Gen. Math. Notes, Vol. 28, No. 2, June 2015, pp. 21-29 ISSN 2219-7184; Copyright © ICSRS Publication, 2015 www.i-csrs.org Available free online at http://www.geman.in On Some Integrals Involving Laguerre Polynomials of Several Variables Fadhle B.F. Mohsen 1, Ahmed Ali Atash 2 and Salem Saleh Barahma 3 1Department of Mathematics Faculty of Education-Zingibar, Aden University, Yemen E-mail: [email protected] 2Department of Mathematics Faculty of Education-Shabowh, Aden University, Yemen E-mail: [email protected] 3Department of Mathematics Faculty of Education-Aden, Aden University, Yemen E-mail: [email protected] (Received: 1-1-15 / Accepted: 10-6-15) Abstract The main object of the present work is to derive some general integral formulas (single, double and multiple) involving Laguerre polynomials of several variables. A number of known and new integral formulas involving Laguerre polynomials of two and three variables are obtained as special cases of our general formulas. Keywords: Laguerre polynomials, Hypergeometric functions, Integral formulas, Lauricella's function, Kampé de Fériet function, Exton's functions, Chandel function. 22 Fadhle B.F. Mohsen et al. 1 Introduction In 1991, Ragab [7] defined the Laguerre polynomials of two variables (,)α β Ln (,)x y as follows: n r ()α α β Γ(n +α +1)Γ(n + β +1) (−y) L ()x L(,) (,)x y = n− r (1.1) n ∑ Γ α + − + Γ β + + n! r=0 r! ( n r 1) ( r 1) ()α where Ln ()x is the Laguerre polynomials of one variable [8] The definition (1.1) is equivalent to the following explicit representation of (,)α β Ln (,)x y , given by Ragab [7]: n n− r r s α β (α +1) ( β +1) (−n) y x L(,) (,)x y = n n r+ s (1.2) n 2 ∑∑ α + β + (n !) r=0s= 0 ( 1)s ( 1) r r!! s It may be remarked that (1.2) can be written as α β (α +1) (β +1) L(,) (,)x y = n n Ψ []− n; α +1, β +1;x , y (1.3) n (n !)2 2 Ψ where 2 is the confluent hypergeometric function of two variables [11, p.62] ∞ ()a r s Ψ()m []= r+ s x y 2 a;,;,b c x y ∑ , (1.4) r, s= 0()b r ()c s r! s! 1 , if n = 0 λ = where ()n (1.5) λ(λ +1)(λ + 2)....