The aim of the present paper is to derive some multiple integral formulas involving Jacobi and Laguerre polynomials of several variables. These results are established with the help of a known and interesting integrals given in Edwards [2].Furthermore, some special cases are also derived.
On Some Multiple Integral Formulas Involving Jacobi and Laguerre Polynomials of Several Variables
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 5 || June. 2016 || PP-01-06
www.ijmsi.org 1 | Page
On Some Multiple Integral Formulas Involving Jacobi and
Laguerre Polynomials of Several Variables
1
Ahmed Ali Atash, 2
Salem Saleh Barahmah
1
Department of Mathematics, Faculty of Education-Shabwah, Aden University, Yemen
2
Department of Mathematics, Faculty of Education -Aden, Aden University, Yemen
ABSTRACT: The aim of the present paper is to derive some multiple integral formulas involving Jacobi and
Laguerre polynomials of several variables. These results are established with the help of a known and
interesting integrals given in Edwards [2].Furthermore, some special cases are also derived.
KEYWORDS: multiple integral formulas, Jacobi polynomials, Laguerre polynomials Kamp´e de F´eriet
function
I. INTRODUCTION
The Jacobi polynomials of several variables rn
xxP rr
,,1
),;;,( 11
of Shrivastava [7]are defined as follows
:
r
nrnn
rn
n
xxP rr
)!(
)1()1()1(
,,
21
1
),;;,( 11
2
1
,,
2
1
;
;
1
1
;
;
;
;
1
1
:
: 1
1
11
1
1
;
;
;
;
1
1
:
:
0
1 r
r
rr xxnnn
F
(1.1)
where n
qqp
mml
xxF n
n
,,1
;;:
;;:
1
1
is the multivariable extension of the Kamp´e de F´eriet function [8], see also [9]
nn
m
n
q
m
q
l
p
n
n
n
n
n
zz
d
b
d
b
c
a
mml
qqp
Fzz
mml
qqp
F
n
n
,,
;
;
)(
)(
;
;
;
;
)(
)(
:
:
)(
)(
;;:
;;:
,,
;;:
;;:
1)(
)(
'
'
1
1
1
1
1
1
1
,
!!
,,
0, 1
1
1
1
1
n
n
ss n
s
n
s
n
s
z
s
z
ss
(1.2)
where
n
nn
n
nn
m
j
s
n
jsj
m
j
ssj
l
j
q
j
s
n
jsj
q
j
ssj
p
j
n
ddc
bba
ss
1
)(
11
1
)(
11
1
)()'()(
)()()(
,,
1
1
1
1
1
1
. (1.3)
And for convergence of the multivariable hypergeometric series in (1.3)
;,,1,01 nkqpml kk
The equality holds when, in addition, either
;1
11
1
lplp
n
xxandlp
or
1},,max{ 1
n
xxandlp .
The Laguerre polynomials of several variables rn
xxL r
,,1
),,( 1
of khan and Shukla [3] are defined as
follows:
2. On Some Multiple Integral Formulas Involving Jacobi And Laguerre Polynomials Of Several…
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rr
r
r
nrn
rn
xxn
n
xxL r
,,;1,,1;
)!(
)1()1(
,, 11
)(
2
1
1
),,( 1
(1.4)
where
)(
2
r
is the confluent hypergeometric function of r-variables [9]
!!)()(
)(
,,;,,;
1
1
0,, 1
11
)(
2
1
1 1
1
r
m
r
m
mm mrm
mm
rr
r
m
x
m
x
cc
a
xxcca
r
r r
r
(1.5)
In our investigation we require the following integrals [2]:
),()1()1()1(
1
0
1
0
111
baBdxdyxyyxy
babaa
, (1.6)
Provided 0)Re( a and 0)Re( b .
),()1()1()1(
1
0
11
baBcxdxcxxx
ababa
(1.7)
Provided 0)Re(,0)Re( ba and 1c , where
)(
)()(
),(
ba
ba
baB
, is the well known Beta
function.
