Abstract and Applied Analysis Volume 24, Article ID 327429, 6 pages http://dx.doi.org/.55/24/327429 Research Article Exact Evaluation of Infinite Series Using Double Laplace Transform Technique Hassan Eltayeb, Adem KJlJçman, 2 and Said Mesloub Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 45, Saudi Arabia 2 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 434Serdang,Selangor,Malaysia Correspondence should be addressed to Adem Kılıçman; akilic@upm.edu.my Received October 23; Accepted 28 December 23; Published 2 January 24 Academic Editor: S. A. Mohiuddine Copyright 24 Hassan Eltayeb et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 Laplacetransformsinconjunctionwiththegeometricseries. Efthimiou presented a method [] 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 complexnumberswereappliedtoderiveformulas.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 powerofthetechniqueinthecaseofdoubleseries. The infinite series, whose summand Q(n, m) = Q n Q m which is given by n= m= Q nq m, can be realized as double Laplace transform as Q n Q m = e nt mx f (x) f (t) dt dx. () 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 Q n Q m n=m= = = f (x) m= (e x ) m dx f (t) (e t ) n dt n= f (x) ( e x e x )dx f (t) ( e t e t )dt. (2) 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.
2 Abstract and Applied Analysis Consider the series Q (α, β; c, d) = n=m=(m+α) (m + β) (n+c)(n+d), (3) where α =βand c =dandneither α nor β is a negative integer and also c, d.sumsofthisformariseofteninproblemsdealing with Quantum Field Theory. By using partial fractions, we have Q(α,β;c,d)= ( α βm= (m + β) (m+α) ) (4) ( c d (n+d) (n+c) ). n= By using double Laplace transform of function f(t, x) =, e At Bx dt dx =, A,B >. (5) AB Therefore, for α, β > and c, d >,wehave Q(α,β;c,d) =( (α β) lim M M m= ( (c d) lim =( lim M ( lim N N N m= ( e βt e αt α β e mt (e βt e αt )dt) ( e dx e cx c d e mx (e dx e cx )dx) ( e )e t e t =( ( e βt e αt α β ) e t dt) e t ( ( e dx e cx c d Mt ) dt) )e x ( e Nx ) e x dx) ) e x dx). e x Using change of variables y=e t and z=e x in the integral of (6)givesamoresymmetricresult: Q(α,β;c,d)= (α β) (c d) y β y α y (6) z d z c dy dz. z (7) The integral in (7) convergesforα =β, c=d,andα, β >, c, d >. Incaseα=/2,c=/2,andβ=,d=,(7) becomes mn (m+/2) (n+/2) n=m= =4 y y z dy dz z =4 dy dz. ( + y) ( + z) (8) By using substitution method, we have n= m= mn (m+(/2)) (n+(/2)) = (4 ( ln 2))2. (9) For another example, take α=,c=,andβ, d as positive integers; then (7) becomes m(m+β) n (n+d) n=m= = βd y β y z d dy dz z = βd ( + y + y 2 + +y β ) (+z+z 2 + +z d )dydz = βd ( + 2 + 3 + + β )(+ 2 + 3 + + d ). () Example. Find closed-form expressions for series of the form m=2(m ) n=2(n ). () We observe from the definition of double Laplace transform that (p 2 )(s 2 ) = e st px sinh (t) sinh (x) dt dx. (2) The hyperbolic sine of t and x can be expressed exponentially as follows: sinh t= et 2 e t 2, sinh x=ex 2 e x 2. (3) Then (2)canbewrittenas (p 2 )(s 2 ) =( 2 e t e st dt 2 e t e st dt) ( 2 e x e px dx 2 e x e px dx) By using the general summation for (4), this becomes (s 2 ) (p 2 ) p=2 =( 2 ( 2 e t (e t ) s dt 2 e x (e x ) p dx 2 e t (e t ) s dt) (4) e x (e x ) p dx). (5)