(λ + n −1) , if n =1,2,3,...... Khan and Shukla [4,p. 163] defined the Laguerre polynomials of several α α (,,)1 ⋯ m variables Ln (,x1 ⋯, xm ) as follows: α α (,1 ⋯⋯,)m Ln (,x1 ⋯, xm ) m m r Π α + − − − − − Π j ( j 1)n n n r1 n r1 ⋯⋯ rm−1 ( n)r +⋯⋯+r xm+1 − j = j=1 1 m j=1 m ∑∑⋯⋯ ∑ m m (1.6) = = = (n!) r1 0r2 0rm 0 Π Π α + rj! ( j 1)m+1 − j j=1 j=1 On Some Integrals Involving Laguerre… 23 m Π α + ( j )1 n = j =1 Ψ()m []− n; α + ,1 ⋯⋯,α + ;1 x ,⋯⋯, x , (1.7) (n !)m 2 1 m 1 m Ψ ()m where 2 is the confluent hypergeometric function of m-variables [11, p.62] ∞ r r ()a + + x 1 x m Ψ()m []= r1 ⋯ rm 1 m 2 a;c1 ,⋯,cm ; x1 ,⋯, xm ∑ ⋯ (1.8) = (c ) ⋯ (c ) r ! r ! r1 ,,⋯ rm 0 1 r1 m rm 1 m The object of this paper is to obtain certain integral formulas involving Laguerre polynomials of several variables ,these integrals are evaluated in terms of Chandel function (c.f.[2, p.90]) and the generalized Kampé de Fériet function of several variables [3, p.28 ] which are defined as follows: ()()k n [ ′ ] )1( EC a,,a b ;c1 ⋯,cn ; x1,⋯, xn ∞ m1 mn (a)m + ⋯+m (')a m + +⋯ +m ()b m +⋯ +m x1 ⋯xn = ∑ 1 k k 1 n 1 n (1.9) m ,,⋯ m =0 (c ) ⋯(c ) m !⋯m ! 1 n 1 m1 n mn 1 n and ′ ()n ′ ()n ()a :(b′) ;⋯ ; ()n ; A:B ;⋯; B [] = A:B ;⋯; B (b ) F x1 ,⋯, xn F x1 ,⋯, xn C: D′;⋯; D()n C: D′;⋯; D()n ()c :(d′);⋯ ;(d ()n ) ; ∞ ′ ()n m1 mn (( a)) + + (( b )) ⋯(( b )) x ⋯x = m1 ⋯ mn m1 mn 1 n ∑ ′ ()n (1.10) = (( c)) + + (( d )) ⋯(( d )) m !⋯m ! m1 ,,⋯ mn 0 m1 ⋯ mn m1 mn 1 n A Π where ((a)) m mean the product (a j) m . j=1 2 Integral Formulas For Re ( λ) > 0; Re ( σ) > 0, we have the following integral formulas involving Laguerre polynomials of several variables: ∞ −σ λ− (α ,,)⋯⋯α β β x 1 1 r γ γ (,,)1 ⋯⋯ s δ δ ∫ e x Lm ( 1x,⋯⋯, r x)Ln ( 1x,⋯⋯, s x) dx 0 Γ()(λ α + )1 ⋯(α +1) (β + )1 ⋯(β + )1 = 1 m r m 1 n s n σ λ r s (m !) (n !) γ γ δ δ ()(r r+ s) − − λ α + α + β + β + 1 r 1 s )1( EC m, n, ; 1 ,1 ⋯, r ,1 1 ,1 ⋯, s ;1 ,⋯, , ,⋯, (2.1) σ σ σ σ 24 Fadhle B.F. Mohsen et al. t σ+ λ −1 σ − λ− (α ,,)⋯⋯α (α + )1 ⋯(α + )1 B(,)σ λ t x 1(t − x) 1 L 1 r (β x,⋯⋯, β x) dx = 1 n r n ∫ n 1 r (n !)r 0 ;0:2 ⋯⋯ 0; − n,σ : − ;⋯⋯ ; − ; F β t ,⋯⋯, β t σ + λ α + α + 1 r (2.2) ;1:1 ⋯⋯ 1; : 1 1;⋯⋯ ; r 1 ; t α α σ −1 − λ−1 ( 1,,⋯⋯ ,r ) γ − γ − ∫ x (t x) Ln ( 1(t x ,) ⋯⋯, r (t x )) dx 0 (α + )1 ⋯(α + )1 B(,)σ λ t λ+ σ −1 = 1 n r n (n !)r ;0:2 ⋯ 0; − n,λ : − − ;⋯; − − ; F γ t ,⋯,γ t (2.3) λ +σ +α +α 1 r ;1:1 ⋯ 1; : 1 1 ;⋯;1 r ; t s r δ ⋯⋯δ α − λ−1 β − µ−1 γ − ν −1 ( 1,,)m ∫∫∫ x (r x) y (s y) z (t z) Ln ( xyz ,⋯⋯, xyz ) dx dy dz 0 0 0 δ + ⋯ δ + α + λ β + µ γ + ν α+ λ β+ µ γ+ ν = ( 1 )1 n ( m )1 n B( 1, )B( ,1 )B( 1, )r s t r (n !) ;0:4 ⋯⋯ 0; − n,α + ,1 β + ,1 γ +1 : − − ;⋯ ; − − ; F rst ,⋯,rst α + λ + β + µ + γ +ν + δ + δ + (2.4) ;1:3 ⋯⋯ 1; ,1 ,1 1: 1 1;⋯ ; m 1 ; tr t1 µ λ − µ λ − (α ,,)⋯α 1 − 1 1 r − r 1 1 r ∫⋯ ∫ x1 (t1 x1) ⋯x1 (tr xr ) Ln ( 1x 1 ,⋯, xr ) dx 1 ⋯dx r 0 0 µ+ λ µ+ λ α + ⋯ α + µ + λ ⋯ µ + λ 1 1 ⋯ r r = ( 1 )1 n ( r )1 n B( 1 ,1 1) B( r ,1 r )t1 tr r (n !) 1: ;1 ⋯⋯ 1; − n: µ +1 ;⋯⋯ ; µ +1 ; F 1 r t ,⋯⋯,t − α + µ + λ + α + µ + λ + 1 r (2.5) 0: ;2 ⋯⋯ 2; : 1 ,1 1 1 1;⋯⋯ ; r ,1 r r 1 ; tr t1 µ λ − µ λ − (α ,,)⋯α 1 − 1 1 r − r 1 1 r γ − γ − ∫⋯ ∫ x1 (t1 x1) ⋯x1 (tr xr ) Ln ( 1(t 1 x1 ,) ⋯, r(t r xr )) dx 1 ⋯dx r 0 0 On Some Integrals Involving Laguerre… 25 µ+ λ µ+ λ α + ⋯ α + µ + λ ⋯ µ + λ 1 1 ⋯ r r = ( 1 )1 n ( r )1 n B( 1 ,1 1) B( r ,1 r )t1 tr r (n !) 1: ;1 ⋯⋯ 1; − n: λ ;⋯⋯ ; λ ; F 1 r γ t ,⋯⋯,γ t − α + µ + λ + α + µ + λ + 1 1 r r (2.6) 0: ;2 ⋯⋯ 2; : 1 ,1 1 1 1;⋯⋯ ; r ,1 r r 1 ; Following integral can be obtained readily from (2.6) as follows: tr t1 µ λ − µ λ − (λ −1,⋯ ,λ − 1) 1 − 1 1 r − r 1 1 r γ − γ − ∫⋯ ∫ x1 (t1 x1) ⋯xr (tr xr ) Ln ( 1(t 1 x1 ,) ⋯, r(t r xr )) dx 1 ⋯dx r 0 0 µ+ λ µ+ λ λ ⋯ λ µ + λ ⋯ µ + λ 1 1 ⋯ r r = ( 1)n ( r) n B( 1 ,1 1) B( r ,1 r )t1 tr µ + λ + µ + λ + ( 1 1 )1 n ⋯( r r )1 n µ+ λ µ +λ × ( 1 1,,⋯⋯ r r ) γ γ Lm ( 1t 1,⋯⋯, rt r ) . (2.7) To obtain the main integral formula (2.1), we consider the left-hand side of (2.1) and using (1.2), then expressing Ψ()m in series forms and changing the order of 2 integration and summation to get (α + )1 ⋯(α +1) (β + )1 ⋯(β + )1 L..H S = 1 m r m 1 n s n (m !)r (n !)s ∞ p1 pr q1 qs (−m) + + (−n) + + γ ⋯γ δ ⋯δ ∑ p1 ⋯ pr q1 ⋯ qs 1 r 1 s = (α + )1 ⋯ (α + )1 (β + )1 ⋯ (β + )1 p !⋯ p !q !⋯q ! p1,⋯ ,pr , q1 ,⋯ ,qs 0 1 p1 r pr 1 q1 s qs 1 r 1 s ∞ (2.8) −σ λ+ + + + + + × ∫ e x x p1 ⋯ pr q1 ⋯ qs dx 0 In (2.8), using the definition of Gamma function and considering the definition (1.4), we get the right- hand side of (2.1). The integrals (2.2) to (2.6) are similarly established and we using the definition of Beta function . 3 Special Cases It is important to note that the above integrals are capable of yielding a number of other integrals formulas, these integral are evaluated in terms of certain 26 Fadhle B.F.