II. MAIN INTEGRAL FORMULAS
In this section ,we have proved the following integral formulas:
1
0
1
0
1
0
1
11
1
1
1
11
1
0
11111
)1()1()1(
babaa
yxyxy
rrrrr ba
rr
b
r
a
r
a
r
yxyxy
111
)1()1()1(
rryx
xy
yx
xydcdc
n
dydxdydxP
rr
rrrr
111
)1(2
1
)1(2),;;,(
1,,1
11
1111
r
rrnrn
n
baBbaBcc
)!(
),(),()1()1( 111
1,,1
;
;
,1
,1
;
;
;
;
,1
,1
:
:
2;;2
2;;2
:
:
0
1
111
111
rrr
rrr
bac
andc
bac
andcn
F (2.1)
1
0
1
0
1
0
1
11
1
1
1
11
1
0
11111
)1()1()1(
babaa
yxyxy
rrrrr ba
rr
b
r
a
r
a
r
yxyxy
111
)1()1()1(
rryx
xy
yx
xycc
n
dydxdydxL
rr
rrr
111
)1(
1
)1(),,(
,,
11
111
r
rrnrn
n
baBbaBcc
)!(
),(),()1()1( 111
1,,1
;
;
,1;
;
;
;
,1:
:
2;;2
1;;1
:
:
0
1
111
1
rrr
r
bac
a
bac
an
F (2.2)
1
0
11
1
0
11
1
1
1
1
)1()1()1()1( 1111 rrrr ba
rr
b
r
a
r
baba
xexxxexx
3. On Some Multiple Integral Formulas Involving Jacobi And Laguerre Polynomials Of Several…
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rxe
x
xe
xdcdc
n
dxdxP
rr
rrr
11
2
1
2),;;,(
1,,1
11
111
r
rrnrn
a
r
a
n
baBbaBccee r
)!(
),(),()1()1()1()1( 1111
1
rrrr
rrr
eebac
andc
bac
andcn
F
1
1
,,
1
1
;
;
,1
,1
;
;
;
;
,1
,1
:
:
2;;2
2;;2
:
:
0
1
1111
111
(2.3)
1
0
11
1
0
11
1
1
1
1
)1()1()1()1( 1111 rrrr ba
rr
b
r
a
r
baba
xexxxexx
rxe
x
xe
xcc
n
dxdxL
rr
rr
111
),,(
,,
11
11
r
rrnrn
a
r
a
n
baBbaBccee r
)!(
),(),()1()1()1()1( 1111
1
rrrr
r
eebac
a
bac
an
F
1
1
,,
1
1
;
;
,1;
;
;
;
,1:
:
2;;2
1;;1
:
:
0
1
1111
1
(2.4)
Further, if we take rr
caca 1,,1 11
in (2.1), (2.2),(2.3) and (2.4) respectively we have
1
0
1
0
1
0
11
1
11
1
1
1
0
11111
)1()1()1(
bcbcc
yxyxy
rrrrr bc
rr
b
r
c
r
c
r
yxyxy
)1()1()1(
11
rryx
xy
yx
xydcdc
n
dydxdydxP
rr
rrrr
111
)1(2
1
)1(2),;;,(
1,,1
11
1111
1,,1),1(),1(
),;;,(
11
1111
rrrr bdbcbdbc
nrr
PbncBbncB
(2.5)
1
0
1
0
1
0
11
1
11
1
1
1
0
11111
)1()1()1(
bcbcc
yxyxy
rrrrr bc
rr
b
r
c
r
c
r
yxyxy
)1()1()1(
11
rryx
xy
yx
xycc
n
dydxdydxL
rr
rrr
111
)1(
1
)1(),,(
,,
11
111
1,,1),1(),1(
),,(
11
11
rr bcbc
nrr
LbncBbncB
(2.6)
1
0
11
1
0
1
11
1
11
)1()1()1()1( 1111 rrrr bc
rr
b
r
c
r
bcbc
xexxxexx
rxe
x
xe
xdcdc
n
dxdxP
rr
rrr
11
2
1
2),;;,(
1,,1
11
111
),1(),1()1()1( 11
11
1
1
rr
c
r
c
bncBbncBee r
1
1
1
1),;;,(
,,
1
11111
r
rrrrr
e
e
e
ebdbcbdbc
n
P
(2.7)
1
0
11
1
0
1
11
1
11
)1()1()1()1( 1111 rrrr bc
rr
b
r
c
r
bcbc
xexxxexx
rxe
x
xe
xcc
n
dxdxL
rr
rr
111
),,(
,,
11
11
r
rrr
ee
bcbc
nrr
c
r
c
LbncBbncBee
1
1
1
1),,(
11
11
1
,,),1(),1()1()1(
1
111
(2.8)
4. On Some Multiple Integral Formulas Involving Jacobi And Laguerre Polynomials Of Several…
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Proof of (2.1): Denoting the left hand side of (2.1) by I, using the definition (1.1), expanding
1
1
;
;
;
;
1
1
:
:
0
1
F in a
power series and changing the order of summation and integration, we get:
r
nrn
n
cc
I
)!(
)1()1( 1
n
p
pn
p
ppn
p rprp
prrppp
r
r r
rr
ppcc
ndcndcn
0 0 0 11
11
1
1
2
1
1
11
!!)1()1(
)1()1()(
1
0
1
0
11
1
11
1
1
1
11
11111111
)1()1()1( dydxyxyxy
bpabpapa
1
0
1
0
111
)1()1()1( rr
bpa
rr
b
r
pa
r
pa
r
dydxyxyxy rrrrrrrr
Finally, evaluating the double integral with the help of the result (1.6),we arrive after some simplification to the
right hand sideof (2.1).This completes the proof of (2.1).The result (2.2) can be established similarly .The two
results (2.3) and (2.4) can be established by applying the same method with the help of the result (1.7).
