ABSTRACT
Chapter 1
DEFINITION, REPRESENTATIONS AND
EXPANSIONS OF THEH-FUNCTION
1.1. Denition of the H-Function
For integersm;n; p; q such that 0 5 m 5 q; 0 5 n 5 p; for a
i
; b
j
2 C with C ; the set of complex
numbers, and for
i
;
j
2 R
+
= (0;1) (i = 1; 2; ; p; j = 1; 2; ; q); theH-functionH
m;n
p;q
(z)
is dened via a Mellin{Barnes type integral in the form
H
m;n
p;q
(z) H
m;n
p;q
"
z
(a
i
;
i
)
1;p
(b
j
;
j
)
1;q
#
H
m;n
p;q
"
z
(a
p
;
p
)
(b
q
;
q
)
#
H
m;n
p;q
"
z
(a
;
); ; (a
p
;
p
)
(b
;
); ; (b
q
;
q
)
#
=
2i
Z
L
H
m;n
p;q
(s)z
s
ds (1.1.1)
with
H
m;n
p;q
(s) H
m;n
p;q
"
(a
i
;
i
)
1;p
(b
j
;
j
)
1;q
s
#
H
m;n
p;q
"
(a
p
;
p
)
(b
q
;
q
)
s
#
=
m
Y
j=1
(b
j
+
j
s)
n
Y
i=1
(1 a
i
i
s)
p
Y
i=n+1
(a
i
+
i
s)
q
Y
j=m+1
(1 b
j
j
s)
: (1.1.2)
Here
z
s
= exp[sflog jzj+ i arg zg]; z 6= 0; i =
p
1; (1.1.3)
where log jzj represents the natural logarithm of jzj and arg z is not necessarily the principal
value. An empty product in (1.1.2), if it occurs, is taken to be one, and the poles
b
jl
=
b
j
l
j
(j = 1; ; m; l = 0; 1; 2; ) (1.1.4)
of the gamma functions (b
j
+
j
s) and the poles
a
ik
=
1 a
i
+ k
(i = 1; ; n; k = 0; 1; 2; ) (1.1.5)
i
i
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 poles b
jl
in (1.1.4) to the left and
all the poles a
ik
in (1.1.5) to the right of L; and has one of the following forms:
(i) L = L
is a left loop situated in a horizontal strip starting at the point 1 + i'
and terminating at the point 1+ i'
with 1 < '
< '
< +1;
(ii) L = L
+1
is a right loop situated in a horizontal strip starting at the point +1+ i'
and terminating at the point +1+ i'
with 1 < '
< '
< +1:
(iii) L = L
i 1
is a contour starting at the point i1 and terminating at the point
+ i1; where 2 R= (1;+1).
