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Meijer-G/Fox-H Representation-Based Calculus  |  |
Our discussion in this note will center on the evaluation of integrals of the form
∞
∫
0
ℋ
1
x
t
ℋ
2
t
t
where the are given functions of a single, real variable, and x > 0. Defining the Mellin transform, , of ℋ via
ℋ
i
*
ℋ
*
ℋ
∞
∫
0
s-1
τ
it can be shown that that the value ℋ(x) of eq.1 is related to the input functions by
ℋ
i
*
ℋ
1
*
ℋ
2
*
ℋ
Assuming that the required Mellin transforms can be computed (an assumption which is justified for a large class of functions that we will later define), our task of evaluating the original integral reduces to finding the ℋ(x) corresponding to (s), a task which can be accomplished, for a large class of functions, using a result known as Slater's theorem to be described now.
*
ℋ
i
*
ℋ
The conditions of Slater's theorem are as follows. Let
*
ℋ
(a)+s | (b)-s |
(c)+s | (d)-s |
A
∏
k=1
a
k
B
∏
k=1
b
k
C
∏
k=1
c
k
D
∏
k=1
d
k
where Γ is the gamma function, and the a, b, c, d, A, B, C, and D are parameters.
m,n
G
p,q
a 1 a n a n+1 a p |
b 1 b m b m+1 b q |
1
2π
∫
ℒ
Γ(s+)Γ(1--s)
m
∏
k=1
b
k
n
∏
k=1
a
k
p
∏
k=n+1
a
k
q
∏
k=m+1
b
k
-s
z
and
m,n
H
p,q
( a 1 α 1 a n α n ( a n+1 α n+1 a p α p |
( b 1 β 1 b m β m ( b m+1 β m+1 b q β q |
1
2π
∫
ℒ
m
∏
j=1
b
j
β
j
n
∏
i=1
a
i
α
i
p
∏
i=n+1
a
i
α
i
q
∏
j=m+1
b
j
β
j
-s
z
Note that reduces to when the are all equal to one.
m,n
H
p,q
m,n
G
p,q
α
i
Much of the utility of these functions stems from the fact that a huge group of algebraic and special functions can be written as special cases of them when appropriate choices of the parameters are taken.
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