An application on the second-order generalized difference equations
© Manuel et al.; licensee Springer. 2013
Received: 21 November 2012
Accepted: 29 January 2013
Published: 14 February 2013
In this paper, we study the solutions of the second-order generalized difference equation having the form of
where . Then we provide applications on and .
AMS Subject Classification:39A12, 39A70, 47B39, 39B60.
1 Introduction and preliminaries
however, no significant progress took place on this line. Recently, equation (2) was reconsidered and its inverse was defined by , and many interesting results in applications such as in number theory as well as in fluid dynamics were obtained; see, for example, . By extending the study for sequences of complex numbers and ℓ to be real, some new qualitative properties like rotatory, expanding, shrinking, spiral and weblike were studied for the solutions of difference equations involving . The and solutions of the second-order difference equation of (1) when were discussed in  and further generalized in . In this paper, we discuss some applications of in the finite and infinite series of number theory.
Lemma 1.2 (Product formula)
If for all , then is said to be in the -space.
In what follows, we have the summation formula for finite and infinite series.
Now, the proof follows from and Definition 1.1. □
The next lemma is an expansion of Lemma 1.5 and its proof is straightforward.
Proof The proof follows from Lemma 1.3 and Lemma 1.5 by taking and . □
Proof Since , the proof follows from Theorem 1.7 by taking . □
2 Applications of in number theory
In this section, we present some formulae and examples to find the values of finite and infinite series in number theory as an application of . The following theorem and example were given in . In fact, the example is to illustrate Theorem 2.1.
where for , for and each is a constant for all , . In particular is obtained from (14) by substituting .
and (15) follows by Lemma 1.5 and as . □
The following is the illustration for Theorem 2.3.
and one can take any value .
and (16) follows by (10) and as . □
The following theorem generates the formula to find the sum of first partial sums of an infinite series.
and hence (17) follows by Lemma 1.6 as when . □
The following example illustrates Theorem 2.6.
Similarly, one can take any value for and .
The following example shows that and when .
Since for all , Definition 1.4 yields .
3 Concluding remarks
In the present work, we provide an application on and and solutions of the second-order some generalized difference equation.
The authors express their sincere gratitude to the referees for their valuable suggestions and comments. The second author also acknowledges that this project was partially supported by University Putra Malaysia under the ERGS Grant Scheme having project number 5527068.
- Agarwal RP: Difference Equations and Inequalities. Dekker, New York; 2000.MATHGoogle Scholar
- Mickens RE: Difference Equations. Van Nostrand-Reinhold, New York; 1990.MATHGoogle Scholar
- Elaydi SN: An Introduction to Difference Equations. 2nd edition. Springer, Berlin; 1999.MATHView ArticleGoogle Scholar
- Kelley WG, Peterson AC: Difference Equations. An Introduction with Applications. Academic Press, San Diego; 1991.MATHGoogle Scholar
- Manuel MMS, Xavier GBA, Thandapani E: Theory of generalized difference operator and its applications. Far East J. Math. Sci. 2006, 20(2):163-171.MATHMathSciNetGoogle Scholar
- Popenda J, Schmeidal E: Some properties of solutions of difference equations. Fasc. Math. 1981, 13: 89-98.MATHGoogle Scholar
- Manuel MMS, Kılıçman A, Xavier GBA, Pugalarasu R, Dilip DS: On the solutions of second order generalized difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 105. doi:10.1186/1687-1847-2012-105Google Scholar
- Ferreira RAC, Torres DFM: Fractional h -difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 2011, 5(1):110-121. doi:10.2298/AADM110131002F 10.2298/AADM110131002FMATHMathSciNetView ArticleGoogle Scholar
- Manuel MMS, Xavier GBA, Dilip DS, Chandrasekar V: General partial sums of reciprocals of products of consecutive terms of arithmetic progression. Int. J. Comput. Appl. Math. 2009, 4(3):259-272.Google Scholar
- Manuel MMS, Xavier GBA: Generalized difference calculus of sequences of real and complex numbers. Int. J. Comput. Numer. Anal. Appl. 2004, 6(4):401-415.MATHMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.