Certain Indefinite Integrals Involving Laguerre Polynomials

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1 Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 2 Issue 8 Version. Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. USA Online ISSN: & Print ISSN: Certain Indefinite Integrals Involving Laguerre Polynomials By Salahuddin Department of Mathematics, University of Rajasthan, India. Abstract - In this paper we have established certain indefinite integrals involving Polylogarithm and Laguerre Polynomials. The results represent here are assume to be new. Keywords : Polylogarithm; Laguerre polynomials; Gaussian Hypergeometric Function. GJSFR- F Classification : 2 MSC NO: 33C5,33C45,33C5,33D5,33D6 Certain Indefinite Integrals Involving Laguerre Polynomials Strictly as per the compliance and regulations of :. Salahuddin. This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3. Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Notes Certain Indefinite Integrals Involving Laguerre Polynomials Salahuddin Abstract - In this paper we have established certain indefinite integrals involving Polylogarithm and Laguerre Polynomials. The results represent here are assume to be new. Keywords and Phrases : Polylogarithm; Laguerre polynomials; Gaussian Hypergeometric Function. Laguerre polynomials I. Introduction and Preliminaries In mathematics, the Laguerre polynomials, named after Edmond Laguerre , are solutions of Laguerre s equation: xy + xy + ny =. which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. The associated Laguerre polynomials also named Sonin polynomials after Nikolay Yakovlevich Sonin in some older books are solutions of xy + α + xy + ny =.2 The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form fxe x dx These polynomials, usually denoted L, L,..., are a polynomial sequence which may be defined by the Rodrigues formula L n x = ex d n n! dx ne x x n.3 They are orthonormal to each other with respect to the inner product given by < f, g >= The sequence of Laguerre polynomials is a Sheffer sequence. fxgxe x dx.4 43 Global Journal of Science Frontier Research F Volume XII Issue VIII V ersion I Author : P.D.M College of Engineering, Bahadurgarh, Haryana, India. s: vsludn@gmail., sludn@yahoo.com Global Journals Inc. US

3 Notes 44 Global Journal of Science Frontier Research F Volume XII Issue VIII V ersion I Figure : The first six Laguerre polynomials The first six Laguerre polynomials arealso in fig. Polylogarithm L x = L x = x + L 2 x = 2 x2 4x + 2 L 3 x = 6 x3 + 9x 2 8x + 6 L 4 x = 24 x4 6x x 2 96x + 24 L 5 x = 72 x6 36x x 4 24x x 2 432x + 72 The polylogarithm also known as Jonquire s function is a special function Li s z that is defined by the infinite sum, or power series: Li s z = k= z k k s.5 It is in general not an elementary function, unlike the related logarithm function. The above definition is valid for all complex values of the order s and the argument z where z <.The polylogarithm is defined over a larger range of z than the above definition allows by the process of analytic continuation. The special case s = involves the ordinary natural logarithm Li z = ln z while the special cases s = 2 and s = 3 are called the dilogarithm also referred to as Spence s function and trilogarithm respectively. The name of the function comes from the fact that it may alternatively be defined as the repeated integral of itself, namely that Li s+ z = z Li s t dt.6 t Global Journals Inc. US

