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1. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
Simultaneous Triple Series Equations
Associated With Laguerre Polynomials With Matrix Argument
Kuldeep Narain
School of Quantitative Sciences, UUM College of Arts and Sciences, University Utara Malaysia, Sintok –
06010, Malaysia
E-mail: kuldeep@uum.edu.my
Abstract
Integral and Series equations are very useful in the theory of elasticity, elastostatics , diffraction theory and
acoustics. Particularly these equations are very much useful in finding the solution of crack problems of fracture
mechanics. In this paper solution of simultaneous triple series equations associated with Laguerre polynomials
with matrix argument has been obtained , which arises in the Crack problems of Fracture Machanics.
Keywords: Integral equations, Series equations , Laguerre polynomials, Matrix argument.
1. Introduction
In the present paper, we have considered the following simultaneous triple series equations of the form
¥
s
n=0
j =1
¥
s
n=0
j =1
S Sa
ij
S S
bij cnj .
Lni (a + b ¥
s
n=0
j =1
S S
for
G m (a + b + ni)
cnj
Lni (a : x) = fi (x), 0 £ x £ D,
m +1
)G m (a + b + ni)
2
II m (a + b + ni)
G m (a + ni +
m +1
, y ) = gi ( x), D £´£ E ,
2
(1.2)
cnj G m (a + b - m + 1) = Lni (a , x) = hi ( x), E £´£ ¥ ,
a +b >
2
m +1
- 1,0 £ b £1
2
where,
Ln (a , x) =
(1.1)
II m (a + n)
m +1
1F1 (-n, a +
), x)
P m (a )
2
88
(1.3)

2. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
is Laguerre polynomial of matrix argument ,
for
R (a ) > - 1,
R (n + a ) > 1, and II m (a) = G m (a +
m
G m (a) = p m ( m-1)/4 Õ (a i =1
m +1
)
2
i -1
)
2
n =0,1,2,……………………., J = 1,2,3,……………s ,
f(x), g(x) h(x) are known functions of non – singular matrix x of order m ; aij , bij and cij are known
constant .By using multiplying factor technique [ Srivastava], the unknown function C nj is determined .
2 . Some Useful Results
(i) The following integrals are required from Erdelyi et al with matrix argument
y
ò0
y
a
y-x
B -(m +1)/2
Ln (a , x)dx
m +1
)
a +b
2
=
y
Ln (a + b ; y )
m +1
G m (a + b + n +
)
2
G m ( b ) G m (a + n +
for
a > - 1, b >
¥
ò x- y
y
= Gm (
for
-b
m +1
-1
2
(2.1)
and
etr (- x) Ln (a , x)dx
m +1
m +1
- b ) etr (- y) Ln (a + b ; y)
2
2
b<
m +1
m +1
,a + b >
-1 .
2
2
(ii) The orthogonality relation for Laguerre polynomial with matrix argument
89
(2.2)

3. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
a
ò x >0 x etr (- x) Lp (a ; x) Lq (a ; x)
m +1
)
2 d
pq
m +1
)
G m (p+
2
G m (a + b +
=
for a > - 1 and
d pq
(2.3)
being the kronecker delta .
(iii) The differential formula with matrix argument due to Erdelyi et al
a
Dx [ x Ln (a ; x)] = x
a-
m +1
2
Ln (a -
m +1
; x)
2
(2.4)
3. Solution
Multiply eq. (1.1) by
y-x
b - ( m+1) 2
and eq. (1.2) by
x- y
-b
etr (- x) and then integrating w.r.t. x
m +1
) G m (a + b + ni)
2
Lni (a + b ; y ) =
II m (a + b + ni )
G m (a + ni +
s
åå
a
(0, y) and ( y, ¥) respectively, on using the result (2.1) and (2.2), we obtain
over the range
¥
x
aij cnj
n = 0 j =1
-a - b
y
a
b - ( m +1)/2
y
ò0 x y - x
fi ( x)dx ,
Gm (b )
b>
for
(3.1)
m +1
, a > -1, 0 < y < D
2
and
¥
s
S S bij cnj
n = 0 j =1
=
m +1
) G m (a + b + ni)
m +1
2
Lni (a + b ; y)
II m (a + b + ni)
2
G m (a + ni +
etr ( y )
-b
ò ¥ etr (- x) x - y gi ( x)dx ,
y
m +1
Gm ( 2 - b )
for b <
(3.2)
m +1
m +1
, a +b >
- 1, D < y < ¥ .
2
2
90

4. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
If we new multiply eq.(3.1) by by y
¥
s
S S bij cnj
for
a
, differentiating w.r.t. Y and using the formula (2.4), we find
m +1
m +1
(
) -a - b
) G m (a + b + ni)
s
y 2
m +1
2
. Lni (a + b .
; y ) = å eij
2
Gm (b )
II m (a + b + ni)
j =1
G m (a + ni +
n = 0 j =1
y
Dy ò 0 x
a +b
www.iiste.org
y-x
b -(m +1)/2,
0 < y < D, b >
fi ( x)dx ,
(3.3)
m +1
- 1, a > -1, and eij are the element of the matrix [bij ][aij ]-1 and
2
i = 1, 2,.....s.
The left- hand side of eq.. (3.2), (3.3) and (1.2) are identical and hence on using the orthogonality relation (2.3),
we find the solution of eq. (1.1), (1.2) and (1.3) for
a +b >
m +1
- 1, 0 < b < 1
2
m +1
) G m (a + b + ni )
2
Cnj = S dij
Bni (a + b ; D) (3.4)
j =1
m +1
2
)[G m (a + b + ni)]
G m (a + ni +
2
G m (ni +
s
Where ,
n = 0,1, 2.......; j = 1, 2,3,.......,s and d ij are the element of the matrix [bij ]-1 and
s
Bni (a , b ; D) = å eij
j =1
D
1
ò etr (- y)Lni (a + b - m2+1 ; y) .
Gm (b ) 0
m +1
, y )Gi ( y )dy
2
¥
m +1
1
a + b - ( m +1)/2
+
ò y
Ln (a + b ; y ) H ( y )dy
m +1
E
2
Gm (
- b)
2
E
Fi ( y )dy + ò etr (- y ) y
a + b - (m +1)/2
D
y
Fi ( y ) = Dy ò x
a
y-x
-b
etr (- x)hi ( x)dx
0
b - ( m +1)/2
Ln (a + b -
fi ( x)dx
(3.6)
Gi ( y) = gi ( y)
¥
H i ( y) = ò x - y
y
(3.5)
91

5. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
References
T.W. Anderson ; (1958) , An Introduction to Multivariate statistical Analysis, John Wiley and Sons, New York .
A. Erdelyi , et al ; (1953) , Higher Transcendental functions, Vol, II, Mc Graw Hill Book co, Inc; New York.
A. Erdelyi , et al ; (1954) ,Tables of Integral Transforms , Vol, II, Mc Graw Hill Book co, Inc; New York .
A.M. Mathoi and R.K. Saxena; (1978) , The H-function with Applications in Statistics and other Disciplines,
Wiley Eastern Limited, New Delhi , India, 96-132 .
H.M. Srivastava,; (1969) , Notices Am. Math. Soc. 16 , 568, (See also p. 517) .
H. M. Srivastava ; (1969) , Pacific J. Math , 30 ( 1969) , 525 -27 .
H.M. Srivastava ; (1970) , J. Math . Anal . Appl. 31, 587-94 (see also p.587) .
92

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