Some identities related to Dedekind sums and the second-order linear recurrence polynomials
© Li and Zhang; licensee Springer. 2013
Received: 7 August 2013
Accepted: 9 September 2013
Published: 8 November 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 .
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:
where , and .
In this paper, we shall give an exact computational formula for . That is, we shall prove the following two theorems.
From the theorems, we may immediately deduce the following two corollaries.
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
This proves Theorem 1.
This completes the proof of Theorem 2.
from our theorems, we may immediately deduce Corollary 2.
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.
- Yi Y, Zhang W: Some identities involving the Fibonacci polynomials. Fibonacci Q. 2002, 40: 314-318.MATHMathSciNetGoogle Scholar
- Ma R, Zhang W: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 2007, 45: 164-170.MATHGoogle Scholar
- Ohtsuka H, Nakamura S: On the sum of reciprocal Fibonacci numbers. Fibonacci Q. 2008/2009, 46/47: 153-159.Google Scholar
- Rademacher H:On the transformation of . J. Indian Math. Soc. 1955, 19: 25-30.MATHMathSciNetGoogle Scholar
- Carlitz L: The reciprocity theorem of Dedekind sums. Pac. J. Math. 1953, 3: 513-522. 10.2140/pjm.1953.3.513MATHView ArticleGoogle Scholar
- Jia C: On the mean value of Dedekind sums. J. Number Theory 2001, 87: 173-188. 10.1006/jnth.2000.2580MATHMathSciNetView ArticleGoogle Scholar
- Zhang W: A note on the mean square value of the Dedekind sums. Acta Math. Hung. 2000, 86: 275-289. 10.1023/A:1006724724840MATHView ArticleGoogle Scholar
- Zhang W: On the mean values of Dedekind sums. J. Théor. Nr. Bordx. 1996, 8: 429-442. 10.5802/jtnb.179MATHView ArticleGoogle Scholar
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