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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

g (x) dµ1 (x)= g (x) dµ1 (x)

ZZp ZX for g ∈ UD (Zp, Cp) ([24], [26]). Let Tp = Cpn = lim Cpn , n→∞ n>1 [ pn n where Cpn = ζ : ζ =1 is the cyclic of order p . For ζ ∈ Tp, the function x 7→ ζx is a locally constant function from Z to C ([21],  p p [28]). By using q-Volkenborn integration, the second author [46] defined (h) of twisted (h, q)-extension of Bernoulli numbers Bn,ζ (q) (h) and polynomials Bn,ζ (x, q) by means of ∞ ∞ h log q + t tn h log q + t tn = B(h) (q) , and ext = B(h) (x, q) , (1.2) ζqhet − 1 n,ζ n! ζqhet − 1 n,ζ n! n=0 n=0 X X (h) respectively. Note that the numbers Bn,ζ (q) are given by [46] h log q n B(h) (q)= and ζqh B(h) (q)+1 − B(h) (q)= δ , 0,ζ ζqh − 1 ζ n,ζ n,1   j (h) (h) with the usual convention about replacing Bζ (q) by Bj,ζ (q) in the bi-   (h) nomial expansion, where δn,1 is the Kronecker symbol. If ζ → 1, Bj,ζ (q) → (h) Bj (q), which are the numbers defined by Kim [27].

5 In p-adic case, by using p-adic q-Volkenborn integral and twisted (h, q)- Bernoulli functions, we construct p-adic (h, q)-higher order Dedekind type sums as follows:

Definition 1.5 Let h, a and b be fixed integers with (a, b)=1, and let p be an odd prime such that p|b. For ζ ∈ Tp, we define twisted (h, q)-Dedekind type sums as

− b 1 j ja m s(h) (a, b : q)= qhxζx x + dµ (x) , m,ζ b b 1 j=0 Z    X Zp where {t} denotes the fractional part of a real number t.

(1) Observe that when h = 1, q → 1 and ζ → 1, the sum sm,1 (a, b : 1) reduces m to p-adic analogue of higher order Dedekind sums b sm+1 (a, b), defined by (h) Apostol [1]. The main properties of sm,ζ (a, b : q) will be given in Section 5. Dedekind sums were generalized by various mathematicians. Here, we list some of them. Apostol [1] defined generalized Dedekind sums sn (h, k) by − k 1 a ha s (h, k)= B , (1.3) n k n k a=1 X   where n, h, k are positive integers and Bn (x) is the nth Bernoulli function, which is defined as follows:

Bn (x)= Bn (x − [x]G) , if n> 1, (1.4) B (x − [x] ) , if x∈ / Z B (x)= 1 G 1 0, if x ∈ Z,  where Bn (x) is the Bernoulli polynomial [1], [44], [49]. For odd values of n, these generalized Dedekind sums satisfy a reciprocity law, first proved by Apostol [1]:

n n (n + 1) {hk sn(h, k)+ kh sn(k, h)} n+1 n +1 j j n+1−j = (−1) B h B − k + nB , j j n+1 j n+1 j=0 X  

6 where (h, k) = 1 and Bn is the nth . Berndt [4] gave a character transformation formula similar to those for the Dedekind η-function and defined Dedekind sums with character s(h, k; χ) by

kf−1 ha a s(h, k; χ)= χ(a)B B , 1,χ k 1 kf a=0 X     for (h, k) = 1. Here, χ denotes a primitive character of conductor f and Bn,χ (x) is the character Bernoulli function defined as Bn,χ (x)= Bn,χ (x) for 0

f−1 ∞ χ(a)te(a+x)t tn = B (x) . eft − 1 n,χ n! a=0 n=0 X X In [19], Gunnells and Sczech defined certain higher-dimensional Dedekind sums that generalize the classical Dedekind sums. By using Barnes’ dou- ble zeta function Ota [35] defined derivatives of Dedekind sums and proved their reciprocity laws. Using similar method, Nagasaka et.al [34] gave fur- ther generalizations of generalized Dedekind sums. Cenkci et.al [15] gave degenerate analogues of classical Dedekind sums and exact generalizations of Berndt’s character Dedekind sums to the case of any positive number. By using the p-adic interpolation of certain partial zeta functions, Rosen and Snyder [36] defined p-adic Dedekind sums in the sense of Apostol [1]. They also established the reciprocity law for these new p-adic Dedekind sums via interpolation of corresponding law for generalized Dedekind sums. In [31] and [32], Kudo extended the results of Rosen and Snyder. He defined p-adic continuous function which interpolates higher-order Dedekind sums. In [22],

[29], Kim defined q-Bernoulli numbers βn (q) ∈ C and q-Bernoulli polyno- mials βn (x, q) which are different Carlitz’s q-Bernoulli numbers [10], [30]. By using these polynomials and an invariant p-adic q-Volkenborn integral on Zp, he constructed a p-adic q-analogue of generalized Dedekind sums m b sm+1 (a, b). In [44], the second author defined new generating functions. By using these functions, he constructed q-Dedekind type sums related to Apostol’s Dedekind type sums [1]. By using p-adic q-Volkenborn integral, he [45] con- structed p-adic q-higher-order Hardy type sums.

7 In [27], Kim constructed the new (h, q)-extension of the Bernoulli num- bers and polynomials. By applying Mellin transformation to the generating function of the (h, q)- Bernoulli numbers, he defined (h, q)-zeta functions and (h, q)-L-functions, which interpolate (h, q)- Bernoulli numbers at negative in- tegers. By using p-adic q-Volkenborn integral, the distribution property of twisted (h, q)-Bernoulli polynomials is given by the following theorem:

Theorem 1.6 ([46]) For any positive integer m,

m−1 (h) n−1 a ha (h) a + x m B (x, q)= m ζ q B m , q (1.5) n,ζ n,ζ m a=0 X   for all integers n ≥ 0.

Observe that for ζ → 1, q → 1 and h = 1, we have

− m 1 j mn−1 B x + = B (mx) . (1.6) n m n j=0 X   The second author [46] gave generating function for twisted (h, q)-extensions of generalized Bernoulli numbers and polynomials associated with a Dirichlet character χ as follows:

f ∞ χ (a) ζaqhaeat (h log q + t) tn = B(h) (q) , f hf tf n,ζ,χ n! a=1 ζ q e − 1 n=0 f X X∞ χ (a) ζaqhae(a+x)t (h log q + t) tn = B(h) (x, q) . f hf tf n,ζ,χ n! a=1 ζ q e − 1 n=0 X X Note that

f j B(h) (q) = f n−1 χ (j) ζjqhjB(h) , qf , n,ζ,χ n,ζ f j=1 X   f j + x B(h) (x, q) = f n−1 χ (j) ζjqhjB(h) , qf . (1.7) n,ζ,χ n,ζ f j=1 X  

8 Using q-Volkenborn integration, Witt’s type formulas for these numbers and polynomials were also given by [46]

(h) x hx n Bn,ζ (q) = ζ q x dµ1 (x) , (1.8)

ZZp

(h) t ht n Bn,ζ (x, q) = ζ q (x + t) dµ1 (t) ,

ZZp

(h) x hx n Bn,ζ,χ (q) = χ (x) ζ q x dµ1 (x) . ZX

(h) (h) (h) We note that, if ζ → 1 then, Bn,ζ,χ (q) → Bn,χ (q) and Bn,ζ,χ (x, q) → (h) Bn,χ (x, q) which are defined by Kim [27]. Now we summarize our paper as follows: In Section 2, we construct new generating functions of Frobenius-Euler numbers and polynomials. We give relations between these numbers and polynomials. We also define generating functions of Barnes’ type multiple Frobenius-Euler numbers and polynomials. By applying Mellin transforma- tion to these functions, we construct Barnes’ type multiple l-functions. We define Dedekind type sums related to the Frobenius-Euler functions. We prove reciprocity laws of these sums. In Section 3, by using Dirichlet char- acter, we give generalizations of the Frobenius-Euler polynomials and num- bers. We construct generalized Dedekind type sums and prove correspond- ing reciprocity law. In Section 4, we give twisted versions of new Dedekind type sums and corresponding reciprocity law. In Section 5, by using p-adic q-Volkenborn integral and twisted (h, q)-Bernoulli functions, we construct p-adic (h, q)-higher order Dedekind type sums. By using relation between Bernoulli and Frobenius-Euler functions, we also define new Hardy-Berndt type sums. We give some new relations related to to these sums as well.

9 2 New Dedekind Type Sums in the Complex Case

Let Fu (t) be the generating function of Frobenius-Euler numbers Hn (u), that is, ∞ tn 1 − u F (t)= H (u) = , (2.1) u n n! et − u n=0 X ([2], [3], [12], [37], [41], [49], [50]). The generating function of Frobenius-Euler polynomials Hn (x, u) can be defined in a natural way by ∞ tn 1 − u F (x, t)= F (t) ext = H (x, u) = ext. (2.2) u u n n! et − u n=0 X Now rewriting Fu (x, t), we have ∞ ∞ tn 1 − u F (x, t) = H (x, u) = ext = u−1 (u − 1) ext u−1et n u n n! et − u n=0 n=0 X∞ X  = u−n−1 (u − 1) e(n+x)t. n=0 X By applying Mellin transform to Fu (x, t),

∞ ∞ ∞ 1 1 ts−1F (x, −t) dt = u−n−1 (u − 1) ts−1e−(n+x)tdt Γ(s) u Γ(s) Z0 n=0 Z0 X∞ u − 1 u−n = , u (n + x)s n=0 X where Γ (s) is the Euler gamma function. The l-function which interpolates Frobenius-Euler numbers at negative integer values, is defined by ∞ u−n l (s; u)= (2.3) ns n=1 X for Re(s) > 1 and u ∈ C with |u| ≥ 1. Two-variable l-function is defined by ∞ u−n l (s, x; u)= (2.4) (n + x)s n=0 X 10 for x =6 zero or negative integer, Re(s) > 1 and u ∈ C with |u| ≥ 1. So defined two-variable l-function interpolates Frobenius-Euler polynomials Hn (x, u). Indeed, we have

