Internat. J. Math. & Math. Sci.
VOL. ii NO.
(1988) 167-176
167
TWO DIMENSIONAL LAPLACE TRANSFORMS OF
GENERALIZED HYPERGEOMETRIC FUNCTIONS
R.S. DAHIYA
I.H. JOWHAR
Department of Mathematics
Iowa State University
Ames, Iowa 50011
Department of Mathematics
Florida State University
Tallahassee, Florida 32306
(Received October 16, 1986)
ABSTRACT.
The object of this paper is to obtain new operational relations between
the original and the image functions that involve generalized hypergeometric G-
functions.
KEY WORDS AND PHRASES:
Multidimensional Laplace transforms, Meijer’s G-function.
1980 AMS SUBJECT CLASSIFICATION CODE:
44A30, 33A35
IrRODUCTIOI.
The integral equation
(p,q)
f f
pq
exp(-px-qy)f(x,y) dy dx,
>
Re(p,q)
(1.1)
0
O0
represents the classical Laplace transform of two variables and the functions
(p,q)
and
other.
(p,q)
f(x,y)
related by (1.1), are said to be operationally related to each
f(x,y)
is called the image and
the original.
Symbolically we can write
(p,q)
and the symbol
f(x,y)
f(x,y)
or
(p,q),
(1.2)
is called "operational".
Meijer’s G-function [5] is defined by a Mellin-Barnes type integral
m
au
z
Gm’n(
u,v
b
n
II
-i
v
L/
--t
a.3
m, n, u, v
and
pole of
path
L
those of
b.3
are integers with
v
are such that no poles of
F(1-ak+s)
k
goes from
-i
F(bj-s);
F(1-ak+s)
k
u+v (2(re+n)
1,2
to
n.
+i
F( 1-aj +s)
II
j=1
u
z
v
j =m+
where
r(bj -s)
)
F( l-bj +s
I; 0 <
(ak-bj)
--
r(aj
< u; 0
n
r(bj-s);
Thus
j =n+
j
1,2
S
(1.3)
ds
-s
4 m
< v,
,m
the parameters
coincides with any
is not a positive integer.
1,2,...,m lie on one side of the contour L and those of
1,2,...,n must lie on the other side. The integrand converges if
and
j
larg
z
< (m
+ n
u
v).
For sake of brevity
au
,a
In the present paper, we propose to establish a couple of formulae for
al,a 2,
u.
The
so that all poles of integrand must be simple and
denotes
168
R.S. DAHIYA and I.H. JOWHAR
calculating Laplace transform pairs of two dimensious that involve Meijer’s Gfunction.
-
,
TIlE MAIN RESULTS.
2.
m + n
(i)
(ii)
(iii)
0
Re(bj+ E) >
(iv)
Re(a k
(v)
ak-b k
(vi)
r
>
m
v,
v
1,2
.....
0
u,
n
(u+v)
0,
j
o
2
+ E
)
<
=I
larg
0
I,
m
1,2,...,n,
k
0,
<
is not a positive integer,
1,2,...,m,
j
represents the non-negative integer,
r
x (xy) o-E-1 m,n+l
v
1-E’
(’
Uu+2,
b
k
1,2,...,n,
0,1,2,3,...,
then
au’
V
p-r(pq)E-o+l Gm+l,n+l
u+l,v+l (a
(2.1)
Pq
and
x -r-I
.
()
P
E-o+ _re,n+
(’
Gu+2’v
r--I
I- E
b
a
u
o+r- E
V
(p)-o+l Gu+3,v+
_m+l,n+2
o-E+r-, l-E,
(-p
a
U
o-E+r
(2.2)
valid under the conditions:
bj) >
ak) <
(a)
Re( + E
o- r
+
(b)
Re( + E
o- r
+
(c)
along with (i), (ii), (v) and (vi).
-l,
j
1,2
k
1,2
.....
m,
n,
From (2.1) and (2.2), we propose to prove the following relations.
r
x (xy) o-1
p
where
Re(o)
>
r
0,
E(bl
(Pq) 1-o
v
-r-I x l-o
x
()
pq
bv
>
au ,o+r -)
a
E(o,bl
u+l
E(E+bl
and
b
r
a
v
a
u
a
Pq),
(2.3)
is a positive integer.
..... E+bv
E+al
E[l+i-r, l+E+i-o-r+b
2+6-o-r, l+[+-o-r+a
E+au’
o+r
l+E+-o-r+b v
l++6--r+a u, 1+6
:p)
(2.4)
valid under the same conditions as (2.3).