4 Thus the dilogarithm is an integral of the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function. The polylogarithm also arises in the closed form of the integral of the FermiDirac distribution and the BoseEinstein distribution and is sometimes known as the Fermi- Dirac integral or the BoseEinstein integral. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral which has a similar notation. Notes Generalized Gaussian Hypergeometric Function Generalized ordinary hypergeometric function of one variable is defined by a, a 2,, a A ; AF B z a k a 2 k a A k z k = b b, b 2,, b B ; k b 2 k b B k k! k= or AF B a A ; b B ; II. z A F B a j A j= ; b j B j= ; Main Indefinite Integrals z = k= a A k z k b B k k! where denominator parameters b, b 2,, b B are neither zero nor negative integers and A, B are non-negative integers. coshx L x 2 dx = cosx cosx 25 4ι e ι 2 x sin x [ 4 + 2ιe 2x F 2 2 ι, 2 ι, ; 2 ι, 2 ι; e 4 + 2ιe 3F ι, 2 + ι, ; ι, ι; e + +5xe 2x 2F 2 ι, ; 2 ι; e 5xe 2F 2 + ι, ; ι; e 3 4ιe 2+ 2F 2 ι, ; 3 2 ι; e + 5e 2F 2 + ι, ; ι; e coshx L2 x cosh x dx = 5xe 2x ιe 2x] + Constant 2. sinh x [ 4x 2Li 2 e x 2 4x 2Li2 e x 2 +8Li3 e x 2 coshx 2 8Li 3 e x 2 + x 2 log e x 2 x 2 log + e x 2 + 2x 2 cosh x 2 4x log e x x log+e x 2 8x sinh x 2 +6 sinh x 2 8x cosh x 2 +2 cosh x 2 +2 logtanh x 4 ]+Constant 2.2 sinx L2 x sin x dx = sin x 2 + ι cos x 2 2 sin x [ Li e 2 +4ιx 2Li 2 3 +x 2 log e + 4x +4ι tan cosh x L3 x 8Li 3 3 log 3 4 e 4x } ιx 2 8x+2 sin x 2 + +ιx2 4 cos x 2 dx = coshx 24 coshx sinh x 2 { 4ιx 2Li x 2 log 3 4 e + log e + ] + Constant 2.3 [ 48x 2 Li 2 e x x 2 6x + 6Li 2 e x xLi 3 e x 2 92xLi3 e x x Li2 e x 2 576Li3 e x Li3 e x Global Journal of Science Frontier Research F Volume XII Issue VIII V ersion I Global Journals Inc. US

5 +384Li 4 e x Li4 e x 2 x 4 8x 3 log + e x 2 + 8x 3 log e x 2 + 6x 3 cosh x 2 72x 2 log e x x 2 log + e x 2 96x 2 sinh x 2 44x2 cosh x x log e x 2 44x log + e x x sinh x sinh x x cosh x cosh x 2 48 logtanh x 4 + 8π4] + Constant Global Journal of Science Frontier Research F Volume XII Issue VIII V ersion I III. Derivation of The Integrals Involving the same method of ref[8], one can derive the integrals. IV. Conclusion In our work we have established certain indefinite integrals involving Laguerre polynomials,jonquire s function and Hypergeometric function.we hope that the development presented in this work will stimulate further interest and research in this important area of Computational Mathematics. References Références Referencias. Abramowitz, Milton., A and Stegun, Irene ; Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Arfken, G.,Hans Weber ; Mathematical Methods for Physicists, Academic Press,2. 3. B Spain, M G Smith; Functions of mathematical physics, Van Nostrand Reinhold Company, London, Guillera, J., Sondow, J. ;Double integrals and infinite products for some classical constants via analytic continuations of Lerch s transcendent, The Ramanujan Journal,628, Jahnke, E., Emde, F. ; Tables of Functions with Formulae and Curves 4th ed., Dover Publications,New York, Olver, Frank, W. J., Lozier, Daniel, M., Boisvert, Ronald F.,Clark, C. W.; NIST Handbook of Mathematical Functions,Cambridge University Press,2. 7. Ricci, Paolo Emilio ;Generalized Lucas polynomials and Fibonacci polynomials, Rivista di Matematica della Universit di Parma, 4995, Salahuddin; Hypergeometric Form of Certain Indefinite Integrals, Global Journal of Science Frontier ResearchF, 2, Stein, Elias, Weiss, Guido ; Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press,97.. Temme, Nico ; Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, 996. Whittaker, E. T.; Watson, G. N. ; A Course of Modern Analysis, London: Cambridge University Press, Wiener, Norbert ; The Fourier Integral and Certain of its Applications, New York: Dover Publications, Zhi-Wei Sun; Hao Pan.;Identities concerning Bernoulli and Euler polynomials, Acta Arithmetica, 2526,2-39. Ref. 8. Salahuddin; Hypergeometric Form of Certain Indefinite Integrals, Global Journal of Science Frontier ResearchF, 2, Global Journals Inc. US