∞ ∞ u − 1 u−n u − 1 1 = l (s, x; u)= ts−1F (x, −t) dt. u (n + x)s u Γ(s) u n=0 X Z0 For s = −n, n ∈ Z, n ≥ 0, by using Cauchy residue theorem, we have u − 1 l (−n, x; u)= H (x, u) . (2.5) u n In [25], Kim gave the definition of rth Frobenius-Euler polynomials of x with parameters a1,...,ar as

∞ (1 − u)r tn ext = H(r) (x, u|a ,...,a ) , r n 1 r n! (eaj t − u) n=0 j=1 X Q for complex numbers x, a1,...,ar,u such that aj =6 0 for each j = 1,...,r and |u| > 1. For x = 0, the rth Frobenius-Euler polynomials are called (r) as the rth Frobenius-Euler numbers and denoted by Hn (0,u|a1,...,ar) = (r) Hn (u|a1,...,ar). Let x be a complex number, Re(x) > 0 and a1,...,ar be real numbers such that aj =6 0 for each j = 0,...,r. We modify the definition of rth Frobenius-Euler polynomials of x with parameters a1,...,ar as

∞ r aj n 1 − u xt t e = Hr,n (x, u|a1,...,ar) . eaj t − uaj n! j=1 n=0 Y X

a1 Note that for r = 1, H1,n (x, u|a1) = Hn (x, u ). We have the following identity about Hr,n (x, u|a1,...,ar):

1 a1 r ar n Hr,n (x, u|a1,...,ar)= H (u ) a1 + ··· + H (u ) ar + x ,

n 1 a1 r ar where in the multinomial expansion of ( H (u ) a1 + ··· + H (u ) ar + x) we mean that

i j i j l k H (u) = Hj (u) but H (u) H (u) =6 Hj+k (u) if i =6 l.    11 This identity can be shown by using the definition of Frobenius-Euler num- bers (2.1):

∞ ∞ r aj r nj n 1 − u xt aj (ajt) (xt) e = Hn (u ) eaj t − uaj  j n !  n! j=1 j=1 n =0 j n=0 ! Y Y Xj X ∞ N (1H (ua1 ) a + ··· +r H(uar ) a + x) tN = 1 r . N! N=0 X

Let Fr,u (x, t) be the generating function of Hr,n (x, u|a1,...,ar). Then, we have ∞ n r aj t 1 − u xt Fr,u (x, t) = Hr,n (x, u|a1,...,ar) = e n! eaj t − uaj n=0 j=1 Xr ∞ Y = 1 − u−aj ext u−aj nj e−aj nj t

j=1 nj =0 Yr  ∞X − − ··· − ··· = 1 − u aj u (n1a1+ +nrar)e (x+n1a1+ +nrar )t.

j=1 n1,...,nr=0 Y  X

By applying Mellin transformation to Fr,u (x, t), we obtain the following in- tegral representation:

∞ − − ∞ − ··· 1 ts 1e xt u (n1a1+ +nrar) r dt = s . (2.6) Γ(s) (1 − u−aj e−aj t) (x + n a + ··· + n a ) j=1 n ,...,n =0 1 1 r r Z0 1 Xr Q By (2.6), we give the definition of multiple Frobenius-Euler function lr (s, x; u|a1,...,ar) as follows:

Definition 2.1 For s ∈ C with Re(s) >r, we define

∞ − ··· u (n1a1+ +nrar) l (s, x; u|a ,...,a )= r 1 r (x + n a + ··· + n a )s n ,...,n =0 1 1 r r 1 Xr for Re(x) > 0, a1,...,ar positive real numbers and u ∈ C, |u| ≥ 1.

12 Remark 2.2 If we take r = 1 and a1 = 1 in above definition, we get Frobenius-Euler l-function (2.4), and in addition if x =0, we get Frobenius- Euler l-function (2.3). If r = 1 and u = 1, we have . If u =1, r =1 and x =0, Riemann zeta function is obtained (cf. [22], [25], [26], [39], [41], [42], [43], [49]).

Substituting s = −n, n ∈ Z, n > 0 in (2.6), by Cauchy residue theorem, we arrive at the following theorem:

Theorem 2.3 For n ∈ Z, n ≥ 0, we have

r −aj 1 − u lr (−n, x; u|a1,...,ar)= Hr,n (x, u|a1,...,ar) . j=1 Y 

We now list some theorems and definitions for the polynomials Hn (x, u), which are needed in the following sections.

Lemma 2.4 For n ∈ Z, n ≥ 0, we have

n Hn (x +1,u) − uHn (x, u)=(1 − u) x .

By using (2.1) and (2.2), and after some elementary calculations, we have

∞ − − m 1 m−j n m 1 m−j m u j t u 1 − u x+ j mt mn H x + ,um = e( m ) um − 1 n m n! um − 1 emt − um n=0 j=0   j=0 X X X ∞ u tn = H (mx, u) . u − 1 n n! n=0 X Therefore, we easily arrive at the following lemma.

Lemma 2.5 For real x and a positive integer m,

− m 1 j um − 1 mn um−1−jH x + ,um = H (mx, u) n m u − 1 n j=0 X   for all integers n ≥ 0.

