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 βα ),( Ln yx ),( as follows:

n r α )( βα Γ(n +α + )1 Γ(n + β + )1 (−y) L x)( L ),( x y),( = −rn (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 βα ),( Ln yx ),( , given by Ragab [7]:

n −rn sr βα (α + ()1 β + )1 (−n) y x L ),( x y),( = n n +sr (1.2) n 2 ∑∑ α + β + n )!( r=0s= 0 ( s ()1 )1 r sr !!

It may be remarked that (1.2) can be written as

βα (α + ()1 β + )1 L ),( x y),( = n n Ψ []− n; α + ,1 β + x,;1 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)( []= +sr x y 2 a b c x,;,; y ∑ , (1.4) sr =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 = ,3,2,1 ......

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 rn 1 rn 1 ⋯⋯ 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 rm m 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 A:B′;⋯; B n)( A:B′;⋯; B n)(  a)( :(b′) ;⋯ ; (b n)( ) ;  F []x ,⋯, x = F  x ,⋯, x  ′ ⋯ n)( 1 n C: D′;⋯; D n)( c)( :(d′);⋯ ;(d n)( ) ; 1 n C: D ; ; D  

∞ ′ 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 )mj . 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 ()( +sr ) − − 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 λσ −+ 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 : − ;⋯⋯ ; − ;  β β F  1t ,⋯⋯, r t ;1:1 ⋯⋯ 1; σ + λ :α +1;⋯⋯ ;α +1 ; (2.2)  1 r  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 σλ −+ 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 + + +νγµβλα = ( 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 ( x11 ,⋯, xr ) dx 1 ⋯dx r 0 0

+λµ +λµ α + ⋯ α + µ + λ ⋯ µ + λ 11 ⋯ rr = ( 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 ( (t11 x1 ,) ⋯, (trr xr )) dx 1 ⋯dx r 0 0 On Some Integrals Involving Laguerre… 25

+λµ +λµ α + ⋯ α + µ + λ ⋯ µ + λ 11 ⋯ rr = ( 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 + 11 rr  (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 − ⋯ λ − )1,,1 1 − 1 1 r − r 1 1 r γ − γ − ∫⋯ ∫ x1 (t1 x1) ⋯xr (tr xr ) Ln ( (t11 x1 ,) ⋯, (trr xr )) dx 1 ⋯dx r 0 0

+λµ +λµ λ ⋯ λ µ + λ ⋯ µ + λ 11 ⋯ rr = ( 1)n ( )nr B( 1 ,1 1) B( r ,1 r )t1 tr (µ + λ + )1 ⋯(µ + λ + )1 1 1 n r r n +λµ µ +λ × ( 1 1 ⋯⋯,, r r ) γ γ Lm ( t11 ,⋯⋯, trr ) . (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 + ⋯ α + β + ⋯ β + = ( 1 )1 m ( r m ()1 1 )1 n ( s )1 n L H.. S r s m n )!()!(

∞ 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 ⋯ r qp 1 ⋯ qs 0,,,,, 1 p1 r pr 1 q1 s qs 1 r 1 s

∞ (2.8) −σ λ ++++++ × e x x p1 ⋯ r qp 1 ⋯ 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. Mohsen et al.

hypergeometric function for example the generalized hypergeometric function n)( functions Fqp [8,p.42], Appell's function F2 [8,p. 53] , Lauricella's function FC ;: DBA [8,p. 60] , Kampé de Fériet function of two variables F ;: HGE [8,p. 63] Saran's function F [8, p. 66] and Exton's functions K and K [2, p.78] . E 2 5 = On setting r 0 in (2.1), we get

∞ −σ λ − β β x 1 1 ,( ⋯⋯ s ), δ δ ∫ e x Ln ( 1x,⋯⋯, s x) dx 0 δ δ Γ λ β + β + s)(   ()( 1 )1 n ⋯( s )1 n − λ β + β + 1 s = FC  n, ; 1 ,1 ⋯, s ;1 ,⋯,  (3.1) σ λ s  σ σ  n )!(

On setting r = s =1, integral (2.1) reduces to a known result [6, p. 94(12)] see also [9,p. 1132]

∞ −σ x λ − α )(1 γ β )( δ ∫e x Lm ( x)Ln ( x)dx 0

Γ λ ()( α + ()1 β + )1  γ δ  = m n λ − − α + β + λ F2  , m, n; ,1 ;1 ,  (3.2) σ m n!!  σ σ 

On setting r = ,3 s =1 in (2.1), we get

∞ αα α −σ x λ −1 ( 1, 2 3 ), γ γ γ β )( δ ∫ e x Lm ( 1x, 2x, 3x) Ln ( x) dx 0

Γ λ α + α + α + β + = ()( 1 m ()1 2 m ()1 3 m ()1 )1 n σ λ 3 m )!( n!

