Internat. J. Math. & Math. Scl.
Vol. 8 No. 2 (1985) 359-365
359
A GENERALIZED MEIJER TRANSFORMATION
G. L. N. RAO
Department of Mathematics
Jamshedpur Co-operative College of the
Ranchi University
Jamshedpur, India
and
L. DEBNATH
Department of Mathematics
Unlvreslty of Central Florida
Orlando, Florida 32816, U.S.A.
(Received on May 13, 1983 and in revised form January 15, 1985)
In a series of papers [I-6], Kratzel studies a generalized version of the
+ I; (st)q).
(q,
classical Meljer transformation with the Kernel function (st)
ABSTRACT.
This transformation is referred to as GM transformation which reduces to the classical
I.
Meijer transform when q
He also discussed a second generalization of the Meijer
transform involving the Kernel function
(n)(x)
(or ML) transformation.
which reduces to the Meijer function
I.
2 and the Laplace transform when n
when n
This is called the Meijer-Laplace
This paper is concerned with a study of both GM and ML trans-
Several properties of these transformations in-
forms in the distributional sense.
cluding inversion, uniqueness, and analytlclty are discussed in some detail.
Distributional GM and
KEY WORDS AND PHRASES.
transforms, Meijer Transform.
1980 AMS MATHEMATICS SUBJECT CLASSIFICATION COD,S.
I.
46F12, 44A20.
INTRODUCTION.
In Zemanlan’s book [7, plT0] the Meijer transformation is defined by means of the
integral
K
[f(t)]
2
[
(st) /2
K
(2 s/) f(t) dt,
o
K
where
(z) is the modified Bessel function of third kind of order
(I.I)
,
and has the
integral representation [7, p148]
K(z)
r( + )
()
(t 2
I) -1/2 e -zt dt,
-I, Re z 0.
An alternative form of (1.2) is
for Re
z2
Ku(z)
()x)
t
o
-)-I
e
4t
dt
(1.2)
_
360
G. L. N. RAO AND L. DEBNATH
Kratzel [I, p149] has introduced a generalization of the Meijer trnasformation in
the form
F(s)
where q
and
t
K (q)
larg
=I (st)
f(t)}
s
(q’ s + I; (st) q
(1.4)
f(t) at,
o
(I +
-
).
In his other paper [3, p143], Kratzel considered an integral representation of
n(O, B; z) in the form
B; z)
n(P
where p > o and
n(l,
larc
<
z
z2
t
.
zt-O
e -t
(1.5)
at
+ I,
I, B
When p
z2
-)= 2(-) /2 K(z)
+ I;
Result (1.4) reduces to (I.I) when q
(1.6)
i.
Also, Kratzel introduced a second generalization of the Meijer transformation
p 369],
([!, p148], [, p328], [,
F(s)
e (n)
!
where Re v >
n)
[, plOD])
(2) -2--
r
(+
n (z)
tn
I)
I-)
I, Re z > 0 and n
I.
by Dimovski
,
kv(n)(z)
-n
(1.7)
is given by
-zt
e
(1.8)
dt,
1,2,3
It is noted that (1.7) reduces to (i.I) when n
when n
in the form
{n(st) I/n} f(t) dt
n
(n)(z)
n
p 383] and
Re {n(st) i} > 0 and the Kernel
n
n-I
with Re
=o
{f(t)}
[!,
2, and to the Laplace transform
Also, (1.7) is a special case of a more general transformation studied
[,
p23;
p141; I_0, p156].
The purpose of this paper is to study both (1.4) and (1.7) in the distributional
sense and establish theorems concerning complex inversion, uniqueness and analyticity.
2.
DIFFERENTIAL OPERATORS.
we use the notation and the terminology similar to those of Kratzel
Zemanian
[7, pplTO-200].
[!
]
and
The following differential operators will be needed for this
study:
S,q
(t)
[t -q+l
Dt{tq- Dtq
(t)}] k, k
O, I, 2
(2.1)
where (t) is a complex smooth function.
M,n[%(n)
where
l(n)(t)
(t)]
tn
Dtn-I
[t l-n D t
% (n)(t)],
n
I, 2,.
is defined in (1.8).
The operators (2.1) and (2.2) will be used to in=tlgae (1.4) and (1.7)
respectively.
(2 2)
361
GENERALIZED MEIJER TRANSFORMATION
3.
FUNCTION SPACE
K, a
AND ITS DUAL.
We define the following seminorms on certain complex smooth functions (t)
(Zemanian [7, p176]):
k
,a
where
S,q
(3.1)
(t)
real number, v is a complex number with Re v
Kv, a
We next define
as the linear space of all functions
y ,a
which the seminorms
Kv, a which
Kv,a- The
le at t v-
Sup
o < t <
a
is
a
()
0, I, 2,
exist for each k
self [7, p171].