13 Definition 2.6 ([12]) Let Hn (x, u) denotes the nth Frobenius-Euler polyno- mial and let Hn (x, u) be defined recursively by

Hn (x, u)= Hn (x, u) , (0 ≤ x< 1) , Hn (x +1,u)= uHn (x, u) .

With this definition of Hn (x, u), it is easily verified that Lemma 2.5 hold for Hn (x, u).

Lemma 2.7 For real x and a positive integer m,

− m 1 j um − 1 mn um−1−jH x + ,um = H (mx, u) n m u − 1 n j=0 X   for all integers n ≥ 0.

Lemma 2.8 For all integers n ≥ 0 and (h, k)=1,

− − k 1 h 1 a b u (hk)n uhk−(kb+ha)H + ,uhk = uhk − 1 H (u) . n k h u − 1 n a=0 b=0   X X  Proof. Using Lemma 2.7, the left hand side of the above equation be- comes

h−1 − a b (hk)n k 1 uhk−(ha+kb)H + ,uhk a=0 n k h b=0   P X − uh h 1 kb = uhk − 1 hn u−kbH ,uh . uh − 1 n h b=0    X For b =0, 1,...,h−1, the residues kb modh are c =0, 1,...,h−1. Therefore,

− uh h 1 kb uhk − 1 hn u−kbH ,uh uh − 1 n h b=0    X− u h 1 c u = uhk − 1 hn uh−c−1H ,uh = uhk − 1 H (u) , uh − 1 n h u − 1 n c=0  X    by Lemma 2.7.

14 In the theory of Dedekind sums, the famous relation is reciprocity law, which plays a major role in this theory and other related topics. We now give the proof of main theorem for this section, which is related to reciprocity law for Sn,u (h, k). We use similar methods of Ota ([35]) and Nagasaka et.al ([34]) for proving Theorem 1.2. Proof of Theorem 1.2. For r = 2 and a1 = k, a2 = h,

∞ u−(km+hn) l (s; u|k, h) = 2 (km + hn)s m,n=0 (m,nX)=(06 ,0) − − ∞ k 1 h 1 −(kb+ha+hk(m′+n′)) ′′ u = (kb + ha + hk (m′ + n′))s a=0 ′ ′ X Xb=0 mX,n =0 ′′ by writing n = a+kn′, m = b+hm′, where means that the summation is taken over all positive integers m′, n′ except (m′, n′)=(0, 0) when a = b = 0. Then for M = m′ + n′, P

− − ∞ k 1 h 1 −(kb+ha+hkM) ′ (M + 1) u l (s; u|k, h)= 2 (kb + ha + hkM)s a=0 b=0 M=0 − − X X X∞ k 1 h 1 −hkM 1 ′ u = u−(kb+ha) (hk)s kb+ha s−1 a=0 b=0 M=0 hk + M X−X− X∞ k 1 h 1 −hkM 1 − ′ a b u + u (kb+ha) 1 − − , (2.7) (hk)s k h kb+ha s a=0 hk + M X Xb=0 MX=0   ′  where means that the summation is taken over all positive integers M except M = 0 when a = b = 0. By using (2.4), we obtain P − − 1 k 1 h 1 kb + ha l (s; u|k, h) = u−(kb+ha)l s − 1, ; uhk 2 (hk)s hk a=0 b=0   X−X− 1 k 1 h 1 a b kb + ha + u−(kb+ha) 1 − − l s, ; uhk . (hk)s k h hk a=0 X Xb=0    

15 By substituting s = −n, n ∈ Z, n ≥ 0, into (2.7) and using (2.5), we have

− − (hk)n k 1 h 1 a b l (−n, u|k, h)= uhk−(kb+ha)H + ,uhk 2 uhk − 1 n+1 k h a=0 b=0   − − X X (hk)n k 1 h 1 a b a b + uhk−(kb+ha) 1 − − H + ,uhk , uhk − 1 k h n k h a=0 X Xb=0     where the values a and b in the sums satisfy a b a b 0 ≤ + < 2 and + =16 . k h k h Let B be the set defined by

a b B = (a, b) ∈ Z × Z :0 ≤ a ≤ k − 1, 0 ≤ b ≤ h − 1, + > 1 . (2.8) k h   Then by Lemma 2.4 and Definition 2.6, we obtain

− − (hk)n k 1 h 1 a b l (−n; u|k, h)= uhk−(kb+ha)H + ,uhk 2 uhk − 1 n+1 k h a=0 b=0   n X X n+1 (hk) hk hk−(kb+ha) a b + hk 1 − u u + − 1 u − 1 ∈ k h (a,bX) B   − −  (hk)n k 1 h 1 a b a b + uhk−(kb+ha) 1 − − H + ,uhk uhk − 1 k h n k h a=0 X Xb=0     (hk)n a b a b n + 1 − uhk uhk−(kb+ha) 1 − − + − 1 . uhk − 1 k h k h (a,b)∈B      X

