Internat. J. Math. & Math. Scl.
VOL. 15 NO. 4 (1992) 653-658
653
INTEGRAL REPRESENTATIONS OF
GENERALIZED LAURICELLA HYPERGEOMETRIC FUNCTIONS
VU KIM TUAN
AND
Institute of Mathematics
P.O. Box 631, B H5
R.G. BUSCHMAN
Department of Mathematics
Box 3036, University Station
Laramie, WY 82071, USA
Hanoi, Vietnam
(Received August I0, 1990 and in revised form March 24, 1992)
ABSTRACT. The generalized hypergeometric function was introduced by Srivastava and Daoust. In the present
paper a new integral representation is derived. Similarly new integral representations of Lauricella and Appell
function are obtained.
KEY WORDS. Lauricella, Appell, integral representations, Mellin transformation.
1991 AMS SUBJECT CLASSIFICATION CODES. 33A35, 44A30
1. INTRODUCTION.
The generalized Lauricella function of several complex variables was introduced by Srivastava and Daoust
[7,8].
The only integral representation which seems to be known for this function is in terms of a Mellin-Barnes inte-
gral [6,8]. From a fundamental result about Mellin transforms in n-dimensions, we obtain a new representation for
the generalized Lauricella function under suitable restrictions on the parameters. In a similar manner new integral
F("), F(") and F(n), and consequently for the Appell funcFrom these derivations it is clear that the method does not provide representations for F(sn)
representations are obtained for the Lauricella functions
tions F1, F2, and
and
2.
F4.
F3.
THE FUNDAMENTAL THEOREM.
Although the following theorem is quite simple, nevertheless it has basic importance.
THEOREM. Let f(z)z ’-1 E L(0, oo) and
()
M. (m={x,, .-., x.})l(s, .--,s.)
s
+---+sf.( +’"+s)
81
He
/(max{x,,
$n
Mn psen the n-dmensional Melhn integral tmsfoation [].
PROOF. In fact
we have
,x.})
z,
(2)
656
V.K. TUAN AND R.G. BUSCHHAN
i=1
Thus the formula
(2)
$152
+
81
f(z’)’’+ +’)"-ldz*
Sn
8n+
$1
Sn
f*($1
""""" " sn)
is proved.
The only related rults which we have found in the literature are those for the twdimensional Laplace transformation in the tabi of Voelker and Doetsch
[;
(30),(32)] and
p. 15,
in the work of
ernov [4; p. 145].
In
an
analogous manner to our threm, thee results eily can be derived and extended to higher dimensions.
3.
THE INTEGRAL REPSENTATION OF THE GENERALIZED LAURICELLA FUNCTION.
.
The generalized hypergeometric function of Srivtava and Dust
’"
b;
c.
d
,,...,,
sucient that
l+9+q-p-pk 0;
The function in
(3)
H,,, (i).,
H=,(.,)-,+.+-.,
where for beolute convergence it
[6] is defined by
: 1,9,...,n.
speci c of the H-functions of sever wriabl which were defined in
(3)
mentl theorem les to the following )ult which involv the G-function of Meijer
(
,)
),.
[3,]. The fund-
[8]. Let
’ (+
/+ ni=
)
P
/’()
Therefore
(2) becomes
-
_
s)
)
Gamma(b, + s, +
> 0.
(4)
P
+ s +... + s,)
I’[=11’(a
H,":,
/=1
j=l
(s; +... + s,)
+ s,)
s, ...s,
Resl,... Res, > 0.
The representation for the generalized Lauricella function can now be obtained as follows. Let Pi
P
q and Re
1, 2,..., n, and either p
a < Re
=1
!
"
b
or p
> q. Then
j-1
p
+m.(n)m,+..+,,.(n+ mx +-.-+m,,)
rl.=1(aj).,,+
(1"1 4" 1re,4- +,. ()’1’i + 1) (ran + 1)
..))0 H=(,),,+ +,.