S,q
y ,a
< t
for
is a seminorm on
is a continuous linear mapping of
Kv, a
into it-
It is noted that the differential operator is slightly different from
].
that used in the book
LEMMA 3.1:
If
zv
(q, v + I; z q)
o and q _> I, larg z <
(z)
where Re v
(t) on o
Each
We note that D(1) is subspace of
is complete and hence a Frechet space.
differential operator
o.
(3.2)
(I +
),
then (st) E
Kv,a
for every t in (o,)
and for every fixed nonzero s.
PROOF:
We have from (3.1)
k
k
feat t-1/2 Sv,
q
Sup
v,a(St)
< t <
O
(st)
Re v > 0.
Making reference to [I, p153], we use the fact
k
S v,q (st)
(-I) k(q+l) s k (st)
(3.3)
combined with the asymptotic property of (t)
t
0
the seminorms
,
kv,a
(st) are finite for v
y
DEFINITION i:
We prove that, as
O.
0 which follows
om
Also,
the
n [1, p 149].
The distributional generalized Meijer transform of f(t) is defined by
K {s;
q)
F(s)
for every s in
O.
and for every fixed s
are finite for a
it can be shown that
,a
asymptotic property of the function
as t
[I, p 153] as t
f
(q
<f(t) (st) q
,a f(t)
(I +
s # 0, larg
sl
the number assigned to some
I
+ I; (st) q
and q t I}, where
(3 4)
< f,
represents
in a testing function space by a member of the dual space.
In short, we call it as the distributional GM
transform of f.
;
v >
a for every fixed nonzero s, and for
E
to
(st)
definition (3.4) has a sense as the application of f(t) E
a where
a
a is any negative real number and
aa is the dual space of
Since by Lemma 3.1,
(st)
E
Kv,
K,
DEFINITION 2:
Kv,
Kv,
Kv,
A distribution f is called a GM-transformable distribution if f
Kv, a
for some real number a.
NOTE:
4.
O, v # 0; (ii) v
Lemma 3.1 is not true for (i) Re v
O; and (ill) Re v
< 0.
ANALYTIClTY OF F (s)
The analyticity of F(s) can be expressed in the following theorem:
THEOREM 4.1:
F(s)
for s
If
q
f(t), (st) q n (q, v + I; (st)
then F(s) is analytic on
f(t), D s (st) q
D s F(s)
9f,
9f;
(4.1)
and
q
(q, v + I; (st)
(4.2)
G. L. N. RAO AND L. DEBNATH
362
However, we
A fairly standard procedure can be used to prove this theorem.
PROOF:
state some initial steps for the proof.
F (s + As) -F(s)
< f(t)
As
(q, + I; (st) q
(st)q
Ds
<
>
As(t)
f(t)
>
(4.3)
where
s [(st + Ast)
1
?As (t)
n (q,9 + I; (st + Ast) q
(st) q n (q,9 + I; (st)q)]
(4.4)
We use the series expansion of n from [6, p 142] as
(q,; z)
n=o
(-----)
m=o
n
and then asymptotic behavior of
[,
as given in
m-o+
.o-l-m.
E
nq) (-z) n +
(i
q
(-I) m z
(4.5)
p 142]. After some calculation, it
can be shown that
D s (st) q n (q,v + I; (st) q
e
(4.6)
We next follow the arguments given in [7, pp
(4.2) and (4.3) have a sense.
so that
Kv, a
185-186] combined with the use of Cauchy’s integral formula to complete the proof of
the theorem.
FUNCTION SPACE
5.
We now define
=.
#(t) on o < t
n
Gv, a
Gv, a
AND ITS DUAL
as the linear space of all complex-valued smooth functions
as
Or,a, n
nv,a,n ’(n)
nn
M,
where
(t)
Gv, a
by
Gv, a-
If
Mv,nn %n)(t)l,
(5.1)
It is noted that (5.1) exists.
(n)
{v(st)N}
#(st) e Gg, a
(5.2)
^.
o, then
provided
PROOF:
t
is the differential operator defined by (2.2).
#(st)
for Re
le at
<SP<
o
We denote the dual space of
LEMMA 5.1:
-
The topology of this space is generated by a set of seminorms
for t in (o,=)and for every fixed
s
such that s # o
n
It follows from [3, p 371] that
k
M9,n
%(n)(z)
z
n d
n-I
dzn-I
[z
l-nv d
dz
Using the following asymptotic property of
(n)
n-I
(z)
o
n) (z)]
"(n)(z)
A9
F( + ) + 0(I) as z
n
(-I)
n
.(n)(z),
A
z
(k=o)
given in [3, p 371] in the form
o, Re
(5.3)
o,
we obtain
on
-
v,n,a % v {n(st) I/n}
Sup
o<t<
which are finite for each n=l,2
le
at t
are finite provided
>
Sup
O
< t
Je at
as t
t -I n(st) I/n
%n) {n(st)I/n}l
o if
.(n) {n(st) I/n}
n sff
n
We next consider the case for t
equation (7) of [3, p 372] to obtain
=.