Remark 1.
Similar eight multiple integral formulas, involving Jacobi and Laguerre polynomials of several
variables rr
rrr
rr
rrr
xe
x
xe
xdcdc
nyx
y
yx
ydcdc
n
PP
1
)1(2
1
)1(2),;;,(
1
)1(2
1
)1(2),;;,(
1,,1,1,,1
11
111
11
111
rr
rr
yx
y
yx
ycc
n
L
1
1
1
1),,(
,,
11
11
and rr
rr
xc
x
xc
xdd
n
L
1
1
1
1),,(
,,
11
11
,can be also obtained by applying the same method.
III. SPECIAL CASES AND APPLICATIONS
1. In (2.1) and if we take 2r , we get
1
0
1
0
1
0
1
0
1
11
1
1
1
11
11111
)1()1()1(
babaa
yxyxy
22222 1
22
1
2
1
22
)1()1()1(
babaa
yxyxy
22111
)1(2
1
)1(2),;,(
22
22
11
112211
1,1 dydxdydxP yx
xy
yx
xydcdc
n
1,1,,,,),(),( 221121
),;,(
2211
2211
babaaaHbaBbaB
dcdc
n
, (3.1)
where rn
xxppH ,,,,, 12121
),;,( 2211
is the generalized Rice polynomials of two variables defined by
2
21
12121
),;,(
)!(
)1()1(
,,,,,2211
n
xxppH
nn
rn
21
22
222
11
111
,
;
;
,1
,1
;
;
,1
,1
:
:
2;2
2;2
:
:
0
1
xx
p
n
p
nn
F
(3.2)
2. in (2.2), if we take 111
1,2 dcar and 222
1 dca , we get
1
0
1
0
1
0
1
0
11
1
11
1
1
11111111
)1()1()1(
bdcbdcdc
yxyxy
22222222
)1()1()1( 22
1
22
1
2
bdcbdcdc
yxyxy
5. On Some Multiple Integral Formulas Involving Jacobi And Laguerre Polynomials Of Several…
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22111
)1(
1
)1(),(
22
22
11
1121
, dydxdydxL yx
xy
yx
xycc
n
2
22211121
)!(
),1(),1()1()1(
n
bdcBbdcBcc nn
1,;1, 222111
),;;,( 2211
bdcbdcZ
ndcndc
n
(3.3)
where 2211
),;,(
,;,2211
xbxbZ n
is the generalized Batman’s polynomials of two variables [1]
21
22
22
11
11
2211
),;,(
,
;
;
1,1
1
;
;
1,1
1
:
:
2;2
1;1
:
:
0
1
,;,2211
xx
b
n
b
nn
FxbxbZ n
. (3.4)
Remark 2.
Similar two other double integrals involving Jacobi and Laguerre polynomials of two
variables 22
2
11
12211
1
2
1
2),;,(
1,1 xe
x
xe
xdcdc
n
P
and 22
2
11
121
11
),(
, xe
x
xe
xcc
n
L
can be also obtained as special cases of our
main results (2.3) and (2.4).
3. In (2.1), if we take 1r , we get
dxdyPxyyxy xy
xydc
n
babaa
1
)1(2),(
1
0
1
0
111
1)1()1()1(
1
;
;
,1
,1,
!
),()1(
23
bac
andcn
F
n
baBc n
(3.5) .
where )(
),(
xPn
is the classical Jacobi polynomials [6]
2
1
;
;
1
1,
!
)1(
)( 12
),( xnn
F
n
xP
n
n
(3.6)
In view of the definition of the generalized Rice polynomials [4]
x
p
nn
F
n
xpH
n
n
;
;
,1
,1,
!
