Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 5 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials By Baghdadi Aloui Abstract- In this paper, we introduce a connection formula between the monomial basis and the shifted Laguerre basis. As an application, some integral representations in terms of Laguerre polynomials for certain sequences are obtained. Keywords: Laguerre polynomials, special functions, integral formulas. GJSFR-F Classification : MSC 2010: Primary 33C45; Secondary 42C05 Certain SequencesanditsIntegralRepresentationsinTermsofLaguerrePolynomials Strictly as per the compliance and regulations of : © 2015. Baghdadi Aloui. This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Notes Certain Sequences and its Integral Representations in Terms of Laguerre 2015 r ea Polynomials Y 1 Baghdadi Aloui Abstract- In this paper, we introduce a connection formula between the monomial basis and the shifted Laguerre basis. V As an application, some integral representations in terms of Laguerre polynomials for certain sequences are obtained. V Keywords: Laguerre polynomials, special functions, integral formulas. ue ersion I s s I. Introduction and Main Results I By using some special functions and some particular integrals, we recall some integral rep- XV resentations for certain integer (or real) sequences. a) Some special functions ) The Gamma function is defined by the definite integral F ) Z +1 Γ(z) = xz−1e−x dx; <e(z) > 0: 0 Research Volume We can see directly, that Γ(1) = 1, and using integration by parts, that Γ(z + 1) = zΓ(z). Notice that, for z = n 2 n f0g, the following formulas hold N n! = Γ(n + 1) Frontier +1 Z n −x = x e dx; (1) Science 0 of (2n)! p 1 2n π = Γ n + 2 n! 2 Journal Z +1 e−x = xn p dx: (2) x Global 0 The Betâ function is given in terms of the integral Z 1 s−1 t−1 B(s; t) = x (1 − x) dx; <e(s); <e(t) > 0: 0 Author: Faculté des Sciences de Gabès, Département de Mathématiques, Cité Erriadh, 6072 Gabès, Tunisie. e-mail: [email protected] ©2015 Global Journals Inc. (US) Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials which is symmetric in s and t, i.e., B(s; t) = B(t; s). Notice that, after a change of variable 1 x = 1+y , we get Z +1 xs−1 B(s; t) = s+t dx: 0 (1 + x) This function also admits the following representation in terms of the Gamma function [3] Γ(s)Γ(t) B(s; t) = : Γ(s + t) Ref In particular, if s and t are non-zero integers, then we have 2015 n!p! = B(n + 1; p + 1) 3. (n + p + 1)! ear P. de Y Z 1 n p Maroni, 2 = x (1 − x) dx; (3) l'Ing 0 +1 n Z x é nieur, Trait = n+p+2 dx; n; p ≥ 0: (4) 0 (1 + x) F V onc The monic Hermite polynomials Hn(x) are orthogonal in the interval (−∞; +1) with V 2 respect to the weight function e−x and fulfil the following orthogonality relation [2] tions Eul ue ersion I s s p é Z +1 I 2 π −x G e Hn(x)Hm(x) dx = n!δn;m; n; m ≥ 0; m é XV −∞ 2 n é é ralit riennes where δn;m is the Kronecker delta. The canonical moments, (H) , of the Hermite polynomials have the representation [3] n≥0 é s (Sciences Fondamentales) A 154 Paris, 1994. 1 1994. Paris, Fondamentales) A 154 (Sciences s ) n , F 1 + (−1) n! ) = (H) P n+1 n n o 2 Γ( 2 + 1) lyn Z +1 ô = xne−x2 dx; n ≥ 0: (5) m Research Volume e −∞ Classiques s Orthogonaux b) Some other integrals Frontier The Wallis integral is given by π Z 2 n In = sin x dx; n ≥ 0: Science 0 of By a simple integration by parts, we can obtain (2n)!π 22n(n!)2 Journal I = ;I = ; n ≥ 0: 2n 22n+1(n!)2 2n+1 (2n + 1)! . Te Global By the change of variable t = sin x, this gives the following formulas c hnique (2n)!