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Special Functions Related to Dedekind Type DC-Sums and Their SPECIAL FUNCTIONS RELATED TO DEDEKIND TYPE DC-SUMS AND THEIR APPLICATIONS Yilmaz SIMSEK Akdeniz University, Faculty of Arts and Science, Department of Mathematics 07058 Antalya, Turkey [email protected] Abstract In this paper we construct trigonometric functions of the sum Tp(h, k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric represen- tations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations between the Clausen functions, Polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums and the sum Tp(h, k). We also give some applications related to these sums and functions. 2000 Mathematics Subject Classification. 11F20, 11B68, 11M35, 11M41, 11S80, 33C10, 33E20. Key Words and Phrases. Dedekind sums, Hardy-Berndt sums, Bernoulli Functions, Euler Functions, Euler numbers and polynomials, Riemann zeta function, Hurwitz zeta function, Lerch zeta function, Dirichlet series for the polylogarithm function, Dirichlet’s eta function, Legendre chi function, trigonometric functions and generalized Lambert series. arXiv:0902.0380v1 [math.CV] 2 Feb 2009 1 Introduction, Definitions and Notations In this section, we give some definitions, notations and results related to the Dedekind sums. Firstly we start with the definition of the classical Dedekind sums. Let x [x] 1 , if x / Z ((x)) = − G − 2 ∈ 0, if x Z, ∈ 1 [x]G being the largest integer x. Let h and k be coprime integers with k > 0, the classical Dedekind sum≤ s(h, k) is defined as follows k−1 a ha s (h, k)= . k k a=1 X The reciprocity law of the classical Dedekind sums is given by 1 1 h k 1 s(h, k)+ s(k, h)= + + + , −4 12 k h hk where (h, k) = 1 and h, k N := 1, 2, 3, ... , and N0 := N 0 . The classical Dedekind∈ sums s{(h, k) arise} in the transformation∪{ } formula the Dedekind-eta function. By using this transformation formula, Dedekind proved reciprocity law of the classical Dedekind sums cf. [18]. For other proofs of reciprocity law of the classical Dedekind sums, see cf. ([23], [41], [1], [3], [7]-[9], [19], [22], [26]), see also the references cited in each of these earlier works. In the literature of the Dedekind sums, there are several generalizations of the Dedekind sums that involve higher order Bernoulli functions and Euler functions, the reader should look at [23], [1], [7]-[9], [19], [4], [5] and [36] for references and, see also the references cited in each of these earlier works. In 1950, Apostol ([1], [3]) generalized Dedekind sums as follows: a ah S (h, k)= B ( ), (1) p k p k aXmod k where (h, k) = 1 and h, k N and Bp(x) is the p-th Bernoulli function, which is defined as follows: ∈ Bp(x)= Bp(x [x]G) (2) − ∞ = p!(2πi)−p m−pe2πimx, − m=−∞ Xm=06 where Bp(x) is the usual Bernoulli polynomials, which are defined by means of the following generating function tetz ∞ tn = B (z) , t < 2π et 1 n n! | | n=0 − X 2 where Bn(0) = Bn is denoted the Bernoulli number cf. [1]-[59]. Observe that when p = 1, the sums S1(h, k) are known as the classical Dedekind sums, s(h, k). The following theorem proved by Apostol [1]: Theorem 1 Let (h, k)=1. For odd p 1, we have ≥ p! ∞ 1 e2πimh/k e−2πimh/k S (h, k)= ( ). (3) p (2πi)p mp 1 e2πimh/k − 1 e−2πimh/k mX=1 − − m 0(k) 6≡ In [3], Apostol established a connection between the sums Sp(h, k) and certain finite sums involving Hurwitz zeta functions. By using this relation, he proved the reciprocity law of the sum Sp(h, k). By using same motivation of the Dedekind sums,in this paper, we study on infinite series representation of the Dedekind type DC-sum, reciprocity law of this sum and some special functions. In [31] and [32], Kim defined the first kind n-th Euler function Em(x) as follows: 2(m!) ∞ e(2n+1)πix E (x)= , (4) m (πi)m+1 (2n + 1)m+1 n =X −∞ n =0 6 where m N. Hoffman [25] studied on Fourier series of Euler polynomials. He also expressed∈ the values of Euler polynomials at any rational argument