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Some special finite sums related to the three-term polynomial relations and their applications
© Cetin et al.; licensee Springer. 2014
Received: 7 September 2014
Accepted: 18 October 2014
Published: 3 November 2014
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.
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 , Goldberg  and also Simsek  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.
A proof of (1) was given by Apostol  and the references cited in each of these earlier works.
The reciprocity law for the is given by the following theorem.
(cf. [1, 2, 5, 8, 10, 17]and the references cited in each of these earlier works). In the following theorem, Sitaramachandraro showed that the Hardy-Berndt sum can be expressed explicitly in terms of Dedekind sums.
The next theorem will be useful for the further sections.
1.1 Three-term polynomial relations for the Hardy sums
The following corollary was given by Pettet and Sitaramachandrarao .
Corollary 1 (Three and two-term polynomial relations)
Corollary 2 
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.
where is a positive integer.
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.
- (1)For , find the reciprocity laws of the sums and . That is, find
- (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 , Beck defined generalized the Carlitz polynomials as follows.
Definition 2 (The Carlitz polynomial)
Theorem 5 (Berndt-Dieter)
Proof of Theorem 5 was given by Beck in . Now we will give a new definition:
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 .
Finally, if we also take times partial derivative of , with respect to w, then we obtain the desired result. □
and by (22) we arrive at the following corollary.
We can also have some results from  by using the same method as follows.
where a and are even and .
where b is even.
(cf. [, (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.
Now we can give the generalization of Theorem 6 as follows.
Remark 10 If we substitute in Theorem 7, then Theorem 7 reduces to Theorem 6.
A proof of Corollary 5 was given by Beck [, Corollary 3.1].
Substituting into Definition 4 and Theorem 7, then we arrive at the following result.
where be positive integers, relatively prime in pairs (cf. also [, 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.
We are now ready to give a proof of Theorem 4.
Hence, we arrive at the desired result. □
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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