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Some special finite sums related to the three-term polynomial relations and their applications
Advances in Difference Equations volume 2014, Article number: 283 (2014)
Abstract
We define some finite sums which are associated with the Dedekind type sums and Hardy-Berndt type sums. The aim of this paper is to prove a reciprocity law for one of these sums. Therefore, we define a new function which is related to partial derivatives of the three-term polynomial relations. We give a partial differential equation (PDE) for this function. For some special values, this PDE reduces the three-term relations for Hardy-Berndt sums (cf. Apostol and Vu in Pac. J. Math. 98:17-23, 1982; Berndt and Dieter in J. Reine Angew. Math. 337:208-220, 1982; Simsek in Ukr. Math. J. 56(10): 1434-1440, 2004; Simsek in Turk. J. Math. 22:153-162, 1998; Simsek in Bull. Calcutta Math. Soc. 85:567-572, 1993; Pettet and Sitaramachandraro in J. Number Theory 25:328-339, 1989), to the generalized Carlitz polynomials, which are defined by Beck (Diophantine Analysis and Related Fields, pp. 11-18, 2006), to the Gauss law of quadratic reciprocity (cf. Beck in Diophantine Analysis and Related Fields, pp. 11-18, 2006; Berndt and Dieter in J. Reine Angew. Math. 337:208-220, 1982; Simsek in Turk. J. Math. 22:153-162, 1998), and also to the well-known identity on the greatest integer function which was proved by Berndt and Dieter (J. Reine Angew. Math. 337:208-220, 1982), p.212, Corollary 3.5. Finally, we prove the reciprocity law for an n-variable new sum which is related to the Dedekind type and Hardy-Berndt type sums. We also raise some open questions on the reciprocity laws of our new finite sums.
MSC:11F20, 11C08.
1 Introduction
The Dedekind sums are very useful in analytic number theory, in combinatorial theory and also in other branches of mathematics. That is, these sums arise in many areas of mathematics and also mathematical physics. Recently, there are many papers on the Dedekind sums which are related to elliptic modular functions, geometry (lattice point enumeration in polytopes, topology (signature defects of manifolds), algorithmic complexity (pseudo random number generators), character theory, the family of zeta functions, the Bernoulli functions, and other special functions. In 1877, Dedekind gave, under the modular transformation, an elegant functional equation for the Dedekind eta function, which contains the Dedekind sums.
On the other hand, Berndt [1], Goldberg [2] and also Simsek [3] gave, under the modular transformation, other elegant functional equations for the theta functions, which contain six different arithmetic sums (Hardy-Berndt sums). These sums are also related to the Dedekind sums and other special functions which have been mentioned before. Motivated largely by a number of recent investigations of the Dedekind sums and the Hardy-Berndt sums, we introduce and investigate various properties of a certain new family of finite arithmetic sums. We are ready to summarize our results in detail as follows.
In this section, some elementary properties and definitions on the Dedekind sums, the Hardy-Berndt sums, and the Simsek sum are given. In Section 2, we define some new finite arithmetic sums which are associated with the Dedekind sums, the Hardy-Berndt sums, and Simsek’s sum. We gave reciprocity laws for one of these sums. We also raise two open questions for the reciprocity laws. In the last section, we give a PDE for three-term polynomial relations. We give many applications for this PDE, which are related to the Dedekind-Rademacher sums, the Hardy-Berndt sums, and other finite arithmetic sums. Finally, by using this equation we give a proof of the reciprocity law of our new sums.
In the customary notation, we have
where denotes the largest integer ≤x (cf. [1–17], and the references cited in each of these earlier works). Let n be a positive natural number and α be a real number, then
and
The Dedekind sum , arising in the theory of the Dedekind eta function, is defined by
where h is an integer and k is a positive integer (cf. [1–17], and the references cited in each of these earlier works). The most important property of Dedekind sums is the following reciprocity theorem: If h and k are coprime positive integers, then
A proof of (1) was given by Apostol [4] and the references cited in each of these earlier works.
