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On the solutions of second order generalized difference equations
Advances in Difference Equations volume 2012, Article number: 105 (2012)
In this article, the authors discuss and solutions of the second order generalized difference equation
and we prove the condition for non existence of non-trivial solution where for . Further we present some formulae and examples to find the values of finite and infinite series in number theory as application of .
MSC:39A12, 39A70, 47B39, 39B60.
Difference equations usually describe the evolution of some certain phenomena over time and are also important in describing dynamics for fundamentally discrete system, see . For example, in the numerical integration, the standard approach is to use the difference equations. Similarly, the population dynamics have discrete generations; the size of the st generation is a function of the k th generation . This can be expressed as difference equation of the form
see for example . Further, the concept of difference equations with many examples in applications such as asymptotic behavior of solutions of difference equations were studied extensively by Elaydi  where the analytic and geometric approaches were also combined in order to studying difference equations. Further, in , both classical and modern treatment of the difference equations were presented in excellent form. For related results on difference equations, see [4–8]. In the present article, we study and solutions of the following second order generalized difference equation
where for . We provide some related definitions and development for the present article.
The basic theory of difference equations is based on the operator Δ defined as
and there are several research took place on this line. By defining and its inverse , many interesting results and applications in number theory as well as in fluid dynamics can be obtained. 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 structures were studied for the solutions of difference equations involving . For similar results, we refer to [9–13].
In particular, the and solutions of second order difference equations of (1) when , were discussed in . In this article, we discuss and solutions for the second order generalized difference Equation (1) and present some applications of in the finite and infinite series of number theory. Throughout this article, we use the following notation:
denotes the integer part of k,
and is the set of all real numbers.
In this section, we present some of the preliminary definitions and related results which will be useful for future discussion. The following three definitions held in .
Definition 2.1 Let and then, the generalized difference operator is defined as
Similarly, the generalized difference operator of the r th kind is defined as
Definition 2.2 For arbitrary the h-factorial function is defined by
where Γ is the Euler gamma function. Note that when , , Definition 2.2 coincides with Definition 2.1.
Definition 2.3 Let , be a real or complex valued function and . Then, the inverse of denoted by and defined as follows.
where is a constant for all , .
Definition 2.4 The generalized polynomial factorial for is defined as
Lemma 2.5 Ifandthen,
for all, andis constant.
Lemma 2.6 ( Product formula)
Letandbe any two functions. Then
Lemma 2.7 ()
Let, , and. Then,
Definition 2.8 A function , is said to be in the space , if
If , for all then is said to be in the space .
Lemma 2.9 ( Summation formula of finite series)
If real valued functionis defined for all, then
whereis a constant for all, . Since, each complex number, () is called an initial value of. Usually, each initial valueis taken from any one of the values, , , etc.
Lemma 2.10 (Summation formula of infinite series)
Proof Assume . Then,
Now, the proof follows from and Definition 2.3. □
Theorem 2.11 Ifand, then
Proof The proof follows by taking on (14). □
Corollary 2.12 Letand. Then
Proof The proof follows from Equation (14) and as . □
The following example illustrates Corollary 2.12.
Example 2.13 Taking , in (16), we obtain
The following example shows that and .
Example 2.14 Assume and . Let . By Lemmas 2.7 and 2.10, we obtain
Since as . Replacing k by , we get
for thus Equation (17) yields
By Definition 2.8, the function . Since
Now taking then is an space function.
3 Main results
In this section, we present the condition for non existence of non-trivial solution of (1).
Lemma 3.1 Letand. Then
Proof We have
Since each positive term is greater than the consecutive negative term in the first expression, we find
since the second term is positive. □
Lemma 3.2 Letand. Then
Proof From the Binomial theorem for rational index, we find
Since each negative terms is greater than the next consecutive positive term and , we get
Lemma 3.3 Let. If
and for all then
Proof From the inequality (19) and for all , we find,
Now (20) follows by taking and . □
The following theorem shows the nonexistence of solutions of (3).
Theorem 3.4 For all, let the functionbe defined and
Then, ifis a solution of (3), there exists a real () such thatfor all.
Proof Since is a solution of (3) and belong to , we have which yields and hence
Now by applying again on both sides, and by Theorem 2.10, we get
Therefore, from (21), we obtain
Obviously for all and .
If , , for some , then for all . Hence for all . In this case, the proof is complete.
Now, we suppose that for all , from (27), we have
From (26), we have
From (27), , , by Schwartz’s inequality, we obtain
By using Corollary 2.12, we get
If , then
Hence we have
By applying Lemma 2.6 to Equation (29) twice, we obtain
Again from Lemma 2.6 and Equation (29), we obtain
From (31), (32) and by Lemma 2.6, we find that
which in view of (28), (30) gives
Since , from , we obtain
From Lemmas 3.1 and 3.2
By Lemma 3.2, we find for all . Thus from Lemma 3.3 and , we find
is decreasing by ℓ steps.
for all , then . From (34) we find and hence is increasing by ℓ steps, but this contradicts (30).
If there exists a real such that
for all , then
for all , that is,
However from (37), since and , it follows that , and hence from (34), we find . Further, since
we get which yields and hence we get
for all , since
But this implies that , and again we get a contradiction to (30).
Thus combining the above arguments, we conclude that our assumption for all is not correct, and this completes the proof. □
Theorem 3.5 For all, let the functionbe defined and
Then, ifis a solution of (3) , there exists an integer () such thatfor all.
Proof Let be a solution of (3) such that . Then,
for all . Thus, for this solution also the relation (24) holds. Further, since there exists a constant such that for all , where , we find that
for all . Therefore, this solution also has the representation (24).
Now as in Theorem 3.4, we define
Since , we find
then it follows that
Hence we define
and applying similar arguments as in the previous theorem one can see that there exists a positive integer such that for all . □
In the next we present some formulae and examples to find the values of finite and infinite series in number theory as application of . First of all we need the following theorem.
Theorem 3.6 Letand. Then
wherefor, forand eachis a constant for all, . In particularis obtained from (40) by substituting. Further
Proof By Definition 2.1, we find
and (40) follows by Lemma 2.9 and
The following example illustrates Theorem 3.6.
Example 3.7 By taking , and in (40), we get and hence (40) becomes
Example 3.8 Taking in (41), we obtain
In particular, when , above series becomes
4 Concluding remarks
In the difference equations there are several interesting development, see for example, [4–6], and [8–16]. Recently, in , the fractional h-difference equations was studied. In the present work we study the and solutions of the second order generalized difference equation
and we prove the condition for non existence of non-trivial solution.
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The authors would also like to thank the referee(s) for valuable remarks and suggestions on the previous version of the manuscript.
The authors declare that they do not have competing interest.
All the authors contributed equally.