)1(
,, 23
),(
, (3.7)
the integral (3.5) can be written in the following form:
dxdyPxyyxy xy
xydc
n
babaa
1
)1(2),(
1
0
1
0
111
1)1()1()1(
1,,),(
),(
baaHbaB
dc
n
(3.8)
Further, in (3.5) replacing n by n2 , putting 2
1
2
1
, hdcba and using the following special case of the
result given by [5]
nn
nn
ha
ah
ah
anhn
F
)()(
)()(
1
;
;
2),1(
,2,2
2
1
2
1
2
1
2
1
2
1
2
1
2
123
, (3.9)
we get
dxdyPxyyxy xy
xyhh
n
aaaa
1
)1(2),(
2
1
0
1
0
2111
1)1()1()1( 2
1
2
1
2
1
2
1
nn
nnn
han
ahaaBh
)()()!2(
)())(,()(
2
1
2
1
2
1
2
1
2
1
2
1
22
1
2
1
(3.10)
6. On Some Multiple Integral Formulas Involving Jacobi And Laguerre Polynomials Of Several…
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Again, in (3.5) replacing n by 12 n , putting 2
1
2
1
,1 hdcab and using the result [5]
nn
nn
haa
ah
ah
anhn
F
)()(12
)()(
1
;
;
12),1(
,12,12
2
1
2
1
2
3
2
1
2
1
2
3
2
123
, (3.11)
we get
dxdyPxyyxy xy
xyhh
n
aaaa
1
)1(2),(
12
1
0
1
0
21
1)1()1()1( 2
1
2
1
2
1
2
1
nn
nnn
haan
ahaaBh
)())(12()!12(
)())(1,()(
2
1
2
1
2
3
2
1
2
1
2
3
122
1
2
1
. (3.12)
In a similar way, a number of double integrals involving Jacobi polynomials )(
),(
xPn
can be also obtained,
we mention here the following examples :
dxdyPxyyxy xy
xyhh
n
aaaa
1
)1(2),1(
2
1
0
1
0
2221
1)1()1()1( 2
1
2
1
nn
nnn
hahan
hanhaaaBh
)())(22()!2(
)1()()422)(1,()(
2
1
2
1
2
1
2
1
22
1
(3.13)
dxdyPxyyxy xy
xyhh
n
aaaa
1
)1(2),1(
12
1
0
1
0
2221
1)1()1()1( 2
1
2
1
nn
nnn
hanaa
ahnhaaaBh
)1()()!12)(12(
)2()()42)(1,()(
2
1
2
1
2
1
2
3
122
1
. (3.14)
REFERENCES
[1]. S.M.Abbas, M.A.Khan and A.H. Khan, On some generating functions of Gener alized Bateman’s and Pasternak’s polynomials of
two variables, Commun. Korean Math. Soc.28 (2013), 87-105.
[2]. J.Edward, A Treatise on the Integral Calculus, Vol. II, Chelsea publishing Company, New York, 1922.
[3]. M. A. Khan and A. K. Shukla, On Laguerre polynomials of several variables, Bull Cal. Math. Soc. 89 (1997),155-164.
[4]. P.R. Khandekar, On generalization of Rice’s polynomials, I. Proc. Nat. Acad. Sci. India Sect., A 34(1964),157-162.
[5]. J. L. Lavoie, F. Grondin and A. K. Rathie,Generalizations of Watson’s theorem on the sum of a 23
F , Indian J. Math.,
34(1992),23-32.
[6]. E. D. Rainville, Special Functions. The Macmillan Company, New York,1960.
[7]. H.S.P. Shrivastava, On Jacobi polynomials of several variables, Integral Trans. Spec. Func. 10 (2000), 61-70.
[8]. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press, New York, 1985.
[9]. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press,New York,1984.
![International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 5 || June. 2016 || PP-01-06
www.ijmsi.org 1 | Page
On Some Multiple Integral Formulas Involving Jacobi and
Laguerre Polynomials of Several Variables
1
Ahmed Ali Atash, 2
Salem Saleh Barahmah
1
Department of Mathematics, Faculty of Education-Shabwah, Aden University, Yemen
2
Department of Mathematics, Faculty of Education -Aden, Aden University, Yemen
ABSTRACT: The aim of the present paper is to derive some multiple integral formulas involving Jacobi and
Laguerre polynomials of several variables. These results are established with the help of a known and
interesting integrals given in Edwards [2].Furthermore, some special cases are also derived.