 γ γ γ δ  K λ λ λ λ;,,, −m,−m,−m,−n ;α + ,1 α + ,1 α + ,1 β + ;1 1 , 2 , 3 , (3.3) 2  1 2 3 σ σ σ σ 

= = On setting r s 2 in (2.1), we get

∞ −σ λ − ( ,αα ) ββ x 1 1 2 γ γ 21 ),( δ δ ∫ e x Lm ( 1x, 2 x)Ln ( 1x, 2 x) dx 0

Γ λ ()( α + ()1 α + ()1 β + ()1 β + )1 = 1 m 2 m 1 n 2 n σ λ m 2 n )!()!( 2 On Some Integrals Involving Laguerre… 27

 γ γ δ δ  K λ λ λ λ;,,, −m,−m,−n,−n,λ ;α + ,1 α + ,1 β + ,1 β + ;1 1 , 2 , 1 , 2 (3.4) 5  1 2 1 2 σ σ σ σ 

γ = γ = γ δ = δ = δ Further, (3.4) for 1 2 and 1 2 and use the result [1, p. 64(3.7)] ( ) K5 a a a a bb d d khfe ,,,;,,,;,,,;,,, vvzz

3;3:1 a :b, 1 (e + f ), 1 (e + f −1);d, 1 (h + k), 1 (h + k −1);  = F 2 2 2 2 z 4,4 v (3.5) − + − + −  3;3:0  : e, f , e f 1 ; h , k , h k 1 ; 

We get

∞ −σ λ − ( ,αα ) ββ x 1 1 2 γ γ 21 ),( δ δ ∫ e x Lm ( x , x) Ln ( x, x) dx 0

Γ λ ()( α + ()1 α + ()1 β + ()1 β + )1 = 1 m 2 m 1 n 2 n σ λ 2 2 m n )!()!(

λ − 1 (α +α + ) 1 (α +α + ) − 1 (β +β + ) 1 (β +β + ) γ δ 3;3:1  : m , 2 1 2 2 , 2 1 2 1; n, 2 1 2 2 , 2 1 2 1 ; 4 4  F  ,  (3.6) 3;3:0 −: α + ,1α + ,1 , α +α +1 ; β + ,1β +1 , β +β +1 ; σ σ  1 2 1 2 1 2 1 2 

On setting r = ,1 s = 2 in (2.1), we get ∞ −σ λ − α ββ x )(1 γ 21 ),( δ δ ∫ e x Lm ( x ) Ln ( 1x, 2x) dx 0

Γ λ ()( α + ()1 β + ()1 β + )1 = m 1 n 2 n σ λ 2 m n )!(!

 γ δ δ  F λ λ λ,,, −m,−n,−n ;α + ,1 β + ,1 β + ;1 , 1 , 2 (3.7) E  1 2 σ σ σ 

= α = α β = β σ = α + Now, on putting r ,1 1 , 1 and 1, integral (2.2) reduces to

t α + α + λ +λα α − λ − α )(1 β = ( )1 m B( ),1 t [− α + λ + β ] ∫ x (t x) Lm ( x) dx F11 m; ;1 t (3.8) 0 m!

On setting r = ,2 β = β = β and using the result [7, p. 28(33)] 1 2

A 0;0:  a)( :−; − ;   a (,)( d + d − (,2/)1' d + d 2/)' ;  = F  x, x A+2FC+3  4x (3.9) C 1;1:  c)( :d d'; ;   c),( d d ,', d + d −1' ; 

28 Fadhle B.F. Mohsen et al.

integral (2.2) reduces to

t λσ −+ 1 σ − λ− ( ,αα ) (α + ()1 α + )1 B σ λ),( t x 1(t − x) 1 L 1 2 (βx, βx) dx = 1 n 2 n ∫ n n )!( 2 0

− n σ ,, (α +α + (,2/)1 α +α + 2/)2 ;  1 2 1 2 β F44  4 t (3.10) σ + λ, α + ,1 α + ,1 α +α +1 ;  1 2 1 2 

= λ = β −α µ = γ − β ν = α −γ On setting m ,3 , , in (2.4), we get

t s r δδδ α − αβ −− 1 β − βγ −− 1 γ − γα −− 1 ( 1 2 ,, 3) σ σ σ ∫∫∫ x (r x) y (s y) z (t z) Ln ( 1 xyz , 2 xyz , 3 xyz ) dx dy dz 0 0 0

δδδ = α + β −α β + γ − β γ + α − γ α β γ ( 1 2 ,, 3) σ σ σ B( ,1 )B( ,1 )B( ,1 ) rt s Ln ( 1 rst , 2 rst , 3 rst )

(3.11)

Finally, setting µ = α , j = ,2,1 ⋯,r in (2.5) and considering the definition (1.2), j j we get a known result of Khan and Shukla [5, p. 115(4.1)].

tr t1 α λ − α λ − (α ⋯α ),, 1 − 1 1 r − r 1 1 r ∫⋯ ∫ x1 (t1 x1) ⋯⋯x1 (tr xr ) Ln ( x1 ,⋯⋯, xr ) dx 1 ⋯⋯dx r 0 0

+λα α +λ α + ⋯ α + α + λ ⋯ α + λ 11 ⋯ rr = ( 1 )1 n ( r )1 n B( 1 ,1 1) B( r ,1 r )t1 tr (α + λ + )1 ⋯(α + λ + )1 1 1 n r r n

α +λ α +λ × ( 1 1 ⋯⋯,, r r ) Ln t1,( ⋯⋯, tr ) . (3.12)

4 Conclusion

The results established in this paper are useful in deriving certain new integral formulas involving Laguerre polynomials of several variables. Further, certain class of known integral formulas involving the product of two Laguerre α )( polynomials Lm x)( can also be obtained in terms of hypergeometric functions

F12 and F23 see for example Mavromatis [6], Shawagfeh [9] and Srivastava et al. [12]. On Some Integrals Involving Laguerre… 29

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