For
<arg z<
,
z
n(st) I/n
we use
GENERALIZED MEIJER TRANSFORMATION
(n)
e at t v-1/2+ E
e at
nsE ^v
(n(st)l/n}
n-1
tv-1/2-
This expression is asymptotically equal to
s
e at n
t
{(n-l)
v-+
v
+
xt
-
n-I
s
(2)--2-I}
v(n-1) + E-
n- {(st)l/n}
sE (2n)--2--
n
363
{(n-l)
+
e_n(st)E ,s#o
-
I)E
which is finite if a < o.
REMARK:
Even if we take a more general differential operator (that is, of a greater
order, say k) it must involve terms
as t
which tends
,
exp[-n(st)I/symptotlcally
to zero as t
.
DEFINITION 3:
A distribution f(t) is called an M-L transformable distribution if f(t)
G, a
for some real number
DEFINITION 4:
I.
and Re
a
The M-L transform of a M-L transformable distribution g a
G
is defined
by
G(s)
6.
g(t),
kn)t
{s, Re s
fl
where s
{n(st)}
<
o;
>
arg s
(5.4)
}
which is given in
[,
p 372].
COMPLEX INVERSION THEOREM FOR THE TRANSFORM (1.4).
Kratzel
[,
p 151] proved an inversion theorem for
(,
p 151) in the classical
sense.
q,
In order to discuss a complex inversion theorem, we need the Wright function
a; z) defined in [II] in the form
E
(q, =; z)
n=o
zn
This reduces to Bessel function for q
+
(I,
(6.1)
n! F(qn + )
z2
()-"
I;--g-)
I, that is,
J(z)
(6.2)
One of the properties of the Wright function [I, p 151] can be expressed as
v
v
which is
+
q
"’
needed
t)}
in proving the following theorem:
THEOREM 6.1:
If
(1) G(s) is holomorphic in
a
(6.3)
q s-1
{s; Re
s
c,
where
[arg s[ 2
(1 +
o -v G(o) 0
(ii) g(t)
1,
---;
q
g
1,
o(t)) do,
(6.4)
(6.5)
where the path of integration L is given by
then
(s)
K
q) {g(t)}
(6.6)
364
G. L. N. RAO AND L. DEBNATH
In other words, we prove that, for any (s)
K(q){g(t)},
K
where
PROOF:
E
D (I) in the sense of convergence in D’(I:
(6.7)
G(s), (s)
(s)
q) is
given by (1.4)
In view of condition (ii) of the theorem, the left hand side of (6.7) can be
written as
---;
G(o) (
2i
__q_
(s
2i
(t)) d, (st)
+ I; (st) q 0
(q’
o
t,
n (q,
+I
---;
+ I; (st) q >, 0(s)
(t)) dt G (o)
d,
>
O(s) >
I (say)
In view of (6.3), this expression yields
I
G(o) do
s------
2i
(6.8)
(s) >
[_I, p 152]
which is equal to, using a relation in
G(s), (s) >
This completes the proof.
We shall give here a weaker version of a uniqueness theorem.
THEOREM 6.2:
If
F(s)
(q) f(t) on
K
(q)
G(s)
K
F(s)
G(s) on
g(t) on
f
g
and
then f(t)
PROOF:
Zf (h g,
g(t) in the sense of equality in D’(1).
By Inversion Theorem (6.1), we have
q)q) [g(t)]
F(s)
G(s)
K
K
This implies that f(t)
7.
CLOSING REMARKS:
Oberchkoff in 1958.
[f(t)]-K
q)
If(t)
g(t) in
f
g(t)]
g
0 in
tip
g.
in the sense of equality in
D’(1).
A transform more general than (1.4) and (1.7) was introduced by
A modified version of that transform was studied by Dimorskl
[
IO] who proved both real and complex inversion theorems. We would like to discuss
some of these thoerems in the sense of distribution in a subsequent paper.
ACKNOWLEDGEMENT:
The first author expresses his grateful thanks to Professors E.
Kratzel and I. H. Dimovski for their help.
Thanks are due to Professor H. J. Glaeske
for his kind invitation and hospitality to the first author at Jena.
Authors would
live to express their thanks to Professors Kratzel and Glaeske for useful discussions
on the subject of matter of this paper.
GENERALIZED MEIJER TRANSFORMATION
365
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