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An application on the second-order generalized difference equations
Advances in Difference Equations volume 2013, Article number: 35 (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
The basic theory of difference equations is based on the difference operator Δ defined as , , which allows the recursive computation of solutions. Later, the following definition was suggested for by [1–3] and :
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.
Definition 1.1 Let , , be a real- or complex-valued function and . Then the generalized difference operator is defined as
Then the inverse of denoted by is defined as follows: If
where is a constant for all , . If , then we can take . Further, the generalized polynomial factorial for is defined as
Lemma 1.2 (Product formula)
Let and , , be any two real-valued functions. Then
Lemma 1.3 Let , , and . Then
Definition 1.4 A function , , is said to be in the -space if
If for all , then is said to be in the -space.
In what follows, we have the summation formula for finite and infinite series.
Lemma 1.5 If a real-valued function is defined for all , then
where is a constant for all , . Since , each complex number () is called an initial value of . Usually, each initial value is taken from any one of the values , , , etc. Further, if and , then
Proof Assume . Then
Now, the proof follows from and Definition 1.1. □
The next lemma is an expansion of Lemma 1.5 and its proof is straightforward.
Lemma 1.6 If and , then
Theorem 1.7 Let , such that . Then
Proof The proof follows from Lemma 1.3 and Lemma 1.5 by taking and . □
Corollary 1.8 Let and . Then
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.
Theorem 2.1 Let and . Then
where for , for and each is a constant for all , . In particular is obtained from (14) by substituting .
Example 2.2 By taking , and in (14), we get and hence (14) becomes
Theorem 2.3 Let and . Then
By Definition 1.1, we find
and (15) follows by Lemma 1.5 and as . □
The following is the illustration for Theorem 2.3.
Example 2.4 Taking in (15), we arrive at
and one can take any value .
Theorem 2.5 For and , then
By Definition 1.1, we find
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.
Theorem 2.6 For the positive integer , and ,
Proof Using Definition 1.1 and operating on (7), we find
and hence (17) follows by Lemma 1.6 as when . □
The following example illustrates Theorem 2.6.
Example 2.7 Substituting , in (17), we obtain
In particular, when , the above series becomes
Similarly, one can take any value for and .
The following example shows that and when .
Example 2.8 Let , , and . Replacing k by , in (12), we get
thus from (18) it follows that
The function follows from Definition 1.4 by taking
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.
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
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Manuel, M.M.S., Kılıçman, A., Xavier, G.B.A. et al. An application on the second-order generalized difference equations. Adv Differ Equ 2013, 35 (2013). https://doi.org/10.1186/1687-1847-2013-35
- generalized difference equation
- generalized difference operator
- finite series and infinite series