E
_<
qi,
655
INTEGRAL REPRESENTATIONS OF GENERALIZED LAURICELLA FUNCTIONS
Thus we have proved
Fpp’’
’
.
F (hi).- .F (b)
r(.) r(..)
,(
%;c-;
c-’zt,.., x.)=
a.
G],p+, (max{t,,.., tn}
(5)
b," d_
,0
Cnt""
..
,1
’
i=l
Formula (5) is valid when Rea/ > n;j 1,2,...,p;pi <_ qi, i= 1,2,.--,n’p >_ q, l+q+qi-p-pi >_ 0;
p
(The restriction Re bi + > Re ai is needed when p q.)
E
1,2,...,n.
E
=I
j=l
4. INTEGRAL REPRESENTATIONS OF LAURICELLA
HYPERGEOMETRIC FUNCTIONS.
Derivations similar to those in the previous section lead to representations for 3 of the Lauricella functions.
(Since in (5) p _> q
we do not get
F(Bn).) We introduce the operator
+zi
rt
(a) Using the formula (2)
we get
oo oo
exp(- max{zz,---,z.})
I
"-’+’- mx ...dz.
z,"
i=I
r(a+ml+.-.+mn)
(+ ,)...( +,),
FA(") we have
Consequently, for the Lauricella function
F(An) (a;bt,"
ml,
mz,
Rea> 1.
,bn" cl,
",cn’z,"
,zn)
,am’-0
r-c
t/’Z+m’-x exp (- max {tl,-.", in})
E
i=1
,m, =0
ix
=
r(.)
(c,), m!
dtz"’dtn
(tl "tn) ’z-1 exp (-- max {tl, ",tn))
Consequently
(7)
(b) From (2)
we have
j01 (I
max {zz,
, z.})’-"
I
z, "’+m
r(a+m +...+ m.)r(b- a+ 1)
r(b+m, +...+m.)(". +m,)...(". +m.)
,-I
dz]."dzn
+ Reb > Rea >
(8)
V.K. TUAN AND R.G. BUSC
656
Therefore for the Lauricella function
F get
F (a; bt,--. ,b,," c; ,
")
we
")
.......
,z,,)
(C)m+
-1
ra)r- a + 1)
(1
max {tt,...,/,})c-,
,=
a_l+m, )
fi Z (
(b,),,(x,t,)"’ dt
dt,
1
t=l m,=0
Hence
F(o")(a;b,--.,b.’c;z,...,x.)
r(a)r(-+x) t=l
t’-t(1-max(t,,...,tn})
v,
(l-x,t,)-b’dt,’"dtn,
t=l
+ Rec > Rea >
(c) Using the formula (2) again, we have
(max {,---, n}) rr K,_,,+
(2 (max {t--.t,}) 1/2)
! r(a+’’’+’’’+’")r(b+’’’+’’’+’’")
-2
(+m,)...(an+mn)
Consequently, for the Laurieella hypergeometrie function
(x,) "’’+m’- dxt...dx,
l,>
lb>-O.
F(c’0, by a similar development we have the result
F(c") (a;b;c,...,cn;za,...,z,)
2
fi
r,)r) i=1 v,
o
(10)
i=1
ofit-t-,(max{t, ,..., ,n})z--/t.,_a+ (2(max{t
fl0F1 dtl dtn
(11)
tn},
1/2)
i=1
(ci ,iti
i=1
Rea>l,
Reb>-l.
For Appell functions these read
(. b,.;
,)
r()
r()r(- + )
(12)
"D,D2
l+Rec>Rea> 1"
F2(a;bl,b;cl,c2;z,x2)
(tt2)(’-s)12exp (- max {tx, t2})
12
r(.)
{1El (bl" el; Zltl)} {1El (b2" c2; z2t2)} dtdt2
Rea >
(13)
INTEGRAL REPRESENTATIONS OF GENERALIZED LAURICELLA FUNCTIONS
F4(a;b;c,c2;cl,z2)
657
(14)
Rea> 1, Reb>0.
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