A general index integral of the product of Meijer`s G- functions

A general index integral of the product of Meijer`s G- functions

A general index integral of the product of
Meijer’s G- functions
Semyon B. YAKUBOVICH∗
November 10, 2008
The purpose of this note is to give a formal proof of the following formula
µ ¯
¶
Z ∞
¯ it, −it, (−αpn1 +1 ), (−αn1 )
1
q
−
m
,
p
−
n
+
2
1
1
1
1
2πt
1
¯
te G
a¯
p1 + 2, q1
(−βqm1 1 +1 ), (−βm1 )
2π 2 −∞
m + m2 , n1 + n2 + 2
×G 1
p1 + p2 + 2, q1 + q2
−1
=a
µ ¯
¶
¯ 1 + it, 1 − it, (cp1 +p2 )
b¯¯
dt
(dq1 +q2 )
m ,n
G 2 2
p2 , q2
¶
µ ¯
b ¯¯ (γp2 )
,
a ¯ (δq2 )
(1)
which is associated with three Meijer’s G-functions of different parameters.
We note, that integral (1) contains most of known integrals with respect to
parameters of hypergeometric functions, which can be found in [1], [2]. Here
as usual, 0 ≤ mi ≤ qi , 0 ≤ ni ≤ pi , i = 1, 2, a 6= 0, b are parameters and
vectors of other parameters are defined on a usual way:
(−αpn11 +1 ) = (−αn1 +1 , . . . , −αp1 ),
(−αn1 ) = (−α1 , . . . , −αn1 ),
(−βqm1 1 +1 ) = (−βm1 +1 , . . . , −βq1 ),
(−βm1 ) = (−β1 , . . . , −αm1 ),
∗
Department of Pure Mathematics, Faculty of Sciences, University of Porto, Campo
Alegre st., 687, 4169-007 Porto, Portugal, E-mail: [email protected]. Work supported
by Fundação para a Ciência e a Tecnologia (FCT, the programmes POCTI and POSI)
through the Centro de Matemática da Universidade do Porto (CMUP). Available as a
PDF file from http://www.fc.up.pt/cmup.
1
(cp1 +p2 ) = (α1 , α2 , . . . , αn1 , γ1 , . . . , γn2 , αn1 +1 , . . . , αp1 , γn2 +1 , . . . , γp2 ),
(dq1 +q2 ) = (β1 , β2 , . . . , βm1 , δ1 , . . . , δm2 , βm1 +1 , . . . , βq1 , δm2 +1 , . . . , δq2 ),
(γp2 ) = (γ1 , . . . , γp2 ),
(δq2 ) = (δ1 , . . . , δq2 ).
We also note, that all parameters satisfy the corresponding convergence conditions, which can be done accordingly.
In order to prove (1), we denote its left hand-side as I(a, b) and by using
the evenness with respect to t of the Meijer G-functions under the integral
sign, we write I(a, b) in the form of the inverse Wimp - Yakubovich transform
[5, Ch. 7], [4] namely
Z ∞
1
I(a, b) = 2
t sinh(2πt)
π 0
µ ¯
¶
¯ it, −it, (−αpn1 +1 ), (−αn1 )
q1 − m1 , p1 − n1 + 2
1
¯
×G
a¯
p1 + 2, q1
(−βqm1 1 +1 ), (−βm1 )
m + m2 , n 1 + n2 + 2
×G 1
p1 + p2 + 2, q1 + q2
µ ¯
¶
¯ 1 + it, 1 − it, (cp1 +p2 )
b¯¯
dt.
(dq1 +q2 )
(2)
Hence the direct Wimp-Yakubovich transform [5] should give reciprocally
µ ¯
¶
¯ 1 + it, 1 − it, (cp1 +p2 )
m1 + m2 , n1 + n2 + 2
¯
G
b¯
p1 + p2 + 2, q1 + q2
(dq1 +q2 )
µ ¯
¶
Z ∞
¯ 1 + it, 1 − it, (αp1 )
m1 , n 1 + 2
=
I(x, b)G
x¯¯
dx,
(3)
p1 + 2, q1
(βq1 )
0
and this will be the case if and only if
I(a, b) = a
−1
m ,n
G 2 2
p2 , q2
µ ¯
¶
b ¯¯ (γp2 )
.
a ¯ (δq2 )
(4)
In fact, substituting the right-hand side of (4) into (3), we treat the corresponding integral of the product of G-functions by the Mellin-Parseval
equality [2], [5], [6]. Hence with the definition of the Meijer G-function we
derive
¶
¶
µ ¯
µ ¯
Z ∞
¯ 1 + it, 1 − it, (αp1 ) dx
b ¯¯ (γp2 )
m 1 , n1 + 2
m2 , n2
¯
G
x¯
G
(βq1 )
p1 + 2, q1
p2 , q2
x ¯ (δq2 )
x
0
2
Q 1
Q n1
Γ(it − s)Γ(−it − s) m
j=1 Γ(βj + s)
j=1 Γ(1 − αj − s)
Qp1
Qq1
L
j=m1 +1 Γ(1 − βj − s)
j=n1 +1 Γ(αj + s)
Qm2
Q n2
j=1 Γ(δj + s)
j=1 Γ(1 − γj − s)
Qp2
× Qq2
b−s ds
j=m2 +1 Γ(1 − δj − s)
j=n2 +1 Γ(γj + s)
¶
µ ¯
¯ 1 + it, 1 − it, (cp1 +p2 )
m1 + m2 , n1 + n2 + 2
¯
,
=G
b¯
(dq1 +q2 )
p1 + p2 + 2, q1 + q2
1
=
2πi
Z
where the corresponding parameters are defined above. Thus we have proved
(3) and consequently, the value of the integral (1).
References
1. A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. Vol. II: Special Functions, Gordon and Breach, New York and
London, 1986.
2. A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. Vol. III: More Special Functions, Gordon and Breach, New York
and London, 1989.
3. A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. Vol. IV: Direct Laplace Transforms, Gordon and Breach, New
York and London, 1992.
4. A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. Vol. V: Inverse Laplace Transforms, Gordon and Breach, New
York and London, 1992.
5. S.B. Yakubovich and Yu.F. Luchko, The Hypergeometric Approach to
Integral Transforms and Convolutions, (Kluwers Ser. Math. and Appl.:
Vol. 287), Dordrecht, Boston, London, 1994.
6. S.B. Yakubovich, Index Transforms, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1996.
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