16 Now, by using Lemma 2.7, Lemma 2.8 and Definition 1.1, we get

− − (hk)n k 1 h 1 a b l (−n; u|k, h)= uhk−(kb+ha)H + ,uhk 2 uhk − 1 n+1 k h a=0 b=0   − − X X (hk)n k 1 h 1 a b + uhk−(kb+ha)H + ,uhk uhk − 1 n k h a=0 b=0   X− X− (hk)n k 1 h 1 a a b − uhk−(kb+ha) H + ,uhk uhk − 1 k n k h a=0 b=0   X− X− (hk)n k 1 h 1 b a b − uhk−(kb+ha) H + ,uhk uhk − 1 h n k h a=0 X Xb=0   1 u u = H (u)+ H (u) hk u − 1 n+1 u − 1 n k h u n u n − k S k (h, k) − h S h (k, h) . (2.9) uk − 1 n,u uh − 1 n,u

By definition of l2 (s; u|k, h), we have

1 k 2 h n H2,n (u|k, h) H u k + H u h l2 (−n; u|k, h)= −k −h = −k −h (1 − u )(1 − u ) (1− u )(1 −u )  k h n u u n k h j n−j = H u H − u k h . (2.10) (uk − 1)(uh − 1) j j n j j=0   X   By (2.9) and (2.10), 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 Thus, we arrive the desired result.

17 3 Generalized Dedekind Type Sums Attached to a Dirichlet Character

Character generalizations of classical Dedekind sums have been studied by many mathematicians. By using generalized Bernoulli functions attached to character, Berndt [4] defined Dedekind sums with characters for n = 1, and proved reciprocity laws by using either Eisenstein series with characters ([4], [5]), integrals such as contour integrals and Riemann-Stieltjes integrals, or the Poisson summation formula ([6]). Nagasaka et.al [34] defined generalized character Dedekind sums which are different from Berndt’s definitions for the case n = 1, and Cenkci et.al [15] gave exact generalizations of Berndt’s sums to the case any positive number. Simsek [38], [40], [44], [47] considered Dedekind sums. He gave several properties of these sums. In [52], Zhang studied the distribution property of a sum analogous to the Dedekind sums by using mean value theorem of the Dirichlet L-function. Xiali and Zhang [51] studied the asymptotic behavior of the Dedekind sums with a weight of Hurwitz zeta function by applying the mean value theorem of the Dirichlet L-function. To prove Theorem 1.4, we need the following definitions.

Definition 3.1 ([37], [49], [50]) For a primitive Dirichlet character χ of conductor f = fχ, generalized Frobenius-Euler numbers attached to χ, Hn,χ (u), n ∈ Z, n ≥ 0, are defined by means of

∞ f−1 tn χ (a) 1 − uf uf−aeat H (u) = . (3.1) n,χ n! eft − uf n=0 a=0  X X With this definition, it is easy to verify that

f−1 a H (u)= f n χ (a) uf−aH ,uf . n,χ n f a=0 X   Also, if F is an integer multiple of f, we have

− 1 − uF F 1 a uf H (u)= F n χ (a) uF −aH ,uF . (3.2) uf − 1 n,χ n F a=0 X  

18 Definition 3.2 Let χ be a Dirichlet character of conductor f = fχ with f|hk. We define the double l-function l2 (s; u; χ|k, h) with parameters (k, h), u and χ by ∞ χ (km + hn) u−(km+hn) l (s; u; χ|k, h)= (3.3) 2 (km + hn)s n,m=0 (n,mX)=(06 ,0) for s ∈ C, Re(s) > 2, u ∈ C, |u| ≥ 1. We observe that for u = 1, (3.3) reduces to double zeta function ∞ χ (km + hn) ζ (s;(k, h) , χ)= , 2 (km + hn)s n,m=0 (n,mX)=(06 ,0) e defined in [34]. Also for primitive character χ = 1, (3.3) reduces to the double l-function l2 (s; u|k, h) defined in Section 2. l2 (s; u; χ|k, h) can be analytically continued to the whole plane by the following identities: − − ∞ k 1 h 1 −(kb+ha+hk(m′+n′)) ′′ χ (kb + ha) u l (s; u; χ|k, h) = 2 (kb + ha + hk (m′ + n′))s a=0 ′ ′ X Xb=0 mX,n =0 k−1 h−1 −(kb+ha) = χ (kb + ha) u l2 (s,kb + ha; u|hk, hk(3.4)) . a=0 X Xb=0 Now we give proof of Theorem 1.4 as follows: Proof of Theorem 1.4. By substituting s = −n, n ∈ Z, n ≥ 0 in (3.4), we have

k−1 h−1 −(kb+ha) l2 (−n; u; χ|k, h)= χ (kb + ha) u l2 (−n, kb + ha; u|hk, hk) a=0 b=0 − − X X k 1 h 1 H (kb + ha, u|hk, hk) = χ (kb + ha) u−(kb+ha) 2,n −hk 2 a=0 (1 − u ) X Xb=0 k−1 h−1 = χ (kb + ha) u−(kb+ha) a=0 b=0 X X n 1H uhk hk + 2H uhk hk + kb + ha × . (3.5) −hk 2  (1 − u )  19 By substituting m′ + n′ = M in (3.4), we get