Abstract and Applied Analysis 3 Let us make substitution of variables y=e x and z=e t ;we have (s 2 ) (p 2 ) p=2 =( 2 z 2 dz z ( z) z 2 z 2 zdz ( z) z ) ( 2 y 2 dy y ( y) y 2 y 2 ydy ( y) y ). Simplification of the above equation yields (s 2 ) (p 2 ) p=2 =( 2 (+z) dz) ( 2 ( + y) dy) = 9 6. In the next example we use the same method. (6) (7) Example 2. As a variation, let us choose a general term with analog function of s and p. A typical function of this type might be 6mn F (m, n) = (m 2 ) 2 (n 2 ) 2. (8) It is desired to evaluate this series from m=2,n=2to m=, n = ; by using table of Laplace transform, we have 4p 4s (p 2 ) 2 (s 2 ) 2 =4 e st px xt sinh (x) sinh (t) dt dx. (9) When the two numbers α, β and c, d differ by an integer k, r, that is, α=β+kand c=d+r, respectively, then the sum of (3)canbeeasilycalculatedfrom(4): k r Q (α, α k; c, c r) = kr j=i= ( j+α )( ). (2) i+c Thepreviousequationcan becheckedfrom(7) as follows: Q (α, α k; c, c r) = kr y k z r y yα z zc dy dz Then = kr = k k h= k r h= l= ( yh+α+ h+α+ k r Q (α, α k; c, c r) = kr j=i= y h+α z l+c dy dz r ( zl+c+ r l=. l+c+ (2) ( j+α )( ). (22) i+c We now generalize Efthirniou s technique to the series of the form Q n P n u m V m, (23) n=m= where it is convenient to write that only Q n = e nt f (t) dt, are Laplace transform as follows: n= Q n P n u m V m = m= f (t) ( n= For example, consider P n e nt )dt u m = e mx f (x) dx (24) f (x) ( V m e mx )dx. m= (25) αn + β am + b, (26) n=m= where r [, ), k [, ), α, a >, β, b. By using Laplace transform e nt mx (e (β/α)t e (b/a)x ) dxdt αa, (27) where r =,k=,thepartialsumof e nt mx (e (β/α)t e (b/a)x ) αa n=m= is dominated above by n= where rn e nt ( α e (β/α)t ) r e t m= km e mx ( a e (b/a)x ) = α e (β/α)t ( r e t ) a e (b/a)x k e x ), x ( k e α e (β/α)t ( r e t r e t )dt αa r e t ( )dt r e t k e x (28) (29) k a e (b/a)x e x ( )dx k e x ( k e x )dx<. (3)
4 Abstract and Applied Analysis We apply the Lebesgue dominated convergence theorem to have αn + β am + b n=m= = n= = e nt ( α e (β/α)t )dt ( α e (β/α)t ) n= m= e nt dt = αa e (β/α)t ( re t re t )dt ru β/α kv b/a e mx ( a e (b/a)x )dx ( a e (b/a)x ) e mx dx m= e (b/a)x ( ke x )dx ke x = αa ru kv du dv. (3) Now, let a=,b=and α=,β=yield n=m= n m = r k du dv ru kv = ln ( r ) ln ( k ), (32) and when r=,k=,a=,α=,andb = /2, β = /2, we have ( ) n ( ) m n+(/2) m+(/2) = u /2 V /2 du dv +u+v n=m= =( π 2 2)2. Consider one more example of this technique Q (α, β; a, b) = (m+a)(m+b) (n+α)(n + β) n=m= =( n= ( m= =( e nt ( e αt e βt β α )dt) ( e αt e βt β α e mx ( e ax e bx )dx) b a e nt dt) ) n= (33) ( ( e ax e bx b a ) e mx dx) m= =( β α (e αt e βt )( re t )dt) re t ( b a (e ax e bx )( ke x )dx) ke x rk = (β α) (b a) u α u β ru V a V b kv du dv. (34) Now, let a=,b=and α=,β=,andr, k (, ); then (34)becomes m (m+) n (n+) n=m= u =rk ru =(+( r r V du dv kv ) ln ( r))(+( k ) ln ( k)). k (35) To apply the method to trigonometric series, we need to be able to handle series of the form S= sin (my) sin (nz) e mx nt, n=m= C= cos (my) cos (nz) e mx nt. n=m= (36) In particular, we assume y and z are real number and x> and t>,where sin (my) sin (nz) e mx nt n=m= = e x sin y e t sin z 2cos ye x +e 2x 2cos ze t +e 2t cos (my) cos (nz) e mx nt n=m= = e x (cos y e x ) e t (cos z e t ) 2cos ye x +e 2x 2cos ze t +e 2t. We start by the series n=m= (37) n ] m μ, (38) where ],μ N,and<y,z<2πif μ, ] =or y,z 2π if μ, ] >.Using n ] m μ = (] )!