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It is well-known that the classical Dedekind sums s(h, k) first arose in the transformation formula of the logarithm of the Dedekind-eta function. If h and k are coprime integers with k > 0, Dedekind sums are defined by
− k 1 a ha s (h, k)= , k k a=1 X where x − [x] − 1 , if x is not an integer ((x)) = G 2 0, if x is an integer, [x]G being the largest integer ≤ x. The most important property of Dedekind sums is the reciprocity law, which is given by
1 h k 1 1 s(h, k)+ s(k, h)= + + − . 12 k h hk 4 For detailed information of Dedekind sums see ([1], [4], [5], [6], [11], [13], [15], [18], [19], [23], [31], [32], [34], [35], [36], [38], [40], [44], [47], [51], [52]). In this paper, we define new Dedekind type sums related to Frobenius- Euler functions as follows:
Definition 1.1 Let n, h and k be positive integers with (h, k)=1. We define k−1 − ha a ha S (h, k)= u k H ,u , n,u k n k a=0 X ha where Hn k ,u denotes Frobenius-Euler function, which is given by Defini- tion 2.6, and u is an algebraic number =16 . The most important properties of these sums is the reciprocity law, which is given by the following theorem.
2 Theorem 1.2 Let n, h and k be positive integers with (h, k)=1. Then, we have k h u n u n k S k (h, k)+ h S h (k, h) 1 − uk n,u 1 − uh n,u n k h n u k j u h n−j = H u k H − u h j 1 − uk j 1 − uh n j j=0 X 1 u u + H (u)+ H (u) , hk 1 − u n+1 1 − u n where Hn (u) denotes Frobenius-Euler numbers given by (2.1). Throughout this paper, χ will denote a Dirichlet character of conductor f = fχ, and χ0 will be a principle character with conductor fχ0 = 1. We also define Dedekind type sums attached to χ as follows: Definition 1.3 Let n, h, k be positive integers with (h, k)=1. Dedekind type sums Sn,uk (h, k|χ) are defined by k−1 h−1 n −(kb+ha) a a b hk S k (h, k|χ)= h χ (kb + ha) u H + ,u . n,u k n k h a=0 X Xb=0 Note that if χ = χ0 (that is, f = 1), then uhk − 1 uk S k (h, k|1) = S k (h, k) . n,u uhk uk − 1 n,u We also note that the Definition 1.3 is different from Nagasaka et.al’s definition [34]. In [34], Dedekind sums with character are defined by us- ing Bernoulli polynomials and Bernoulli function. In our definition, we use Frobenius-Euler function H (x, u). Reciprocity law of Sn,uk (h, k|χ) is given by the following theorem:
Theorem 1.4 Let χ be a Dirichlet character of conductor f = fχ with f|hk. Let n, h and k be positive integers with (h, k)=1. Then, we have n n k Sn,uk (h, k|χ)+ h Sn,uh (k, h|χ) 1 − uhk uf 1 = H (u)+ H (u) uhk 1 − uf hk n+1,χ n,χ − − uhk k 1 h 1 + χ (kb + ha) u−(kb+ha) uhk − 1 a=0 b=0 X X n × 1H uhk hk + 2H uhk hk + kb + ha . 3 Proofs of Theorem 1.2 and Theorem 1.4 are given in Section 2 and Section 3, respectively. In this paper, Zp, Qp, Cp and C will, respectively, denote the ring of p- adic integers, the field of p-adic rational numbers, the p-adic completion of −1 the algebraic closure of Qp normalized by |p|p = p , and the complex field. Let q be an indeterminate such that if q ∈ C then |q| < 1 and if q ∈ Cp then −1/(p−1) x |1 − q|p < p , so that q =exp xlogpq for |x|p ≤ 1, where logp is the Iwasawa p-adic logarithm function ([20, Chap.4], [22], [24], [26], [49]). We use the notation 1 − qx [x] = [x : q]= , 1 − q so that limq→1 [x]= x. The p-adic q-integral (or q-Volkenborn integral) is originally constructed by Kim [24]. Kim indicated a connection between the q-Volkenborn integral and non-Archimedean combinatorial analysis. The p-adic q-Volkenborn in- tegral is used in mathematical physics, derivation of the functional equation of the q-zeta function and the q-Stirling numbers, and the q-Mahler theory of integration with respect to a ring Zp together with Iwasawa’s p-adic q- L-function. Recently, many applications of the q-Volkenborn integral have studied by the authors [14], [45], [46], [49], and many mathematicians. We give some basic properties of p-adic q-Volkenborn integral as follows: For g ∈ UD(Zp, Cp)= {g | g : Zp → Cp is uniformly differentiable function}, the p-adic q-Volkenborn integral is defined by [22], [24], [26]
pN −1 1 x Iq (g)= g (x) dµq (x) = lim g (x) q , N→∞ [pN : q] Z x=0 Zp X where qx µ x + pN Z = q p [pN : q] is the q-analogue of the Haar measure. For the limiting case q = 1,
pN −1 1 I1 (g) = lim Iq (g)= g (x) dµ1 (x) = lim g (x) , q→1 N→∞ pN Z x=0 Zp X with 1 µ x + pN Z = 1 p pN 4 is the Haar measure. If g1 (x)= g (x + 1), then ′ I1 (g1)= I1 (g)+ g (0) , (1.1)
′ d where g (0) = dx g (x) x=0 ([24], [26]). Let f be any fixed positive integer with (p, f) = 1. Then set
X X Z N Z X Z = f =←− lim /fp , 1 = p, N ∗ n X = a + fp Zp, 0