The function appearing in (2.3) and (2.4) is MacRobert’s E-function, whose
LAPLACE TRANSFORMS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
[6], pp. 433-434, and [8].
properties are given in
PROOF:
169
The generalized Stieltjes transform of a G-function is given by (see
[6],
p. 237)
f
0
x-I
b
for
pq
1-r
0
[X
P’q
(x+8)
On writing
a
Gm’n
P
8/-0
F--Yo) Gm+l’n+l
p+l,q+l (a8
)dx
q
B and multiplying both the
a
t -I Gm’n
q
P
(pq+t)
u,v (at
(pq) G-o+l p
r(o)
-r
b
u
I-G,
a
o-,
b
P
(2.5)
q
sides of (2.5) by
p
l-r
q,
we have
)dt
V
,n+l
Gm+l
u+1,v+1
(apq
I-,
a
-,
b
u
(2.6)
V
Now interpreting with the help of the known result ([4], result (2.83), p.
137),
it follows
o+r-
yo-I
2F(O)
[
r
2
o
0f t:
2
2
._ (pq)-o+l Gm+l,n+l
F( O)P
u+l, v+
r
jo+r_l[2#t-Y)u n,v
a
[at
b
u
)dt
V
u
(aPq
(2.7)
V
Evaluating the left-hand side integral of (2.7), we get
_m,n+l
xr(xy) o--I u+2,v
l-G, a u, o+r-$
a
[y
p-r(pq)-o-I
b
V
uu+l,n+l
+l,v+l
(aPq
I-
a
r-[, b
u
(2.8)
V
The following result will be used in the proof of (2.2).
If
F(p,q)
f(x,y),
then
--6_
0
J [2p%)F(,q)d
(2.9)
From (2.8) and (2.9), we have
x 8-r-o+
yO--I Gm,n+l
u+2,v
.q-+l
’/2’’I
p
f
0
(’l
I-$, a u,
b
o+r-
V
: +$-o-r J( 2 Cv^;
:X"’Gm+l’n+l
u+l,v+l
l- G, a
(aq%
u
)d.
(2.10)
o-6, b V
On evaluating the right hand side integral and after some simplification, we
obtain the desired result (2.2).
In (2.1) and (2.2), reducing the Meljer’s G-functlon to MacRobert’s E-functlon
to obtain (2.3) and (2.4).
170
3.
R.S. DAHIYA and I.H. JOWHAR
PRTIGJI
On specializing the parameters, the G-functlon can be reduced to MacRobert’s E-
function, generalized hypergeometric function and other higher transcendental
Therefore, the results (2.1) and (2.2) leads to many new operational
functions.
[4,9,2,3] and other literature.
relations not listed in
3(a).
lamed image functions expressed in ter of the G-function.
The following results are obtained by using the known results from [7] on pages
-
226-334.
1
P
-r
G-o+b+ I)
(pq)2
=’"
exP(2) W1
(Pq)
x r(xy) -G-I
r(o) r(b+G)
>
Re(o)
-1/2
P
-r
P
-r
.
(Pq) E
cos(
-
1-G,
[
G2[I,
(3.1)
Re(b+ G)
O,
-b)
2-b ),
x
(xy) o--1
r
1,1
G2
[H1-2 2) Y1 -2 (2)]
-2
cos
__1 (b+F) + 3
4
2/-)
[I
x
r
(xy)
2
G
SI_2G,2b [2/)
1,1
2
G2
. ’"
0.
(3.2)
l+r-
(3.3)
L
--b
,-1
>
l- G,
2,1
2(7- G)xr(xy) -G 62, 2
(Pq) 2
P-r(pq) G
(o-/-b)
(I-G-o-B),
-G -b
[-y
2 -2
1-G
+ r-G
(3.4)
l-G, b
[r(-b)r(E+b)]
2,1
"G22 (1
-I x r (xy) -G
1-[, l+r- G
(3.5)
b, -b
3
P
(pq)
H(1)
2)
(P)
(/pq)
2-5/2
cos(c
)
x
r
(xy)
(3.6)
LAPLACE TRANSFORMS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
m
(pq)2
P
171
Wk,mf2i/pq)WK, m[-2i/pq)
a
(x)
"__"
F[1/2-K+m)F[-K-m)
/
a
a
3,1
a+l
2
p-r (pq)
u
v
2
2
a
+- + K, +-- K, + + +
+’
H
u
2-5/2(cos
cos v)x
u
2
-T
(2)
V
r(xy)- (u+v+w)w+u +v
w+
2
w
3,1
(3.7)
H(1)
[,/---) H(2)
u
V
(I)
H
r
a+l
a
m’
[ei(v-u)/2
i(u-v)/2
m
+r
2
W--U+V
W+U--V
W--U--V
2
2
2
The following four results are obtained from (2.1) by using Carlson’s results [I]:
P
1-r
a.i (1-2a,1-2c 2ig-)
22c-2a
x
r-1
),
1-2a,1-2c -2ig-)]
+
a, a
-1
2 2
G313 (y
r(l-2a) r(l+2c-2a)
)p-
p-r [/I-2a,I-2c;21/)
-i 2
2c-2a
r
2
(3.9)
c,
c+2
(Pq)
:
n
[exp(21g
2
+
[r(l+2c)] -1
r
c+-f,
0
-%,3 (I
r-1
y-1 G 2,1
2,2
+
P
r(a,x)
-r
O, r
(3.11)
denotes incomplete gamma function.