− − 1 k 1 h 1 kb + ha l (s; u; χ|k, h)= χ (kb + ha) u−(kb+ha)l s − 1, ; uhk 2 (hk)s hk a=0 b=0   − − X X 1 k 1 h 1 a b kb + ha + χ (kb + ha) u−(kb+ha) 1 − − l s, ; uhk . (hk)s k h hk a=0 X Xb=0     Setting s = −n, n ∈ Z, n ≥ 0 in the above equality yields

l2 (−n; u; χ|k, h) − − (hk)n k 1 h 1 a b = χ (kb + ha) uhk−(kb+ha)H + ,uhk uhk − 1 n+1 k h a=0 b=0   X− X− (hk)n k 1 h 1 a b a b + χ (kb + ha) uhk−(kb+ha) 1 − − H + ,uhk uhk − 1 k h n k h a=0 X Xb=0    

− − (hk)n k 1 h 1 a b = χ (kb + ha) uhk−(kb+ha)H + ,uhk uhk − 1 n+1 k h a=0 b=0   n X X n+1 (hk) hk hk−(kb+ha) a b + hk 1 − u χ (kb + ha) u + − 1 u − 1 ∈ k h (a,bX) B   − −  (hk)n k 1 h 1 a b a b + χ (kb + ha) uhk−(kb+ha) 1 − − H + ,uhk uhk − 1 k h n k h a=0 X Xb=0     (hk)n + 1 − uhk χ (kb + ha) uhk−(kb+ha) uhk − 1 (a,b)∈B  X a b a b n × 1 − − + − 1 , (3.6) k h k h     where B is defined by (2.8). From (3.2), we have

− − k 1 h 1 a b (hk)n χ (kb + ha) uhk−(kb+ha)H + ,uhk n k h a=0 X Xb=0   1 − uhk = uf H (u) , (3.7) uf − 1 n,χ

20 since for the values a and b in the sums, we have kb + ha kb + ha i − :0 6 a

4 Twisted Version of Dedekind Type Sums

One of the curious facts about Frobenius-Euler polynomials is the relation- ship between Bernoulli polynomials. This relationship occurs when u = ζ is any . For example, for the generating function of Bernoulli polynomials Bn (x), n ∈ Z, n ≥ 0, ∞ text tn = B (x) , et − 1 n n! n=0 X we have ∞ − − tn m 1 j text m 1 text ζrtext ζ−rjB x + = ζ−rjejt/m = = , n! n m et − 1 ζ−ret/m − 1 et/m − ζr n=0 j=0 j=0 X X   X where ζ is a primitive mth root of unity and m 6|r. This implies

m−1 r n−1 −rj j nζ r m ζ B x + = H − (mx, ζ ) . (4.1) n m 1 − ζr n 1 j=0 X   For m|r, we have (1.6). Furthermore, multiplying both sides of (4.1) by ζrj, summing over r and (1.6), we get

m−1 r j H − (mx, ζ ) mnB x + = B (mx)+ n ζrj n 1 , n m n ζ−r − 1 r=1   X where 0 ≤ j < m ([12]). We define twisted Dedekind type sums as follows.

21 Definition 4.1 Let n be a positive integer, h, k be relatively prime positive integers and ζhk−1 =1, ζ =16 . Then, we define twisted Dedekind sums by

k−1 − ha a ha S (h, k)= ζ k H ,ζ . n,ζ k n k a=0 X   Definition 4.2 Let χ be a Dirichlet character of conductor f = fχ with f|hk, n be a positive integer and ζhk−1 =1, ζ =16 . Then we define

k−1 h−1 n −(kb+ha) a a b S k (h, k|χ)= h χ (kb + ha) ζ H + ,ζ . n,ζ k n k h a=0 X Xb=0   Observe that for ζ = u Definition 4.1 and Definition 4.2 reduce to Defi- nition 1.1 and Definition 1.3, respectively. By substituting u = ζ in Theorem 1.2 and Theorem 1.4 with ζhk−1 = 1,

ζ =6 1, we obtain the reciprocity laws for Sn,ζ (h, k) and Sn,ζk (h, k|χ) as follows:

Theorem 4.3 For positive integer n, relatively prime positive integers h, k and ζhk−1 =1, ζ =16 , we have

k h ζ n ζ n k S k (h, k)+ h S h (k, h) 1 − ζk n,ζ 1 − ζh n,ζ n k h n ζ k j ζ h n−j = H ζ k H − ζ h j k j h n j j=0 1 − ζ 1 − ζ X   1 1 1  + H (ζ)+ H (ζ) . hk 1 − ζ n+1 1 − ζ n Theorem 4.4 Let n, χ and ζ be as in Definition 4.2. Then

n n k Sn,ζk (h, k|χ)+ h Sn,ζh (k, h|χ) −1 1 − ζ f ζ −1 = ζ Hn+1,χ (u)+ ζ Hn,χ (u) 1 − ζf hk   − − ζ k 1 h 1 + χ (kb + ha) ζ−(kb+ha) ζ − 1 a=0 b=0 X X n × 1H (ζ) hk + 2H (ζ) hk + kb + ha . 

