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. Mohsen1, Ahmed Ali Atash2 and Salem Saleh Barahma3
1
Department of Mathematics
Faculty of Education-Zingibar, Aden University, Yemen
E-mail: mfazalmohsen@yahoo.com
2
Department of Mathematics
Faculty of Education-Shabowh, Aden University, Yemen
E-mail: ah-a-atash@hotmail.com
3
Department of Mathematics
Faculty of Education-Aden, Aden University, Yemen
E-mail: salemalqasemi@yahoo.com
(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
1
Fadhle B.F. Mohsen et al.
Introduction
In 1991, Ragab [7] defined the Laguerre polynomials of two variables
L(nα , β ) ( x, y ) as follows:
L(nα , β ) ( x, y ) =
Γ(n + α + 1) Γ(n + β + 1)
n!
(− y ) r L(nα−)r ( x)
∑
r = 0 r !Γ (α + n − r + 1) Γ ( β + r + 1)
n
(1.1)
where L(nα ) ( x ) is the Laguerre polynomials of one variable [8]
The definition (1.1) is equivalent to the following explicit representation of
L(nα , β ) ( x, y ) , given by Ragab [7]:
( α + 1) n ( β + 1) n
( n !) 2
L(nα , β ) ( x, y ) =
( −n) r + s y r x s
∑∑
r = 0 s = 0 (α + 1) s ( β + 1) r r! s!
n n− r
(1.2)
It may be remarked that (1.2) can be written as
L(nα , β ) ( x , y ) =
(α + 1) n ( β + 1) n
Ψ2 [− n ; α + 1, β + 1; x, y ]
( n !) 2
(1.3)
where Ψ2 is the confluent hypergeometric function of two variables [11, p.62]
Ψ2( m) [a ; b , c ; x, y ] =
∞
xr y s
,
∑
r , s = 0 (b) r (c) s r ! s !
( a) r + s
1

where (λ ) n = 
 λ (λ + 1)(λ + 2)....(λ + n − 1)
(1.4)
, if n = 0
, if n = 1,2 , 3, ......
(1.5)
Khan and Shukla [4,p. 163] defined the Laguerre polynomials of several
variables L(nα1 ,⋯, α m ) ( x1 , ⋯ , xm ) as follows:
L(nα 1 , ⋯⋯, α m ) ( x1 , ⋯ , xm )
m
=
Π (α j + 1)n
j =1
( n !)
m
m
n n − r1
n − r1 −⋯⋯− rm−1
∑ ∑⋯⋯ ∑
r1 = 0 r2 = 0
rm = 0
(−n)r1 +⋯⋯+ rm Π xmj+1− j
r
j =1
m
m
j =1
j =1
Π rj ! Π (α j + 1)m +1− j
(1.6)
On Some Integrals Involving Laguerre…
23
m
=
Π (α j + 1) n
j =1
(n !)
m
Ψ2( m ) [− n ; α1 + 1, ⋯⋯, α m + 1; x1 ,⋯⋯, xm ] ,
(1.7)
where Ψ2( m ) is the confluent hypergeometric function of m-variables [11, p.62]
Ψ2( m) [a ; c1 ,⋯, cm ; x1 ,⋯, xm ] =
∞
(a) r1 +⋯+ rm
r1 ,⋯, rm = 0
(c1 ) r1 ⋯ (cm ) rm
∑
x1r1 xmrm
⋯
r1 ! rm !
(1.8)
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:
EC( n ) [a, a′, b ; c1 ⋯ , cn ; x1 ,⋯ , xn ]
(k )
(1)
∞
(a) m1 + ⋯ + mk ( a' ) mk +1 +⋯+ mn (b) m1 +⋯+ mn x1m1 ⋯ xnmn
m1 ,⋯, mn = 0
(c1 ) m1 ⋯ (cn ) mn m1 !⋯ mn !
=
∑
(1.9)
and
F
A : B ′ ;⋯ ; B ( n )
A : B ′ ;⋯ ; B ( n )
=
F
[
x
,
⋯
,
x
]
1
n
C : D ′ ;⋯ ; D ( n )
C : D ′ ;⋯ ; D ( n )
(a ) : (b′) ;⋯ ; (b ( n ) ) ;

x1 ,⋯ , xn 

(n)
 (c) : (d ′) ;⋯ ; (d ) ;