+
[2,2-2c;2t/-)- (2,2._2c;_2t-)
-i,
2
2
2c
[r(l+2c) ]-
In the last six relations,
symbol
exp(-2i/--ic)r(-2c,_21v,-q-)]
(C, C
where
+
(3.,o)
ic)r(-2c,21)
x
2
+-
a, a
2,2
c,
P
r
[I-2a,I-2c;-21/)]
2
r-2) r+z-2)
+-,
3F2
as does
for
x
r
2
O,
+ r
C, C
+--
G2:2 (y
(3.12)
2
3,1
G3, 3 or G3,2,23 provides an
2’I or
I,I
2F0 So does G 2,2
G2, 2
interpretation for the
for
2FI.
R.S. DAHIYA and I.H. JOWHAR
172
3(b).
The G-functlon expressed as a naed original function.
The following results are obtained from (2.1) by using the known results [7] on
pages 228-234.
pO-2K-
.
P
a
a
K-o-
2
4,1
(Pq)
G1,5
-K+
[pq
J
( +K+m
r
2K- o+
x
r(2m+l)
"
(xy) o-Ka
1-o-K- a_2 5,1
(pq)
2K+o-
a+
a
o-K+
(3.13)
+K+I
o+K+
F1/2-K/m F(-K-m)
-m
2___)
MK ,m 2)W_K,m[
,/xy
,/xy
G1,5
’m
a+
a+
+m,-a +1,
a
a+l
"
+m,
"--
a+l
-’---m,
a
"
a+l
+1,--
l-o-2K (xy) o-+Kx
(3.14)
/xy
/xy
w
3
2-o-(Pq)
o-P
.
5,1
GI, 5 pq
W
o+
x
2(cos u- cos v)
e
-
3
5/2
-i
i.(u-v)/2
-I,
(pq)
p
.
w
3
2
2
W--U+V
W+U--V
W--U--V
2
2
2
(xY)-2 [ei(v-u)/2
H(1)
I__.L_)
u
-/xy
o’-
W+U+V
2
H
H(1)v (I) H(2)u Ixy
,/xy
(2)
(3.15)
v
’xy
w+
o
5,1
GI, 5
o+
5
w-
w+u+v
w-u+v
2
2
w+u-v
2
w-u-v
2
3
x
2(cos u + cos
v
(xy)
v
/xy
u
,/xy
(3.16)
+ e
u
Ixy
v
,/xy
3
(pq)2-a-
_.
2,/ x
3
2
GI,53,2 [Pq
(xy)
+a-b, b, c, 2a-c, 2a-b
Ib+c+2a
Kb-c
(3.17)
xy
LAPLACE TRANSFORMS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
3
(pq)2-o
0
GI,55,1
[Pq
o-I
5/2
i
4 sin
a
b
sin
e
x
3
2 -o
a, b, -b, -a
3
(3.8)
[Pq
a, b, -b, -a
o-
"=" 4
"
cos
o
2
a cos b
eib
H
(xy)
o___32
H(1)
a-b
e
[/----
a+b
/xy
H(2)[
(I)I
a-b )]
Cxy
]xy
(3.19)
a+b
3
pO+2a-I
(2)
a-b
2
(pq)7-o GI,55’I
x
H(1)[/)
[e -ib
(xy) -2
a-b
a+b
5
173
(pq)-a-o GI,55’I
+a
2
[Pq
o+a--x
,z
O,
,
b,-b
The following three results are obtained by using Carlson’s results [I] on page 239
in (2.
I).
3
o- (Pq) 2-03,1
P
-
0
GI, 5
r(l+2c)
r(l+2c-2d)
+ e -ic
o-I, c,
(Pq) 2-051
Gll 5
] 2
2d
d,
d+-
3
x
22d+l
F
(xy)-c-2[e ic IFI(I+2c;
I(I+2c;
I+2c-2d;
-2i)
?xy
(3.21)
I+2c-2d;
3
0-
c+
/xy
0
[pq
o-I, c, c+
r(l+2c)r(1+2d)
2
x
,
d, d+
(xy)-c-2[e
I+2c,
I+2c-2d;
)
/xy
+ e -ic
[1+2c,
I+2c-2d;
-2i
Cxy
(3.22)
174
R.S. DAHIYA and I.H. JOWHAR
3
(Pq)
GI, 5
c, c+
o-
7, d,
d+
7
3
i 2
-2d
[e ic ,(I+2c,
3(c)’.