22 5 (h, q)-Approach to p-adic Twisted Dedekind Type Sums

In this section, we define twisted (h, q)-Dedekind type sums by using twisted (h, q)-Bernoulli polynomials. By using p-adic interpolation of certain partial zeta function, we interpolate these sums to construct twisted (h, q)-p-adic Dedekind sums. ¿From Definition 1.5 and binomial expansion, we have − b 1 j ja m s(h) (a, b : q) = qhxζx x + dµ (x) m,ζ b b 1 j=0 Z    X Zp − − b 1 m j m ja m c = qhxζxxcdµ (x) (5.1) b c b 1 j=0 c=0    Z X X Zp By using (1.8), we obtain m m B(h) (x, q)= B(h) (q) xm−c. (5.2) m,ζ c c,ζ c=0 X   By substituting (5.2) into (5.1), we get − − − b 1 m j m ja m c b 1 j ja s(h) (a, b : q)= B(h) (q) = B(h) , q . m,ζ b c c,ζ b b m,ζ b j=0 c=0 j=0 X X     X    Throughout this section, ω will denote the Teichm¨uller character (modp) −1/(p−1) and we assume that q ∈ Cp with |1 − q|p

(h) m (h) Sp,ζ (m; a, b : q)= b sm,ζ (a, b : q) for all positive integers m such that m +1 ≡ 0(mod (p − 1)). Proof. Proof of this theorem is similar to that of Theorem 5 of [23] and Theorem 7 of [45]. Let p be an odd prime, j and b positive integers such that (p, j) = 1 and p|b. Then, we define ∞ hjis s b k T (h) (s; j, b : q)= ω−1 (j) B(h) (q) , (5.3) ζ b k j k,ζ Xk=0     23 −1 for s ∈ Zp, where hxi = xω (x). Since

s b (h) ≤ 1, < 1 and Bk,ζ (q) ≤ 1, k j p   p p

the sum ∞ s b k B(h) (q) k j k,ζ Xk=0     converges to a continuous function of s in Zp. Substituting s = m in (5.3), we have

hjim m m b k T (h) (m; j, b : q) = ω−1 (j) B(h) (q) ζ b k j k,ζ k=0     X − m m j m k = ω−m−1 (j) bm−1 B(h) (q) k b k,ζ Xk=0     j = ω−m−1 (j) bm−1B(h) , q . m,ζ b   If m +1 ≡ 0(mod(p − 1)), then

j T (h) (m; j, b : q)= bm−1B(h) , q . ζ m,ζ b   (h) m−1 (h) j Consequently, Tζ (m; j, b : q) is continuous p-adic extension of b Bm,ζ b , q . Now, since −  b 1 j aj s(h) (a, b : q)= B(h) , q m,ζ b m,ζ b j=0 X    and j T (h) (m; j, b : q)= bm−1B(h) , q , ζ m,ζ b   we have b−1 m (h) (h) b sm,ζ (a, b : q)= jTζ (m;(aj)b , b : q) j=0 X for p|b and m +1 ≡ 0(mod(p − 1)).

24 In the sequel, we construct twisted (h, q)-character Dedekind type sums. These sums are new and generalize the sums defined by Kudo [31], [32], Rosen and Synder [36] and Kim [23]. Generalization of Definition 1.5 is given by the following definition.

Definition 5.2 Let a, b be fixed integers with (a, b)=1, and let p be an odd prime such that p|b. For a primitive Dirichlet character with conductor f = fχ and ζ ∈ Tp, we define twisted (h, q)-Dedekind sums as

fb−1 j aj s(h) (a, b : q, χ)= χ (j) B(h) , q . m,ζ fb m,ζ,χ b j=0 X    We now generalize Theorem 5.1 by character χ. Observe that when χ =

χ0, the principle character, Definition 5.2 reduces to Definition 1.5, and the following theorem reduces to Theorem 5.1.

Theorem 5.3 Let a, b, p, χ and ζ be as in Definition 5.2. Then there exists (h) a p-adic continuous function Sp,ζ,χ (s; a, b : q) of s on X which satisfies

(h) m (h) Sp,ζ,χ (m; a, b : q)= fb sm,ζ (a, b : q, χ) for all positive integers m such that m +1 ≡ 0(mod (p − 1)).

Proof. We follow the similar method in the proof of Theorem 5.1. Let p be an odd prime, j and b positive integers such that (p, j) = 1 and p|b. For an embedding of the algebraic closure of Q, Q, into Cp, we may consider the values of a Dirichlet character χ as lying in Cp. Then we define

∞ hjis s b k T (h) (s; j, b : q)= ω−1 (j) B(h) (q) , (5.4) ζ,χ b k j k,ζ,χ Xk=0     for s ∈ X. Since

s b (h) ≤ 1, < 1 and Bk,ζ,χ (q) ≤ 1, k j p   p p

∞ s b k B(h) (q) k j k,ζ,χ Xk=0     25 converges to a continuous function of s in X. Substituting s = m in (5.4), we have

hjim m m b k T (h) (m; j, b : q) = ω−1 (j) B(h) (q) ζ,χ b k j k,ζ,χ k=0     X − m m j m k = ω−m−1 (j) bm−1 B(h) (q) k b k,ζ,χ Xk=0     j = ω−m−1 (j) bm−1B(h) , q . m,ζ,χ b   If m +1 ≡ 0(mod(p − 1)), then j T (h) (m; j, b : q)= bm−1B(h) , q . ζ,χ m,ζ,χ b   (h) m−1 (h) j Consequently, Tζ,χ (m, j, b : q) is continuous p-adic extension of b Bm,ζ,χ b , q . Now, since  fb−1 j aj s(h) (a, b : q, χ)= χ (j) B(h) , q m,ζ fb m,ζ,χ b j=0 X    and j T (h) (m; j, b : q)= bm−1B(h) , q , ζ,χ m,ζ,χ b   we have

fb−1 m (h) (h) fb sm,ζ (a, b : q, χ)= jχ (j) Tζ,χ (m;(aj)b , b : q) j=0 X for p|b and m +1 ≡ 0(mod(p − 1)).