∞
((a)) m1 + ⋯+ mn ((b′)) m1 ⋯ ((b ( n ) )) mn x1m1 ⋯ xnmn
m1 ,⋯, mn = 0
((c)) m1 +⋯ + mn ((d ′)) m1 ⋯ ((d ( n ) )) mn m1 !⋯ mn !
∑
=
(1.10)
A
where ((a)) m mean the product Π (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:
∞
∫e
−σ x
(α ,⋯⋯,α r )
x λ −1 Lm 1
(γ 1 x,⋯⋯, γ r x) L(nβ1 ,⋯⋯, βs ) (δ1 x,⋯⋯, δ s x) dx
0
=
Γ(λ ) (α1 + 1) m ⋯ (α r + 1) m ( β1 + 1) n ⋯ ( β s + 1) n
σ λ (m!) r (n!) s
(r )
(1)
γ
γ δ
δ 

EC( r + s )  − m,− n, λ ; α1 + 1, ⋯ , α r + 1, β1 + 1, ⋯ , β s + 1; 1 , ⋯ , r , 1 , ⋯ , s  (2.1)
σ
σ σ
σ

24
Fadhle B.F. Mohsen et al.
t
(α ,⋯⋯,α r )
σ −1
λ −1
∫ x (t − x) Ln 1
( β1 x,⋯⋯, β r x) dx =
0
F
t
∫x
σ −1
(α1 + 1) n ⋯ (α r + 1) n B (σ , λ )t σ +λ −1
(n!) r
2 : 0 ;⋯⋯; 0 − n, σ : − ;⋯⋯ ; − ;

β1t , ⋯⋯, β r t 

1: 1;⋯⋯;1 σ + λ :α1 + 1;⋯⋯ ;α r + 1 ;

(α ,,⋯⋯, α r )
(t − x) λ −1 Ln 1
(2.2)
( γ 1 (t − x) ,⋯⋯, γ r (t − x) ) dx
0
(α1 + 1) n ⋯ (α r + 1) n B (σ , λ ) t λ +σ −1
=
( n !) r
F
t s r
∫∫∫x
α
2: 0 ;⋯; 0 − n, λ : − − ;⋯ ; − − ;

γ 1t ,⋯, γ r t 

1: 1;⋯;1 λ + σ : 1 + α1 ;⋯ ;1 + α r ;

(δ ,⋯⋯,δ m )
(r − x) λ −1 y β ( s − y ) µ −1 z γ (t − z )ν −1 Ln 1
(2.3)
( xyz,⋯⋯, xyz) dx dy dz
0 0 0
(δ1 + 1) n ⋯ (δ m + 1) n B (α + 1, λ ) B ( β + 1, µ ) B (γ + 1,ν )r α +λ s β + µ t γ +ν
=
(n!) r
F
4 : 0 ;⋯⋯; 0  − n , α + 1, β + 1, γ + 1 : − − ;⋯ ; − − ;

rst ,⋯, rst 

3: 1;⋯⋯;1 α + λ + 1, β + µ + 1, γ + ν + 1:δ1 + 1;⋯ ; δ m + 1 ;

tr
t1
µ
µ
(α ,⋯,α r )
λ −1
λ −1
∫ ⋯ ∫ x1 1 (t1 − x1 ) 1 ⋯ x1 r (tr − xr ) r Ln 1
0
(2.4)
( 1 x1 ,⋯, xr ) dx1 ⋯ dxr
0
=
F
(α1 + 1) n ⋯(α r + 1) n B( µ1 + 1, λ1 )⋯ B (µ r + 1, λr ) t1
(n!) r
µ1 +λ1
⋯tr
µr +λr
µ1 + 1
µr + 1
1 : 1;⋯⋯;1 − n :
;⋯⋯ ;
;

t1 , ⋯⋯, tr 

0 : 2;⋯⋯;2  − :α1 + 1, µ1 + λ1 + 1;⋯⋯ ;α r + 1, µ r + λr + 1 ;

tr
t1
0
0
µ
µ
(α ,⋯,α r )
λ −1
λ −1
∫ ⋯ ∫ x1 1 (t1 − x1 ) 1 ⋯ x1 r (tr − xr ) r Ln 1
(2.5)
( γ 1 (t1 − x1 ) ,⋯, γ r (t r − xr )) dx1 ⋯ dxr
On Some Integrals Involving Laguerre…
25
(α + 1) n ⋯(α r + 1) n B( µ1 + 1, λ1 )⋯ B (µ r + 1, λr ) t1
= 1
(n!) r
F
µ1 +λ1
⋯tr
µr +λr
λ1
λr
1 : 1;⋯⋯;1 − n :
;⋯⋯ ;
;