π Z 1 x2n = p dx; n ≥ 0; (6) -30. 2n+1 2 2 2 (n!) 0 1 − x s 22n(n!)2 Z 1 x2n+1 = p dx; n ≥ 0: (7) 2 (2n + 1)! 0 1 − x Now, let consider the following integral ©2015 Global Journals Inc. (US) Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials π Z 4 n Tn = tan x dx; n ≥ 0: 0 It is easy to see that 1 T + T = ; n ≥ 0: n+2 n n + 1 We get by iteration the two following formulas n n+k X (−1) n otes T2n+1 = + (−1) T1; n ≥ 0: N 2k k=1 n n+k X (−1) n+1 2015 T2n+2 = + (−1) T0; n ≥ 0: 2k + 1 r k=0 ea Y Then, by the change of variable t = tan x, we get 3 n (−1)n X (−1)k Z 1 x2n+1 ln 2 + = dx; n ≥ 0; (8) 2 k 1 + x2 k=1 0 V n−1 π X (−1)k+1 Z 1 x2n V (−1)n + = dx; n ≥ 0; (9) 2 ue ersion I 4 2k + 1 1 + x s k=0 0 s I P0 P−1 XV with the convention k=1 = k=0 = 0. We also consider the following integral Z 1 n x ) Rn = dx; n ≥ 0: F ) 0 1 + x 1 It is easy to see that Rn + Rn+1 = n ; n ≥ 1, and hence the following formula n Research Volume X (−1)k Z 1 xn (−1)n ln 2 + = dx; n ≥ 0: (10) k 1 + x k=1 0 Frontier Finally, we consider the integral 1 Z 1 B = (1 − x)nex dx; n ≥ 0: Science n n! 0 of 1 For n ≥ 1, integration by parts yields Bn = Bn−1 − n! , and we obtain the formula n Journal X 1 1 Z 1 e − = (1 − x)nex dx; n ≥ 0: k! n! 0 k=0 Global This gives, after a change of variable t = 1 − x, the following relation n 1 X 1 Z 1 n! 1 − = xne−x dx; n ≥ 0: (11) e k! k=0 0 In this paper, we introduce the following connection formula, between the monomial n fx gn≥0 and the shifted Laguerre polynomials, ©2015 Global Journals Inc. (US) Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials Z +1 n 1 m −t (m) x = t e Ln t(x + 1) dt; n ≥ 0; m 2 N n f0g: (n + m)! 0 As an application of our formula, we give the integral representations in terms of Laguerre polynomials for the sequences given by the equations (1)−(11). II. Integral Representations in Terms of Laguerre Polynomials (m) Let fLn gn≥0 be the monic Laguerre polynomial sequence, with parameter m 2 N n f0g, [4] Ref n X n (n + m)! L(m)(x) = (−1)n−ν xν; n ≥ 0: (12) 2015 n ν (ν + m)! ν=0 1. ear Y For any c 2 and any polynomial p, let introduce in the space of polynomials the linear York, 1971. York, C H. 4 isomorphism Sc, called intertwining operator, and given by [1]: Hoc +1 Z hstadt, −t Sc(p)(x) = te p t(x − c) + c dt: 0 V V The operator Sc can be characterized taking into account its linearity as well as the fact T he ue ersion I s Functions of Mathematica Functions of s n n I Sc (x − c) = (n + 1)!(x − c) ; n ≥ 0: (13) XV By (13) and (12), it is easy to prove that m−1 (m) m−1 n S0 x Ln (x) = (n + m)!x (x − 1) ; n ≥ 0: ) F ) Hence, we can obtain the following result. Theorem 2.1 For every integer m 2 N n f0g, the following formula holds Z +1 Research Volume n 1 m −t (m) x = t e Ln t(x + 1) dt; n ≥ 0: (14) (n + m)! 0 l Physics Frontier Now, as an application of the above formula, we can express the sequences given by the equations (1)−(11) by integral representations in terms of Laguerre polynomials. Indeed, . substituting expression (14) into (1)−(11), we can state the following theorem. D ov Science er Publications Inc. New Inc. er Publications of Theorem 2.2 For every integers m ≥ 1, and n; p ≥ 0, the following formulas hold Z +1 Z +1 n!(n + m)! = tme−(x+t)L(m)t(x + 1) dtdx Journal n 0 0 p (2n)!m! π n + m Z +1 Z +1 tm Global p −(x+t) (m) n = e Ln t(x + 1) dtdx 4 n 0 0 x Z 1 Z +1 n!p!(n + m)! p m −t (m) = (1 − x) t e Ln t(x + 1) dtdx (n + p + 1)! 0 0 Z +1 Z +1 m −t n!p!(n + m)! t e (m) = n+p+2 Ln t(x + 1) dtdx (n + p + 1)! 0 0 (1 + x) ©2015 Global Journals Inc.