-L-functions Kyushu J Math 48(1), 73–86 (1994) [23] Srivastava, HM, Choi, J: Series Associated with the Zeta and Related Functions Kluwer Academic Publishers, Dordrecht (2001) [24] Howard, FT: Applications of a recurrence for the Bernoulli numbers J Number Theory 52, 157–172 (1995) [25] Carlitz, L: A note on Euler numbers and polynomials Nagoya Math J . and some properties for the q-Bernoulli, Euler numbers, and polynomials related to the bosonic and the fermionic p-adic integral on Z p . In Section 3, we obtain the generating functions of the. original work is properly cited. Some results for the q-Bernoulli, q-Euler numbers and polynomials Daeyeoul Kim 1 and Min-Soo Kim ∗2 1 National Institute for Mathematical Sciences, Doryong-dong,. 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