1. Introduction
The aim of this paper is to investigate and study the properties of the sum
which is defined by
In Equation (
1),
indicates the greatest integer function, which is also called floor function or integer value, that gives the largest integer less than or equal to
x. Besides, the greatest integer function can also be defined by the help of the sawtooth function
, as follows:
Dedekind sums
, which are defined by Richard Dedekind in the nineteeth century, is given with the below equality:
where
h is an integer, and
k is a positive integer. The basic introduction to the arithmetic properties of the Dedekind sum can be found in [
1,
2,
3,
4,
5]. Dedekind defined these sums with the help of the famous Dedekind eta function. Although Dedekind sums arise in the transformation formula for the eta function, they can be defined independently of the eta function. Dedekind sums have many interesting properties but most important one is the reciprocity theorem: When
h and
k are coprime positive integers, the following reciprocity law holds [
6]:
The first proof of (
2) was given by Richard Dedekind in 1892 [
6]. After R. Dedekind, Apostol [
7] and many authors have given many different proofs [
1]. By using contour integration, in 1905, Hardy [
8], gave another proof of the reciprocity theorem.
In that work, Hardy also gave some finite arithmetical sums which are called Hardy sums. These Hardy sums are also related to the Dedekind sums and have many useful properties.
We are ready to recall some of the Hardy sums which are needed in the further sections: If
h and
with
, the Hardy sums
and
are defined by
We also note that some authors have called Hardy sums as Hardy-Berndt sums. For
, the below equality also holds true:
when
h and
k are odd [
9]. Further, following equations will be necessary in the next section [
10]:
Reciprocity law for the is given by the following theorem:
Theorem 1. Let h and k be coprime positive integers. If h and k are odd, thenand if is odd then (cf. [
9,
10,
11,
12] and the references cited in each of these works).
The proof of the next reciprocity theorem was given by Hardy [
8] and Berndt [
13]:
Theorem 2. Let h and k be coprime positive integers. Then In the light of Equation (
8), Apostol [
14] gave the below result:
Theorem 3. If both h and k are odd and , then The following two theorems give the relations between the Hardy-Berndt sums and the Dedekind sums :
Theorem 4. [10] Let . Then Theorem 5. [15] For is odd and with then we have Next theorem gives infinite series representation of the Hardy-Berndt sums:
Theorem 6. [9] Let h and k denote relatively prime integers with . If is odd, thenand if h and k are odd, then Now we will give the finite series representation of the Hardy-Berndt sums:
Theorem 7. [9] Let h and k be coprime integers with . If is odd, thenand if h and k are odd, then In [
16], Simsek gave the following new sums: Let
h is an integer and
k is a positive integer with
. Then
We observe that
sums are also related to the Hardy sums
. That is
Reciprocity law for this sum was given by Simsek in [
16] (p. 5, Theorem 4) as below:
Simsek gave two different proofs of this reciprocity law. Another proof was also given in [
17].
sums are also related to the three term polynomial relations, [
17,
18,
19,
20].
In this paper we study the Hardy sums, the Simsek sums
and the Dedekind sums
which are related to the symmetric pairs [
21], and the Fibonacci numbers. Before starting our results, we need some properties of the Fibonacci numbers which are given as follows: The Fibonacci numbers are defined by means of the following generating function [
22]:
One can easily derive the following recurrence relation from (
19):
From (
19), we also easily compute the first few Fibonacci numbers as follows:
In [
21], Meyer studied a special case of the Dedekind sums. In that paper, Meyer investigated the pairs of integers
for which
. Meyer defined that
is a symmetric pair if this property holds and he showed that
is a symmetric pair if and only if
and
for
where
is the
th Fibonacci number. In [
21], Meyer proved the following theorem:
Theorem 8. If and is a symmetric pair, then .
In [
17], Cetin et al. defined the sum
as follows:
where
are positive integers with
For the odd values of
k, the below theorem is given in [
23]:
Theorem 9. If , h and k are odd integers with , then we have In [
17], Cetin et al. also defined the sum
as follows:
where
are pairwise positive integers.
Two-term polynomial relation has an important role in the next section. So we need to remind it in the following theorem:
Theorem 10. If a, b, and c are pairwise coprime positive integers, then Equation (
22) is originally due to Berndt and Dieter [
24].
Next corollary, which was given in [
23] (Corollary 7), will be useful in the next section.
