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Some identities related to Dedekind sums and the second-order linear recurrence polynomials
Advances in Difference Equations volume 2013, Article number: 299 (2013)
In this paper, we use the elementary method and the reciprocity theorem of Dedekind sums to study the computational problem of one kind Dedekind sums, and give two interesting computational formulae related to Dedekind sums and the second-order linear recurrence polynomials.
For any positive integer x, we define the generalized Lucas polynomial as follows: , , and for all .
It is clear that this polynomial is a second-order linear recurrence polynomial, it is satisfying the computational formula:
is the well-known Lucas sequence; is the Lucas-Pell sequence. About the properties of this sequence and related contents, some authors had studied them, and obtained many interesting results, see [1–3]. In this paper, we use the elementary method and the reciprocity theorem of Dedekind sums to study the computational problem of one kind Dedekind sums, and obtain some interesting identities related to Dedekind sums and the second-order linear recurrence polynomials. For convenience, we first give the definition of the Dedekind sums as follows:
For a positive integer q and integer h with , the classical Dedekind sum is defined by
This sum describes the behaviour of the logarithm of eta-function (see ) under modular transformations. About its other properties and applications, see [5–8]. For example, Carlitz  proved the reciprocity theorem
where , and .
In this paper, we shall give an exact computational formula for . That is, we shall prove the following two theorems.
Theorem 1 For any integers and odd number , we have the computational formula
Theorem 2 For any integers and odd number , we have the computational formula
From the theorems, we may immediately deduce the following two corollaries.
Corollary 1 For any positive integer n, we have the identities
Corollary 2 For any odd number , we have the limits
In our theorems, x must be a positive odd number. If x is an even number, then . This time, the situation is more complex, it is very difficult for us to give an exact computational formula for .
2 Proof of the theorems
In this section, we shall prove our theorems directly. First, we prove Theorem 1. It is clear that for any positive integer n and odd number x, we have . So, by reciprocity theorem (1), we have
Similarly, we also have the identity
Note that and , from identities (2) and (3), we have
From (4), we may immediately deduce the recurrence formula
Using (5), repeatedly, and note that formula (1) and , we have
This proves Theorem 1.
Now, we prove Theorem 2. From the method of proving (2), we have
Note that , , , from (1), (6) and (7), we have
This completes the proof of Theorem 2.
Note that the definition of , , and the limit
from our theorems, we may immediately deduce Corollary 2.
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The authors would like to thank the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P.S.F. (2013JZ001, 2013KJXX-34) and the N.S.F. (11071194) of P.R. China.
The authors declare that they have no competing interests.
JL carried out the exact computational formula for , . HZ participated in the research and summary of the study. All authors read and approved the final manuscript.
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Li, J., Zhang, H. Some identities related to Dedekind sums and the second-order linear recurrence polynomials. Adv Differ Equ 2013, 299 (2013). https://doi.org/10.1186/1687-1847-2013-299
- Dedekind sums
- the second-order linear recurrence polynomials