84
J. Harnad
Rational Weighting Case
For G(z) = G c,d (z), denote φ k (x) =: φ
(c,d,β)
k
(x). Then
ζ
L
l=1
(D +
1
βc l
)φ
(c,d,β)
k
+ (D + k − 1)
M
m=1
(D − 1 −
1
βd m
)φ
(c,d,β)
k
= 0,
(26)
where
ζ := −κ c,d x, κ c,d := (−1)
M
L
l=1 βc l
M
m=1 βd m
.
(27)
Mellin-Barnes Integral Representation: Meijer G-Functions [2, 7]
It may be shown that φ
(c,d,β)
k
has the Mellin-Barnes integral representation:
φ
(c,d,β)
k
= C
(c,d,β)
k
G
1,L
L,M+1
1 −
1
βc 1
, · · · , 1 −
1
βc L
1 − k, 1 +
1
βd 1
, · · · , 1 +
1
βd M
− κ c,d x
=
C
(c,d,β)
k
2πi
C k
Γ (1 − k − s)
L
Γ
s +
1
βc
−κ c,d x
s
M
m=1 Γ
s −
1
βd m
ds.
∼
βρ −k (c, d)
(κx) k−1 L F M
1 − k +
1
βc 1
, · · · , 1 − k +
1
βc L
1 − k −
1
βd 1
, · · · , 1 − k −
1
βd M
κ c,d x
(28)
where
C
(c,d,β)
k
:=
M
j =1 Γ (−
1
βd j
)
(−β) k−1 L
Γ (
1
βc
)
.
(29)
The contour C k is chosen so that the poles at 1 − k, 2 − k, · · · are to the right and
the poles at {−i −
1
βc j
} j =1,···L, i∈N + to the left. (See Fig. 1.)
Quantum Case Expressed as Mellin-Barnes Integrals [7]
The following is an integral representation of φ
(H q ,β)
k
(x), valid for all x ∈ C,
φ
(H q ,β)
k
=
1
2πi
C k
A H q ,k (s)x
s ds,
(30)