6 Analogues of Hardy-Berndt Type Sums

As mentioned in Section 1, the classical Dedekind sums first arose in the transformation formula of the logarithm of the . The logarithms of the classical are studied by Berndt [7] and Gold- berg [17] derived the transformation formulas for classical theta-functions.

26 Arising in the transformation formulas, there are six different arithmetic sums, which are thus similar to Dedekind sums and called as Hardy sums or Berndt’s arithmetic sums. For h, k ∈ Z with k > 0, these six sums are defined as follows: k−1 k j+1+[ hj ] [ hj ] j S (h, k) = (−1) k G , s (h, k)= (−1) k G , 1 k j=1 j=1 X X   k j hj k hj s (h, k) = (−1)j , s (h, k)= (−1)j , 2 k k 3 k j=1 j=1 X     X   k−1 k [ hj ] j+[ hj ] j s (h, k) = (−1) k G , s (h, k)= (−1) k G . 4 5 k j=1 j=1 X X   The analytic and arithmetical properties of these sums were given by Berndt [7], Berndt and Goldberg [8], Can [9], Goldberg [17], Meyer [33], Simsek [38], Sitaramachandrarao [48]. In this section, we show the sums, defined by Definition 1.1, yield new type sums, which we call analogues of Hardy-Berndt type sums. By taking m = 2, r = 1 and ζ = −1 in (4.1), we obtain 2n+1 x +1 x H (x, −1) = B − B . (6.1) n n +1 n+1 2 n+1 2      From Definition 2.6, it is clear that Hn (x +2, −1) = Hn (x, −1). Since Bn (x) is periodic for any integer, (6.1) can be written in terms of these functions as 2n+1 x +1 x H (x, −1) = B − B . n n +1 n+1 2 n+1 2      From (1.4), it is easy to see that (1.6) is also valid for the functions Bn (x), that is, we have − m 1 j mn−1 B x + = B (mx) . (6.2) n m n j=0 X   From (6.2) for m = 2, we get x +1 x B =21−nB (x) − B . n 2 n n 2     27 We therefore have 2 2n+2 x H (x, −1) = B (x) − B . (6.3) n n +1 n+1 n +1 n+1 2   Now, taking u = −1 in Definition 1.1, we obtain

k−1 ha a ha S − (h, k)= (−1) k H , −1 . (6.4) n, 1 k n k a=0 X   ha Substituting (6.3) into (6.4) with x = k , we have − k 1 n+2 ha a 2 ha 2 ha S − (h, k)= (−1) k B − B . n, 1 k n +1 n+1 k n +1 n+1 2k a=0      X (6.5) By using (6.5), we define the following new sums, which we call analogues of Hardy-Berndt type sums. Definition 6.1 For n, h, k ∈ Z with (h, k)=1 and n > 0, we define

k−1 ha a ha HB (h, k) = (−1) k B , n,0 k n+1 k a=0   Xk−1 ha a ha HB (h, k) = (−1) k B . n,1 k n+1 2k a=0 X   Remark 6.2 Let k be an odd integer. Then, we have the following relations: (i) If h is even, then

HBn,0 (h, k)= sn+1 (h, k) and HBn,1 (h, k)= sn+1 (h, 2k) , where sn+1 (h, k) is given by (1.3). (ii) Let h be an odd integer. Then (a) If n +1 is even, then n +1 HB (h, k)=2−1−ns (h, k)+ H (−1) . n,1 2,n+1 2n+2 n (b) If n +1 is odd, then 1 1 HB (h, k)= 1 − 2−n s (h, k) − s (h, k) − 2−ns (h, k) , n,1 2 n+1 4 5,n+1 3,n+1 where s2,n+1 (h, k), s3,n+1 (h, k) and s5,n+1 (h, k) are generalizations of Berndt’s arithmetic sums s2 (h, k), s3 (h, k) and s5 (h, k), respectively ([16]).

28 Conclusion: The conclusion we can most likely draw from above is that the sums given by Definition 1.1 is different from Carlitz, Apostol, Berndt type Dedekind sums, and Definition 1.3 is different from Ota type Dedekind sums. For instance, Carlitz type Dedekind sums are defined by Frobenius- −1 Euler numbers Hn−1 (u ) as follows [11]:

−1 n H − (u ) S (h, k : n)= n 1 . hn (u − 1)(u−h − 1) u X In Definition 1.1 and Definition 1.3, we use Frobenius-Euler functions, which provides a different and useful approach to the theory of Dedekind sum. Definition 1.5 is different from those of Kim, Rosen and Synder and Kudo. Acknowledgment: This work was supported by Akdeniz University Scientific Research Projects Unit.

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