in terms of tan x and sec x. Observe that if 0 x < 1, then (4) reduces to the first kind n-th Euler ≤ polynomials En(x) which are defined by means of the following generating function 2etx ∞ tn = E (x) , t < π. (5) et +1 n n! | | n=0 X Observe that En(0) = En denotes the first kind Euler number which is given by the following recurrence formula n n E = 1 and E = E , (6) 0 n − k k Xk=0 3 1 1 n 1 Some of them are given by 1, 2 , 0, 4 , , En = 2 En( 2 ) and E2n = 0, (n N) cf. ([28]-[37], [25], [39],− [47], [53],··· [51]) and see also the references ∈ cited in each of these earlier works. In [32] and [31], by using Fourier transform for the Euler function, Kim derived some formulae related to infinite series and the first kind Euler num- bers. For example, (4), and ∞ 1 ( 1)m+1π2m+2E = − 2m+1 . (7) (2n + 1)2m+2 4(2m + 1)! n=1 X Kim [31] gave the following results: 1 2ex ∞ tn π sec hx = = = E∗ , t < , (8a) cosh x e2x +1 n n! | | 2 n=0 X ∗ where Em is denoted the second kind Euler numbers. By (5) and (8a), it is easy to see that m m E∗ = 2nE , m n n n=0 X and m−1 2m E∗ = E∗ cf. [31]. 2m − 2n 2n n=0 X ∗ ∗ ∗ ∗ ∗ From the above E0 = 1, E1 = 0, E2 = 1, E3 = 0, E4 = 5, , and E∗ =0, (m N). − ··· 2m+1 ∈ The first and the second kind Euler numbers are also related to tan z and sec z. eiz e−iz e2iz 2 e−2iz 2 tan z = i − = . − eiz + e−iz 2i e2iz +1 − 2i e−2iz +1 4 By using (5) and Cauchy product, we have 1 ∞ (2iz)n ∞ (2iz)n 1 ∞ ( 2iz)n ∞ ( 2iz)n tan z = E E − − 2i n n! n! − 2i n n! n! n=0 n=0 n=0 n=0 X X X X 1 ∞ n (2iz)k (2iz)n−k 1 ∞ n ( 2iz)k ( 2iz)n−k = E E − − 2i k k! (n k)! − 2i k k! (n k)! n=0 n=0 X Xk=0 − X Xk=0 − 1 ∞ n E 1 ∞ n E = k (2i)nzn k ( 2i)nzn 2i k!(n k)! − 2i k!(n k)! − n=0 k=0 − n=0 k=0 − X X 2j+1 X X ∞ 2j +1 z2j+1 = ( 1)n22j+1 E − k k (2j + 1)! j=0 ! X Xk=0 By using (6), we find that ∞ 22n+1E tan z = ( 1)n+1 2n+1 z2n+1, z < π. (9) − (2n + 1)! | | n=0 X Remark 2 The other proofs of (9) has also given the references cited in each of these earlier work. In [31], Kim gave another proof of (9). We shall give just a brief sketch as the details are similar to those in [31]. eiz e−iz i tan z = − (10) eiz + e−iz 2 4 = 1 + . − e2iz 1 e4iz 1 − − From the above ∞ 4n(1 4n)B z tan z = ( 1)n − 2n z2n. − (2n)! n=1 X By using the above, we arrive at (9). Similarly Kim [31] also gave the fol- lowing relation: ∞ E∗ π sec z = ( 1)n 2n z2n, z < . − (2n)! | | 2 n=0 X Kim [36] defined the Dedekind type DC (Daehee-Changhee) sums as fol- lows: 5 Definition 3 Let h and k be coprime integers with k> 0. Then k−1 j hj T (h, k)=2 ( 1)j−1 E ( ), (11) m − k m k j=1 X where Em(x) denotes the m-th (first kind) Euler function. The sum Tm(h, k) gives us same behavior of the Dedekind sums. Several properties and identities of the sum Tm(h, k) and Euler polynomials were given by Kim [36]. By using these identities, Kim [36] proved many theorems. The most fundamental result in the theory of the Dedekind sums, Hardy- Berndt sums, Dedekind type DC and the other arithmetical sums is the reciprocity law. The reciprocity law can be used as an aid in calculating these sums. The reciprocity law of the sum Tm(h, k) is given as follows: Theorem 4 ([36])Let (h, k)=1 and h, k N with h 1mod2 and k 1mod2. Then we have ∈ ≡ ≡ p p k Tp(h, k)+ h Tp(k, h) k−1 u hu p = 2 kh(E + + k(E + h [ ]) +(hE + kE)p +(p + 2)E , k − k p uX=0 u [ hu ] 1mod2 − k ≡ where n+1 n +1 (hE + kE)n+1 = hjE kn+1−jE . j j n+1−j j=1 X We summarize the result of this paper as follows: In Section 2, we construct trigonometric representation of the sum Tp(h, k). We give analytic properties of the sum Tp(h, k). In Section 3, we give some special functions and their relations. By using these functions, we obtain relations between the sum Tp(h, k), Hurwitz zeta function, Lerch zeta function, Dirichlet series for the polylogarithm function, Dirichlet’s eta function and Clausen functions. In Section 4, we prove reciprocity law of the sum Tp(h, k).
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