γ 1t1 ,⋯⋯, γ r tr 

0 : 2;⋯⋯;2  − :α1 + 1, µ1 + λ1 + 1;⋯⋯ ;α r + 1, µr + λr + 1 ;
 (2.6)
Following integral can be obtained readily from (2.6) as follows:
tr
t1
µ1
∫⋯∫ x
1
0
( λ −1,⋯,λr −1)
µ
(t1 − x1 ) λ1−1 ⋯ xr r (tr − xr ) λr −1 Ln 1
( γ 1 (t1 − x1 ) ,⋯, γ r (t r − xr )) dx1 ⋯ dxr
0
=
(λ1 ) n ⋯(λr ) n B( µ1 + 1, λ1 ) ⋯ B (µ r + 1, λr ) t1
(µ1 + λ1 + 1) n ⋯( µ r + λr + 1) n
( µ +λ1 ,⋯⋯,µ r +λr )
× Lm 1
µ1 +λ1
⋯t r
µr +λr
(γ 1t1 , ⋯⋯ , γ r t 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 Ψ2( m ) in series forms and changing the order of
integration and summation to get
L.H .S =
(α1 + 1) m ⋯ (α r + 1) m ( β1 + 1) n ⋯ ( β s + 1) n
(m!) r (n!) s
∞
∑
p1 ,⋯, pr ,q1 ,⋯,qs =0
(−m) p1 +⋯+ pr (−n) q1+⋯+qs γ 1p1 ⋯γ rpr δ1q1 ⋯δ sqs
(α1 + 1) p1 ⋯ (α r + 1) pr ( β1 + 1) q1 ⋯ ( β s + 1) qs p1!⋯ pr !q1!⋯ qs !
∞
× ∫ e −σx x λ + p1+⋯+ pr +q1+⋯+qs dx
(2.8)
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
functions p Fq [8,p.42], Appell's function F2 [8,p. 53] , Lauricella's function FC(n )
[8,p. 60] , Kampé de Fériet function of two variables FEA:G:B;;HD [8,p. 63] Saran's
function FE [8, p. 66] and Exton's functions K 2 and K 5 [2, p.78] .
On setting r = 0 in (2.1), we get
∞
∫e
−σ x λ −1 ( β1 ,⋯⋯, β s )
n
x
L
(δ1 x, ⋯⋯, δ s x) dx
0
=
Γ(λ ) ( β1 + 1) n ⋯ ( β s + 1) n F ( s ) − n, λ ; β + 1,⋯, β + 1; δ1 ,⋯, δ s 
C 
1
s
σ
σ 

σ λ (n!) s
(3.1)
On setting r = s = 1 , integral (2.1) reduces to a known result [6, p. 94(12)] see
also [9,p. 1132]
∞
∫e
−σ x
x λ −1L(mα ) (γx) L(nβ ) (δx) dx
0
=
Γ(λ ) (α + 1) m ( β + 1) n
γ δ

F2 λ ,−m,− n ; α + 1, β + 1; , 
λ
σ m! n!
σ σ

(3.2)
On setting r = 3, s = 1 in (2.1), we get
∞
∫e
−σ x λ −1 (α1 ,α 2 ,α 3 )
m
x
L
(γ 1 x , γ 2 x, γ 3 x) L(nβ ) (δx) dx
0
=
Γ(λ ) (α1 + 1)m (α 2 + 1) m (α 3 + 1)m ( β + 1) n
σ λ (m!)3 n!
γ γ γ δ

K 2 λ , λ , λ , λ ;−m,− m,− m,− n ; α1 + 1, α 2 + 1, α 3 + 1, β + 1; 1 , 2 , 3 , 
σ σ σ σ

On setting r = s = 2 in (2.1), we get
∞
∫e
−σ x λ −1 (α1 ,α 2 )
m
x
L
(γ 1 x , γ 2 x) L(nβ1 , β 2 ) (δ1x, δ 2 x) dx
0
=
Γ(λ ) (α1 + 1) m (α 2 + 1) m ( β1 + 1) n ( β 2 + 1) n
σ λ (m!) 2 (n!) 2
(3.3)
On Some Integrals Involving Laguerre…
27
γ γ δ δ 

K 5 λ , λ , λ , λ ;−m,− m,−n,−n, λ ; α1 + 1, α 2 + 1, β1 + 1, β 2 + 1; 1 , 2 , 1 , 2  (3.4)
σ σ σ σ

Further, (3.4) for γ 1 = γ 2 = γ and δ1 = δ 2 = δ and use the result [1, p. 64(3.7)]
K 5 (a, a, a, a ; b, b, d , d ; e, f , h, k ; z , z , v , v )
=F
1 : 3 ; 3 a :b , 12 (e + f ), 12 (e + f − 1); d , 12 (h + k ), 12 (h + k − 1);

