Exact Evaluation of Infinite Series Using Double Laplace Transform Technique

and Applied Analysis 3 Let us make substitution of variables y = e−x and z = e; we have ∞ ∑ s=2 1 (s − 1) ∞ ∑ p=2 1 (p − 1) = ( 1 2 ∫ 0 1 1 z −z 2 dz (1 − z) z − 1 2 ∫ 0 1 −z 2 z dz (1 − z) z )

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.

Abstract and Applied Analysis
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 ) .

(4)
By using double Laplace transform of function ( , ) = 1, Therefore, for , > −1 and , > −1, we have Using change of variables = − and = − in the integral of (6) gives a more symmetric result: The integral in (7) converges for ̸ = , ̸ = , and , > −1, , > −1. In case = 1/2, = 1/2, and = 0, = 0, (7) becomes By using substitution method, we have For another example, take = 0, = 0, 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 Abstract and Applied Analysis 3 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 ( , ) = 16 It is desired to evaluate this series from = 2, = 2 to = ∞, = ∞; by using table of Laplace transform, we have When the two numbers , and , differ by an integer , , that is, = + 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 ∈ [−1, 1), ∈ [−1, 1), , > 0, , ≥ 0. By using Laplace transform where ̸ = − 1, ̸ = − 1, the partial sum of where 4