Abstract
Double Laplace transform method was applied to evaluate the exact value of double infinite series. Further we generalize the current existing methods and provide some examples to illustrate and verify that the present method is a more general technique.
Finding the exact value of infinite series is not an easy task. Thus, it is a common way to estimate by using the certain test or methods. It is also known that there are certain conditions to apply the estimation methods. For example, some methods are only applicable for the series having only positive terms. However, there is no general method to estimate values of all the type series. One of the most valuable approaches to summing certain infinite series is the use of Laplace transforms in conjunction with the geometric series.
Efthimiou presented a method [1] that uses the Laplace transform and allows one to find exact values for a large class of convergent series of rational terms. Lesko and Smith [2] revisited the method and demonstrated an extension of the original idea to additional infinite series. Sofo [3] used a forced differential difference equation and by the use of Laplace Transform Theory generated nonhypergeometric type series. Dacunha, in [4], introduced a finite series representation of the matrix exponential using the Laplace transform for time scales. In [5], infinite series and complex numbers were applied to derive formulas. Abate and Whitt in their paper [6] applied infinite series and Laplace transform in study of probability density function by numerical inversion; see [7] for application to summing series arising from integrodifferential difference equations; also see Eltayeb in [8] who studied the relation between double differential transform and double Laplace transform by using power series. Our intention in this note is to illustrate the power of the technique in the case of double series.
The infinite series, whose summand which is given by can be realized as double Laplace transform as
In such a case, what appeared to be a sum of numbers is now written as a sum of integrals. This may not seem like progress, but interchanging the order of summation and integration yields a sum that we can evaluate easily, namely, the way of a geometric series consider We first illustrate the method for a special case. We then describe the general result pointing out further generalizations of the method, and we finally end with a brief discussion.
Consider the series where and and neither nor is a negative integer and also . Sums of this form arise often in problems dealing with Quantum Field Theory. By using partial fractions, we have By using double Laplace transform of function , Therefore, for and , we have Using change of variables and in the integral of (6) gives a more symmetric result: The integral in (7) converges for , and . In case , and , (7) becomes By using substitution method, we have For another example, take , and as positive integers; then (7) becomes
Example 1. Find closed-form expressions for series of the form
We observe from the definition of double Laplace transform that
The hyperbolic sine of and can be expressed exponentially as follows:
Then (12) can be written as
By using the general summation for (14), this becomes
Let us make substitution of variables and ; we have
Simplification of the above equation yields
In the next example we use the same method.
Example 2. As a variation, let us choose a general term with analog function of and . A typical function of this type might be
It is desired to evaluate this series from to ; by using table of Laplace transform, we have
When the two numbers and differ by an integer and , respectively, then the sum of (3) can be easily calculated from (4):
The previous equation can be checked from (7) as follows:
Then
We now generalize Efthirniou’s technique to the series of the form
where it is convenient to write that only
are Laplace transform as follows:
For example, consider
where .
By using Laplace transform
where , the partial sum of
is dominated above by
where
We apply the Lebesgue dominated convergence theorem to have
Now, let and yield
and when , and , we have
Consider one more example of this technique
Now, let and , and ; then (34) becomes
To apply the method to trigonometric series, we need to be able to handle series of the form
In particular, we assume and are real number and and , where
We start by the series where , and if or if . Using we can write (38) as With the change of variables and , (40) becomes In the special case, let ; we have We use the same steps for the series we arrive at the integral representation
In particular, for ,
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful for the very useful comments regarding detailed remarks which improved the presentation and the contents of the paper. Further, the authors would also like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RGP-VPP-117.