KEYWORDS: multiple integral formulas, Jacobi polynomials, Laguerre polynomials Kamp´e de F´eriet
function
I. INTRODUCTION
The Jacobi polynomials of several variables rn
xxP rr
,,1
),;;,( 11
of Shrivastava [7]are defined as follows
:
r
nrnn
rn
n
xxP rr
)!(
)1()1()1(
,,
21
1
),;;,( 11
2
1
,,
2
1
;
;
1
1
;
;
;
;
1
1
:
: 1
1
11
1
1
;
;
;
;
1
1
:
:
0
1 r
r
rr xxnnn
F
(1.1)
where n
qqp
mml
xxF n
n
,,1
;;:
;;:
1
1
is the multivariable extension of the Kamp´e de F´eriet function [8], see also [9]
nn
m
n
q
m
q
l
p
n
n
n
n
n
zz
d
b
d
b
c
a
mml
qqp
Fzz
mml
qqp
F
n
n
,,
;
;
)(
)(
;
;
;
;
)(
)(
:
:
)(
)(
;;:
;;:
,,
;;:
;;:
1)(
)(
'
'
1
1
1
1
1
1
1
,
!!
,,
0, 1
1
1
1
1
n
n
ss n
s
n
s
n
s
z
s
z
ss
(1.2)
where
n
nn
n
nn
m
j
s
n
jsj
m
j
ssj
l
j
q
j
s
n
jsj
q
j
ssj
p
j
n
ddc
bba
ss
1
)(
11
1
)(
11
1
)()'()(
)()()(
,,
1
1
1
1
1
1
. (1.3)
And for convergence of the multivariable hypergeometric series in (1.3)
;,,1,01 nkqpml kk
The equality holds when, in addition, either
;1
11
1
lplp
n
xxandlp
or
1},,max{ 1
n
xxandlp .
The Laguerre polynomials of several variables rn
xxL r
,,1
),,( 1
of khan and Shukla [3] are defined as
follows:](https://image.slidesharecdn.com/a04050106-160912071410/85/On-Some-Multiple-Integral-Formulas-Involving-Jacobi-and-Laguerre-Polynomials-of-Several-Variables-1-320.jpg)
![On Some Multiple Integral Formulas Involving Jacobi And Laguerre Polynomials Of Several…
www.ijmsi.org 2 | Page
rr
r
r
nrn
rn
xxn
n
xxL r
,,;1,,1;
)!(
)1()1(
,, 11
)(
2
1
1
),,( 1
(1.4)
where
)(
2
r
is the confluent hypergeometric function of r-variables [9]
!!)()(
)(
,,;,,;
1
1
0,, 1
11
)(
2
1
1 1
1
r
m
r
m
mm mrm
mm
rr
r
m
x
m
x
cc
a
xxcca
r
r r
r
(1.5)
In our investigation we require the following integrals [2]:
),()1()1()1(
1
0
1
0
111
baBdxdyxyyxy
babaa
, (1.6)
Provided 0)Re( a and 0)Re( b .
),()1()1()1(
1
0
11
baBcxdxcxxx
ababa
(1.7)
Provided 0)Re(,0)Re( ba and 1c , where
)(
)()(
),(
ba
ba
baB
, is the well known Beta
function.
II. MAIN INTEGRAL FORMULAS
In this section ,we have proved the following integral formulas:
1
0
1
0
1
0
1
11
1
1
1
11
1
0
11111
)1()1()1(
babaa
yxyxy
rrrrr ba
rr
b
r
a
r
a
r
yxyxy
111
)1()1()1(
rryx
xy
yx
xydcdc
n
dydxdydxP
rr
rrrr
111
)1(2
1
)1(2),;;,(
1,,1
11
1111
r
rrnrn
n
baBbaBcc
)!(
),(),()1()1( 111
1,,1
;
;
,1
,1
;
;
;
;
,1
,1
:
:
2;;2
2;;2
:
:
0
1
111
111
rrr
rrr
bac
andc
bac
andcn
F (2.1)
1
0
1
0
1
0
1
11
1
1
1
11
1
0
11111
)1()1()1(
babaa
yxyxy
rrrrr ba
rr
b
r
a
r
a
r
yxyxy
111
)1()1()1(
rryx
xy
yx
xycc
n
dydxdydxL
rr
rrr
111
)1(
1
)1(),,(
,,
11
111
r
rrnrn
n
baBbaBcc
)!(
),(),()1()1( 111
1,,1
;
;
,1;
;
;
;
,1:
:
2;;2
1;;1
:
:
0
1
111
1
rrr
r
bac
a
bac
an
F (2.2)
1
0
11
1
0
11
1
1
1
1
)1()1()1()1( 1111 rrrr ba
rr
b
r
a
r
baba
xexxxexx ](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)