Corollary 11. Let h and be positive integers and is a symmetric pair. If ,
and with n is a natural number, where is the th Fibonacci number, thenand 2. The Sum B1(h, k) and Its Properties
In [
17] we defined a new sum as follows:
which
and
. The sum
has the following arithmetic property:
To show that the last equality holds true, we use the definition of the
function, and the fact that
. If we also consider the equation
when
x is not an integer, then we get the Equation (
24). The Equation (
25) is originally reduced from [
15].
Now we will give a relation between the sums and .
Theorem 12. If is odd, , and , then Proof. We consider the two-term relation which is given in Equation (
22). If we take the partial derivative of Equation (
22) with respect to
u, and substitute
then we have
After some elementary calculations and by using Equation (
5), we get
We know from Equation (
7) that
If we use this fact, then we have
From Equation (
8) we can write
If we put Equation (
28) into Equation (
27), then we have the desired result. ☐
In the next theorem, we will give the relation between the sums and the Hardy-Berndt sums :
Theorem 13. If h and k are relatively prime odd numbers with , then Proof. From the definition of the sum
after basic calculations, we get
From [
23], we know that Equation (
21) holds true. If we also use Equation (
9), then we get the desired result. ☐
In the next theorem we will give the relation between the sums and the sums :
Theorem 14. If h and k are relatively prime odd numbers with , then Proof. It can be directly obtained from Theorem 13 and Equation (
17). ☐
Now, we will give a relation for the sums as follows:
Theorem 15. If is an odd positive integer and , then Proof. From Theorem 12, we showed that Equation (
26) holds. Similarly, when
is an odd positive integer with
we can also write
So first, if we multiply Equation (
26) by
k and Equation (
31) by
h respectively, then if we sum the two equations side by side and use (
8), we get the following identity:
Now we will consider the sum
. From (
26) and (
31), we can see that
So from this last equation, we can write
Now we will use the Equation (
8). First, we multiply Equation (
8) by
h, and we multiply Equation (
8) by
k. Then if add these two equations side by side and if we use the Equation (
32), we get the desired result. ☐
In the below theorem, we will give the reciprocity theorem for the sums :
Theorem 16. If h and k are odd positive integers with , then Proof. From Theorem 13, we know that Equation (
29) holds. Similarly, we can also write
If we multiply Equation (
29) by
k, and Equation (
34) by
h respectively, and add these equations side by side, we get
In this last equation if we use Equation (
6), then we get the desired result. ☐
Theorem 17. If is an odd positive integer and , then Proof. In [
17], if we take
in Theorem 4, and use it with Equation (
30) we have desired result. ☐
Theorem 18. If is an odd positive integer and , then Proof. In [
17], if we take
in Theorem 4, and use it with Equation (
36) we have desired result. ☐
In the following three theorems, we will give the relations between the sums and the Dedekind sums :
Theorem 19. Let is odd, with . Then Proof. It can be directly obtained from Theorem 12 and Equation (
10). ☐
Theorem 20. Let is odd, with . Then Proof. It can be directly obtained from Theorem 12 and Equation (
12). ☐
Theorem 21. If h and k are relatively prime odd numbers with then Proof. It can be directly obtained from Equation (
29) and Equation (
11). ☐
Now we will give two different infinite series representations of the sums :
Theorem 22. Let h and k denote relatively prime integers with . If is odd, then Proof. It can be directly obtained from Theorem 12 and Equation (
13). ☐
Theorem 23. Let h and k denote relatively prime integers with . If h and k are odd, then Proof. It can be directly obtained from Theorem 13 and Equation (
14). ☐
Similarly, we give two different finite series representations of the sums below:
Theorem 24. Let h and k denote relatively prime integers with . If is odd, then Proof. It can be directly obtained from Theorem 12 and Equation (
15). ☐
Theorem 25. Let h and k be coprime integers with . If h and k are odd, then Proof. It can be directly obtained from Theorem 13 and Equation (
16). ☐
Now, we will give the relation between the sums and the Fibonacci numbers.
Theorem 26. Let h and be positive integers and is a symmetric pair. If ,
and with n is a natural number, where is the th Fibonacci number, then Proof. It can be obtained similarly with Theorem 16’s proof. From Theorem 16’s proof, we know that Equation (
35) holds. If we also use the Equation (
23) into Equation (
35), then we get desired result. ☐