If h and k are integers with , the Hardy-Berndt sums are defined by
For , the equality below also holds true:
when h and k are odd [8]. Besides, the following equations will be very useful for the remaining sections [11]:
The 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, then
(cf. [1, 2, 5, 8, 10, 17]and the references cited in each of these earlier works). In the following theorem, Sitaramachandraro [17]showed that the Hardy-Berndt sum can be expressed explicitly in terms of Dedekind sums.
Theorem 2 Let h and k be coprime positive integers. If is even, then
and if is odd, then
The next theorem will be useful for the further sections.
Theorem 3 If both h and k are odd and , then
A proof of this theorem was given by Apostol in [5]. In [16], Simsek defined a new sum related to the sums as follows:
Let h and k be integers with
The reciprocity law for the Simsek sum is given by (cf. [[16], p.5, Theorem 4])
1.1 Three-term polynomial relations for the Hardy sums
Here, two and three-term polynomial relations, which were studied thoroughly in [11] and [14], are recalled. In [6, 7, 11], and [14] some new theorems on three-term relations for the Hardy sums were found by applying derivative operator to the three-term polynomial relation. Throughout this section, we assume that a, b, and c are pairwise coprime positive integers and , , and satisfy
The following corollary was given by Pettet and Sitaramachandrarao [11].
Corollary 1 (Three and two-term polynomial relations)
If a, b, and c are pairwise coprime positive integers, then
The identity (9) is originally due to Berndt and Dieter [7]. The next corollary, which is equivalent to (9), was first established by Carlitz [9].
Corollary 2 [9]
If a and b are coprime positive integers, then
We need the following relations, which were proved by Pettet and Sitaramachandrarao [11]:
and also
2 New sums involving the functions and
In this section, we define some new finite sums which are related to not only the functions and , but also the Dedekind sums, the Hardy-Berndt sums, the Simsek sum , and the other finite sums. We also investigate the reciprocity laws of these sums. We also ask two open questions for these reciprocity laws.
Definition 1 Let be pairwise coprime positive integers. We define the following sums , and , respectively:
where is a positive integer.
Substituting and into (17), we have
and
Combining the function with (18) and (19), we get
and
where
and
Thus, our new definitions are related to the Hardy-Berndt sums and also the Simsek sum . The reciprocity laws for the special finite sums, that is, the Dedekind type sums, the Hardy-Berndt type sums, and the Simsek sum, are very important. Therefore, we are ready to give the reciprocity law of the sums by the following theorem.
Theorem 4 Let n be a natural number with and be positive integers, relatively prime in pairs. Then we have
Let be positive integers, relatively prime in pairs. Then we define the following sums:
Substituting and into (20), we arrive at the Dedekind sums
where .
Substituting into (20), we obtain
Open questions
-
(1)
For , find the reciprocity laws of the sums and . That is, find
and
-
(2)
For , find the reciprocity law of . That is, evaluate
3 PDE for the Carlitz polynomials and their applications
In this section, we study on the Carlitz polynomials and their properties (cf. [6, 7, 9, 11, 14], and the references cited in each of these earlier works). We find a PDE for this polynomial. We give many applications for this PDE, which are related to the Dedekind-Rademacher sums, the Hardy-Berndt sums, and the other finite sums. In [6], Beck defined generalized the Carlitz polynomials as follows.
Definition 2 (The Carlitz polynomial)
, where are indeterminants and are positive integers, is defined as the polynomial
Theorem 5 (Berndt-Dieter)
If are pairwise relatively prime positive integers, then
Proof of Theorem 5 was given by Beck in [6]. Now we will give a new definition:
Definition 3 Let u, v, w be indeterminants. Let a, b, c be positive integers, relatively prime in pairs, and let , , be positive natural numbers, which are the orders of the derivatives of u, v, w, respectively. Then the polynomial is defined by
where , , and are not zero simultaneously.
By using (8), we derive the following theorem, which is very important and valuable to obtain some new and old identities related to the function , the Dedekind sums, the Hardy-Berndt sums, and the Simsek sum .
Theorem 6 The following identity holds true:
Proof By using (8), we have
First we take the partial derivative of with respect to u, then we have
Now, we take the partial derivative of with respect to u, and we get
If we continue this process with the mathematical induction method, taking partial derivative of with respect to u, then we get
That is,
Now we need to apply the same procedure to the function . If we calculate times the partial derivative of with respect to v, we get
That is,
Finally, if we also take times partial derivative of , with respect to w, then we obtain the desired result. □
Substituting into Definition 3, we get
and by (22) we arrive at the following corollary.
