p -Adic q -Twisted Dedekind-Type Sums

: The main purpose of this paper is to deﬁne p -adic and q -Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized some of the classical p -adic Dedekind sums and their reciprocity law. It is to be noted that the Dedekind reciprocity laws, is a ﬁne study of the existing symmetry relations between the ﬁnite sums, considered in our study, and their symmetries through permutations of initial parameters.


Introduction
For a positive integer h and an integer k, the classical Dedekind sum is defined as 1 2 , if x / ∈ Z, ((x)) = 0, x ∈ Z, and [x] denotes the greatest integer not exceeding x. Dedekind [1] introduced this sum in connection with the transformation formula for, the well-known modular form of weight 1 2 , the Dedekind η-function given by More precisely, we quote from [2] [p. 52, Theorem 3.4] the η-transformation formula.
From this transformation formula, for h and k are coprime integers, Dedekind deduced his reciprocity formula s(h, k) + s(k, h) = − 1 4 + 1 12 In 1950, Apostol [3] generalized s(h, k) by defining For odd values of m, these higher-order Dedekind sums enjoy a reciprocity law, first proved by Apostol [3]: In 1953, Carlitz [4] generalized s m (h, k) by defining the higher order Dedekind sums as and proved their reciprocity laws similar to (2).
Recently, many mathematicians study on p-adic Dedekind type sums. Using Washington's definition of p-adic Hurwitz zeta functions [5], Rosen and Snyder [6] constructed a p-adic interpolation of Apostol's Dedekind sums s m (h, k) and established new reciprocity laws satisfied by these sums. Their method and techniques were generalized by Kudo [7] who defined a p-adic analogue of the higher order Dedekind sums s (r) m (h, k). In fact, Kudo in terms of the Euler numbers, he defined continuous function which interpolates the Euler numbers and p-adic higher order Dedekind sums.
Using a p-adic q-integral invariant on Z p , Kim in [8-10] constructed a p-adic continuous function that provides a p-adic q-analogue of higher order Dedekind type DC sums.
In 2017, Hu and Min-Soo in [11] using Cohen-Tangedal-Young's theory on the p-adic Hurwitz zeta functions, they construct some analytic Dedekind sums on the p-adic complex plane C p . Their sums interpolate Carlitz's higher order Dedekind sums p-adically. They also proved a reciprocity relation for the special values of their p-adic Dedekind sums.
In this paper our motivation is the same of Rosen-Snyder [6], Kim [8,10] and Kudo [7], Hu and Min-Soo [11] and others. We shall consider p-adic q-analogue to the twisted Dedekind sums s (r) m (h, k). For this, we define continuous functions on Z p , which involves p-adic twisted q-Dedekind type sums. We construct p-adic Dedekind sums, which are defined in Definition 2. We call them p-adic q-analogue Dedekind type sums S (h) p,ξ (s; a, b : q), where the parameters h is an integer number, ξ ∈ T p and s ∈ Z p . The main result of this paper is to prove their Dedekind reciprocity laws.

Preliminaries and Definitions
In this section, we consider p-adic (h, q)-Dedekind type sums on Z p and investigate some of their properties.
We need the following definitions and notations: If q ∈ C p , then for |x| p 1.
If q ∈ C, then we assume that |q| < 1. According to [12,13], for each integer N ≥ 0, C p N denotes the multiplicative group of the primitive p N th roots of unity in C * The dual of Z p , in the sense of p-adic Pontrjagin duality, is T p = C p ∞ , the direct limit (under inclusion) of cyclic groups C p N of order p N with N ≥ 0, with the discrete topology. T p admits a natural Z p -module structure which we shall write exponentially, viz ξ x for ξ ∈ T p and x ∈ Z p . T p can be embedded discretely in C p as the multiplicative p-torsion subgroup, and we choose, for once and all, one such embedding. If ξ ∈ T p , then is the locally constant character, x → ξ x , which is locally analytic character if where v p denotes the valuation. Then φ ξ has continuation to a continuous group homomorphism from (Z p , +) to (C p , .) cf. ( [12][13][14][15]), see also the references cited in each of these earlier works. If ξ ∈ C, then we assume that ξ is an r-th root of 1 with r ∈ Z + the set of positive integers. For where µ q denotes p-adic q-Haar distribution which is defined by Definition 1. Let h, a and b be positive integers with (a, b) = 1, and let p be an odd prime such that p b. For ξ ∈ T p , we define twisted (h, q)-Dedekind type sums as where {t} denotes the fractional part of t.
Let < a > denote the principal unit associated with a, which is the unique unit given by the decomposition where a ∈ Z * p = Z p \ pZ p = x : |x| p = 1 , and w denotes the Teichmüller character (or function), which is defined as follows: w(ab) = w(a)w(b), and |w(a + b) − w(a) − w(a)| p < 1 (cf [23], see also [5,6]). Some properties of the Teichmüller character are given by |a − w(a)| p < 1, w 0 (a) = w(a) (cf [23], see also [5,6]). Using (9), a few values of w(a) are give as follows: = w 2 − 2 5 = 1, and so on (cf [23]). Let a and b be positive integers such that (p, a) = 1 and p|b. We extend Thus, we define where s ∈ Z p , and

Statement of Main Result
We define the following p-adic q-twisted Dedekind sum as follows.

Definition 2.
Let a and b be relatively prime positive integers such that p b. Let s ∈ Z p . The p-adic Dedekind sum is defined by The main result of this paper is the following theorem.
Theorem 2. Let a and b relatively prime positive integers p ≡ 1 (mod ab). Then is a continuous function of s on Z p and for all positive integer n such that n + 1 ≡ 0 (mod (p − 1)) this reciprocity law reduces to the reciprocity law for higher-order q-Dedekind type sum which is given by authors [18]: n,ξ (q) and B (h,2) n+2,ξ (q | (a, b)) are given respectively by (5) and (8).
The above theorem is a generalization of the Rosen and Snyder' [6] p-adic Dedekind sums. If m + 1 ≡ 0 (mod (p − 1)), then we have

Proof. Let
By Witt's type formula of the twisted (h, q)-Bernoulli polynomials (see [21]), we have which is continuous p-adic extension of b m−1 B (h) m,ξ j b , q . By using the above integral equation, we have From the above, we obtain Each term in the above sum is interpolated except for the one term for which a + jN ≡ 0 (mod p). Let (p −1 a) N denote the integer x such that 0 ≤ x < N and px ≡ a (mod p). The exceptional term may be written as Thus, we may p-adically interpolate The p-adic function is given by Thus, we have where (p −1 a) N denotes the integer x such that 0 ≤ x < N and px ≡ a (mod p) and m is integer with m + 1 ≡ 0 (mod (p − 1)). By (11), we interpolate b m s (h) m,ξ (a, b : q) where p|b and (ac, p) = 1 for each c = 0, 1, · · · , b − 1; for all m + 1 ≡ 0 (mod (p − 1)), where (α n ) denotes the integer x such that 0 ≤ x < n and x ≡ α (mod n). If p|b, then s We end this paper by the following remark.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: C the field of complex numbers. p an odd rational prime number.
Z p the ring of p-adic integers.
Q p the field of fractions of Z p . C p the completion of a fixed algebraic closure Q p of Q. v p the p-adic valuation of C p normalized so that |p| p = p −v p (p) = p −1 . C p N denotes the multiplicative group of the primitive p N th roots of unity in C * p . T p denotes the set ∪ N≥0 C p N .