2
Chapter 1. Denition
;
Representations and Expansions of the
H
-function
of the gamma functions ;(1
;
a
i
;
i
s
) do not coincide:
i
(
b
j
+
l
)
6
=
j
(
a
i
;
k
;
1) (
i
=1
;
;n
;
j
=1
;
;m
;
k; l
=0
;
1
;
2
;
)
:
(1.1.6)
L
in (1.1.1) is the innite contour which separates all the p oles
b
jl
in (1.1.4) to the left and
all the poles
a
ik
in (1.1.5) to the rightof
L
;
and has one of the following forms:
(i)
L
=
L
;1
is a left loop situated in a horizontal strip starting at the p oint
;1
+
i'
1
and terminating at the point
;1
+
i'
2
with
;1
<'
1
<'
2
<
+
1
;
(ii)
L
=
L
+
1
is a right lo op situated in a horizontal strip starting at the p oint+
1
+
i'
1
and terminating at the point+
1
+
i'
2
with
;1
<'
1
<'
2
<
+
1
:
(iii)
L
=
L
i
1
is a contour starting at the p oint
;
i
1
and terminating at the point
+
i
1
;
where
2
R
=(
;1
;
+
1
).
The prop erties of the
H
-function
H
m;n
p;q
(
z
) dep end on the numbers
a
;
;;;a
1
and
a
2
which are expressed via
m; n; p; q ; a
i
;
i
(
i
=1
;
2
;
;p
)and
b
j
;
j
(
j
=1
;
2
;
;q
)by the
following relations:
a
=
n
X
i
=1
i
;
p
X
i
=
n
+1
i
+
m
X
j
=1
j
;
q
X
j
=
m
+1
j
; (1.1.7)
=
q
X
j
=1
j
;
p
X
i
=1
i
; (1.1.8)
=
p
Y
i
=1
;
i
i
q
Y
j
=1
j
j
; (1.1.9)
=
q
X
j
=1
b
j
;
p
X
i
=1
a
i
+
p
;
q
2
; (1.1.10)
a
1
=
m
X
j
=1
j
;
p
X
i
=
n
+1
i
; (1.1.11)
a
2
=
n
X
i
=1
i
;
q
X
j
=
m
+1
j
; (1.1.12)
a
1
+
a
2
=
a
;a
1
;
a
2
=; (1.1.13)
=
m
X
j
=1
b
j
;
q
X
j
=
m
+1
b
j
+
n
X
i
=1
a
i
;
p
X
i
=
n
+1
a
i
; (1.1.14)
c
=
m
+
n
;
p
+
q
2
:
(1.1.15)
An empty sum in (1.1.7), (1.1.8), (1.1.10)
{
(1.1.12), (1.1.14) and an empty product in (1.1.9),
if they o ccur, are taken to b e zero and one, resp ectively.
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