2
/ F(t+2c)F(l+2d)
(xy)
x
21)
I+2c-2d;
e
OCI+2c, I+2c-2d;--)]
(3.23)
/xy
--
Partlcular cases of (2.2).
-2i
-icw
xy
The following results are obtained by using the known results from [7] on pages
228-230 in (2.2).
a
a
P
2-0-+21)K-0--+I G3,
4,2
.
5
or-K+
I-O-K-
2-2<-0-6
P
3
5,2
q
6)2
-m)x +-I
w
w
2
K+-{
+I
a+l
a
a+l
a
-’
,2
G3,5
(3.24)
(2/ W_< ,m [2/
,m
a
I+-+
a
F(-+m)r(
3
a
l-K/-{
- --- M
0++
/-
a
()
x
r(l+2m)
a
-6+I,
I+K+
a+l
a
a-K+
a
a+l
a+l
+m, ----m,
a
7+I
(3.25)
y
w
l+w
-T-
w+u+v
w
0+2
2’
w--u-v
2
2
w-u+v
w+u-v
2
2
3
cos(u+v)]
/7 [2
2-0-
+w
;
0+
5/2
cos vw)
2i(cos u
e
i(u-v)/2
w
-’ ’
5,2
x
W
-1,
J-u(
x
)J-v(/ )]
(3.26)
w+l
T
W+U+V
W--U+V
W+U-V
2
2
2
6+ y(y-2
H(1)
u
+
)Jv
[Ju(
x y
Le
iW(v-u)/2
H(2),
v [/7 )l
H
(I)
v
W--U-V
-’-""--
(x H(2)(/x
u
(3.27)
The following three results are obtained by using the known results from [I]
on page 239 in
(2.2).
LAPLACE TRANSFORMS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
5
-a-
P
!)
2-a
3,2
5
-,
q
0,
G3, (
c+-
o-1, c,
1"(1+2c)
1’(1+2c-2d) 2
-1/2
2d+1
+ e
-6
P
q 2-a
c+S-
iFi[i+2c;
2
-2d
5,2
3
c+
+c-
3
(I+2c;
-6,
G3,5
I+2c-2d;-2i
(3.28)
2
ya-C-2
[eiC
(1+2c,
1+2c-2d; 2i
x)
(3.29)
0
2
c,
0-’’"
(I+2c,
I+2c;
I+2c-2d; -2i
=. 22-.- 1’(1+2c)F(1+2d)xC+
e
-
d, d+
/ 1’(1+2c)1’(1+2d) x
P
reich. F
I+2c-2d;
G3, 5 (p
+ e
d+-
d,
ya-C-2
x
o--1, c,
175
a-cy
c+’"
d,
d+’"
3
[e
ic
,(I+2c,
I+2c-2d;
/-f
2i/7)
(3.30)
I+2c-2d;-2i/)]
REFERENCES
[I]
Carlson, B.C., Some extensions of Lardner’s relations between
functions.
SlAM Journal of Mathematical
oF3
and Bessel
Analysis. Vol. I, No. 2, May 1970.
[2]
Dahlya, R.S., Computation of two-dimenslonal Laplace transforms. Rendlcontl
Matematica, Univ. of Rome. Vol. (3) 8, Series VI (1975), pp. 805-813.
[3]
Dahlya, R.S. and Egwurube, Michael, Theorems and Applications of two
dimensional Laplace transforms. Ukrainian Mathematical Journal (accepted for
publication).
[4]
Ditkin, V.A. and Prudnikov, A.P., Operational calculus in two variables and its
Pergamon Press (English Edition), (1962).
applications.
[5]
Erde’lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Hisher
Transcendental Functions, Vol. I, McGraw Hill, New York, pp. 207, 209, 215
(1953).
[6]
Erde’lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Tables of
Integral Transforms, Vol. 2, McGraw Hill, New York (1954).
[7]
Luke, Y.L., The special functions and their approximations, Vol. I, Academic
Press, New York (1969).
[8]
MacRobert, T.M., Functions of a complex variables, 5th edition, London (1962).
[9]
Voelker, D. and Doetsch, G., Die Zweidimensionale Laplace transformation.
Verlag Birkhuser, Basel (1950).