Corollary 3
Corollary 4 If we substitute , , and into (23), we get
if we substitute , , and into (23), we get
and finally if we substitute , , and into (23), we get
Remark 1 If we substitute , in (23), then we have the following well-known reciprocity law of the function , which is very important to prove the Gauss law of quadratic reciprocity:
(cf. [6, 7, 14], and the references cited in each of these earlier works).
Remark 2 If we substitute , and , into (21), we get
We also know from (22) that
By combining (25) and (26), we get
which gives us Theorem 2.1 in [14], so we have
or equivalently
Remark 3 If we substitute , and , into (21), we get
we also know from (22) that
so if we use these two equations together, then we get
which gives us Theorem 2.2 in [14], so we have
or equivalently
Remark 4 If we substitute , and , into (21), we get
We also know from (22) that
so if we use these two equations together, then we get
which gives us Theorem 2.3 in [14], so we have
or equivalently
Remark 5 If we substitute , and , into (21), we get
we also know from (22) that
so if we use these two equations together, then we get
which gives us Theorem 2.4 in [14], so we have
or equivalently
Remark 6 If we substitute , and , into (21), we get
we also know from (22) that
so if we use these two equations together, then we get
which gives us Theorem 2.5 in [14], so we have
or equivalently
We can also have some results from [11] by using the same method as follows.
Remark 7 If we substitute , and , into (21), we get
we also know from (22) that
so if we use these two equations together, then we get
which gives us Theorem 3.3 in [11], so we have
or equivalently
where a and are even and .
Remark 8 If we substitute , and , into (21), we get
From (22), we see that
Therefore
which gives us Theorem 3.4 in [11], so we have
or equivalently
where b is even.
Remark 9 If we substitute , and , into (21), we get
we also know from (22) that
Therefore
(cf. [[11], (4.1)], and the references cited in each of these earlier works).
By the mathematical induction method, we shall generalize Theorem 6. But first we need a new definition.
Definition 4 Let be indeterminants. Let be positive integers, relatively prime in pairs, and let be positive natural numbers, which are the orders of the derivatives of , respectively. Then the polynomial is defined by
Now we can give the generalization of Theorem 6 as follows.
Theorem 7 Let be indeterminants. Let be positive integers, relatively prime in pairs, and let be positive integers. For , the following identity holds true:
Remark 10 If we substitute in Theorem 7, then Theorem 7 reduces to Theorem 6.
Corollary 5 Substituting and into Definition 4 and Theorem 7, we arrive at
A proof of Corollary 5 was given by Beck [[6], Corollary 3.1].
Remark 11 Substituting into (27), we easily have
By substituting into the above equation, one can arrive at the reciprocity law of the Dedekind-Rademacher sums:
(cf. [6, 12], and the references cited in each of these earlier works).
Substituting into Definition 4 and Theorem 7, then we arrive at the following result.
Corollary 6 Let be positive integers, relatively prime in pairs, and let be positive integers. Then we have
Remark 12 Setting in (28), we obtain the following well-known identity, which was proved by Berndt and Dieter [[7], p.212, Corollary 3.5]:
where be positive integers, relatively prime in pairs (cf. also [[6], Corollary 3.1]). Note that this result was also obtained in Corollary 5.
Substituting into Definition 4 and Theorem 7, we obtain the following corollary.
Corollary 7
Remark 13 Substituting and , in Corollary 7, then we arrive at
We are now ready to give a proof of Theorem 4.
Proof of Theorem 4 Substituting into (29), we obtain
Combining the above equation with (17), we get
Hence, we arrive at the desired result. □
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Acknowledgements
The authors are supported by the research funds of Akdeniz University and Uludag University (Uludag University project numbers are 201424 and 201220).
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Cetin, E., Simsek, Y. & Cangul, I.N. Some special finite sums related to the three-term polynomial relations and their applications. Adv Differ Equ 2014, 283 (2014). https://doi.org/10.1186/1687-1847-2014-283
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DOI: https://doi.org/10.1186/1687-1847-2014-283