4 z , 4v

0 : 3; 3  − : e , f , e + f − 1 ; h , k , h + k − 1 ;

(3.5)
We get
∞
∫e
−σ x λ −1 (α1 ,α 2 )
m
x
L
(γx , γx) L(nβ1 , β 2 ) (δx, δx) dx
0
=
Γ(λ ) (α1 + 1) m (α 2 + 1) m ( β1 + 1) n ( β 2 + 1) n
σ λ (m!) 2 (n!) 2
F
1: 3;3 λ :−m , 12 (α1 +α2 + 2), 12 (α1 +α2 +1);−n , 12 (β1 + β2 + 2), 12 (β1 + β2 +1) ; 4γ 4δ 
,
0: 3;3 −: α1 +1,α2 +1, , α1 +α2 +1 ; β1 +1, β2 +1 , β1 + β2 +1 ; σ σ 
(3.6)
On setting r = 1, s = 2 in (2.1), we get
∞
∫e
L (γx ) L(nβ1 , β 2 ) (δ1 x, δ 2 x) dx
−σ x λ −1 (α )
m
x
0
=
Γ (λ ) (α + 1) m ( β1 + 1) n ( β 2 + 1) n
σ λ m!(n!)2
γ δ δ
FE λ , λ , λ ,− m,− n,− n ;α + 1, β1 + 1, β 2 + 1; , 1 , 2 
σ σ σ

(3.7)
Now, on putting r = 1, α1 = α , β1 = β and σ = α + 1 , integral (2.2) reduces to
t
α
λ −1 (α )
∫ x (t − x) Lm (β x) dx =
0
(α + 1) m B(α + 1, λ )t α + λ
1 F1 [− m; α + λ + 1 ; βt ] (3.8)
m!
On setting r = 2 , β1 = β2 = β and using the result [7, p. 28(33)]
F
A : 0; 0 ( a ) : − ; − ;
x,
C : 1;1  (c ) : d ; d ' ;

x =

A+ 2
( a ) , (d + d '−1) / 2, ( d + d ' ) / 2 ; 
FC +3 
4x
(c ), d , d ' , d + d '−1
; 

(3.9)
28
Fadhle B.F. Mohsen et al.
integral (2.2) reduces to
t
(α ,α 2 )
σ −1
λ −1
∫ x (t − x) Ln 1
( βx, βx) dx =
0
4
(α1 + 1) n (α 2 + 1) n B(σ , λ )t σ +λ −1
(n!) 2
− n , σ , (α1 + α 2 + 1) / 2 , (α1 + α 2 + 2) / 2 ;

F4 
4β t 
α1 + 1 , α 2 + 1 , α1 + α 2 + 1 ;
 σ + λ,

(3.10)
On setting m = 3, λ = β − α , µ = γ − β ,ν = α − γ in (2.4), we get
t s r
∫∫∫x
α
(δ ,δ 2 ,δ 3 )
(r − x) β −α −1 y β ( s − y )γ −β −1 z γ (t − z )α −γ −1 Ln 1
(σ 1 xyz , σ 2 xyz , σ 3 xyz) dx dy dz
0 0 0
(δ ,δ 2 ,δ 3 )
= B(α + 1, β − α ) B( β + 1, γ − β ) B(γ + 1, α − γ )t α r β s γ Ln 1
(σ 1 rst , σ 2 rst , σ 3 rst )
(3.11)
Finally, setting µ j = α j , j = 1,2, ⋯ , r in (2.5) and considering the definition (1.2),
we get a known result of Khan and Shukla [5, p. 115(4.1)].
tr
t1
∫⋯∫ x
α1
1
0
α
(α ,⋯,α r )
(t1 − x1 ) λ1−1 ⋯⋯ x1 r (tr − xr ) λr −1 Ln 1
( x1 ,⋯⋯, xr ) dx1 ⋯⋯ dxr
0
α1 +λ1
(α + 1) n ⋯(α r + 1) n B(α1 + 1, λ1 )⋯ B (α r + 1, λr ) t1
= 1
(α1 + λ1 + 1) n ⋯(α r + λr + 1) n
(α + λ1 ,⋯⋯,α r + λr )
× Ln 1
4
(t1 , ⋯⋯ , t r ) .
⋯t r
α r +λr
(3.12)
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
polynomials L(mα ) ( x ) can also be obtained in terms of hypergeometric functions
2 F1 and 3 F2 see for example Mavromatis [6], Shawagfeh [9] and Srivastava et al.
[12].
On Some Integrals Involving Laguerre…
29
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