- Open Access
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.
- generalized difference equation
- generalized difference operator
- finite series and infinite series
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 . □
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 .
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.
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