(μ )! e mx nt t ] x μ dt dx, (39)
Abstract and Applied Analysis 5 we can write (38)as n=m= n ] m μ =( (] )! n= ( (μ )! cos (nz) e nt t ] dt) m= =( (] )! cos (my) e mx x μ dx) ( n= cos (nz) e nt )t ] dt) ( (μ )! ( cos (my) e mx )x μ dx) m= =( (] )! e t (cos z e t ) 2cos ze t +e 2t t] dt) (μ )! e x (cos y e x ) 2cos ye x +e 2x xμ dx. (4) With the change of variables u = e t and V = e x,(4) becomes n=m= n ] m μ =( ( )] (] )! (cos z u) 2cos zu+u 2 (ln u)] du) ( )μ (μ )! (cos y V) 2cos yv + V 2 (ln V)μ dv. In the special case, let μ=,] =;wehave n=m= nm =( 2 ( 2 d( 2cos zu+u 2 ) 2cos zu+u 2 ) d( 2cos yv + V 2 ) 2cos yv + V 2 ) =( 2 [ln ( 2 cos zu+u2 )] ) ( 2 [ln ( 2 cos yv + V2 )] ) = ln (2 sin z 2 ) ln (2 sin y 2 ). We use the same steps for the series n=m= (4) (42) sin (nz) sin (my) n ] m μ ; (43) we arrive at the integral representation n=m= sin (nz) sin (my) n ] m μ =( ( )] (] )! sin z (ln u) ] 2cos zu+u 2 du) ( )μ (μ )! sin y (ln V) μ 2cos yv + V 2 dv. In particular, for μ=,] =, n=m= sin (nz) sin (my) nm =(sin z (u cos z) 2 + sin 2 z du) (sin y (V cos y) 2 + sin 2 y dv) u cos z = tan sin z ] V cos y tan sin y ] =(tan sin z sin y cos z )(tan cos y ) =( π y )( π z 2 2 ). Conflict of Interests (44) (45) 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- 7. References [] C. J. Efthimiou, Finding exact values for infinite sums, Mathematics Magazine,vol.72,no.,pp.45 5,999. [2] J. P. Lesko and W. D. Smith, A Laplace transform technique for evaluating infinite series, Mathematics Magazine,vol.76,no.5, pp. 394 398, 23. [3] A. Sofo, Summation of series via Laplace transforms, Revista Matemática Complutense,vol.5,no.2,pp.437 447,22. [4] J. J. Dacunha, Transition matrix and generalized matrix exponential via the Peano-Baker series, Difference Equations and Applications,vol.,no.5,pp.245 264,25.
6 Abstract and Applied Analysis [5] D. E. Dobbs, Using infinite series and complex numbers to derive formulas involving Laplace transforms, International Mathematical Education in Science and Technology, vol.44,no.5,pp.752 76,23. [6] J. Abate and W. Whitt, Infinite-series representations of Laplace transforms of probability density functions foumerical inversion, JournaloftheOperationsResearchSocietyofJapan, vol. 42, no. 3, pp. 268 285, 999. [7] P. Cerone and A. Sofo, Summing series arising from integrodifferential-difference equations, The Australian Mathematical Society B,vol.4,no.4,pp.473 486,2. [8] H. Eltayeb, Note on relation between double Laplace transform and double differential transform, Abstract and Applied Analysis,vol.23,ArticleID5352,7pages,23.
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