I nn. J. h.
Math. S.
(1978) 145-159
145
ON THE MEIJER TRANSFORMATION
j.
CONLAN and E. L. KOH
Department of Mathematics
University of Regina
Regina, Canada
(Received January 27, 1978)
-
Recently [8], an operational calculus for the operator
ABSTRACT.
B
t
Dt I+ D with-i <
approach [4],
version.
<
[13], [15].
was developed via the algebraic
This paper gives the integral transform
In particular, a differentiation theorem and a convolution
theorem are proved.
INTRODUCTION.
i
Ditkin
[4], and later with Prudnikov [6], developed an opersimilar to the algebraic
ational calculus for the operator
t
approach of Mikusinski [15].
[13], [14] generalized Ditkin’s
calculus to operators B
[i0],
Meller
t-Dtl+D
with
-I <
< i.
Krtzel [9],
[ii], [12] gave an integral transform version to Meller’s
calculus and also generalized the calculus to operators containing.
n
th
order derivatives.
of the form
He developed a theory of integral transforms
J. CONLAN AND E. L. KOII
146
fOw (
’(n){f}(s)
v
Re(v)
i, 2,
where n
(n(st)l/n)f(t)dt,
i
and
>-i,
n-i
n)
v
Here,
(2) 2
(z)
()nv
r(v+l-i/n)
(1)iSv
i
fi
(y -i)
nexp(-zy)my.
(2)v
the Laplace transform and
is the Meljer trans-
form of the form
.(2)v {f}(s) 2f0 (st) v/2 Kv (2/) f(t)dt,
(i)
where K (z) is the MacDonald function of order v.
v
Dimovski
[i],
[2], [3] developed an operational calculus for the operator
B
t
a0
t
al d
d--
t
an-i
d
t
dt
an
using an integral transform that for n
2 reduces to the Meljer
transform of the form
k {f}(s)
V
2s
-v
f0 (st) V/2K
V
(2’)f(t)dt.
In [8], Koh reconsidered Meller’s operator B
but with e(-l,oo).
(2)
.d t l+
t-..=
dt
dt
Following Mikusinski, Ditkin, et. al., he
cons tru.c ted an operational calculus through the field extension
of a commutative convolution ring without zero divisors.
calculus reduces to Ditkin’s when
e(-l,l).
0 and Meller’s when
His
14 7
MEIJER TRANSFORMATION
In this paper, we give an integral transform analogue of [8]
MeiJer transform of the form
via the
k {f}(p)
for Re()
2p.
r(+)
> -i.
(pt)/2K(2 p/)f(t)dt
/0
In particular, we prove a differentiation
theorem and a convolution theorem.
2p
r(+)
in (3)
(3)
The presence of a factor
as opposed to those in (i) and (2)
is essential
in our convolution theorem.
2.
THE MAIN THEOREMS.
We will define the convolution, *, of two functions, f, g
by
(4)
f(x)g[(1-x)(t-)]dxd,
see Koh [8], where D
order
X,
see Ross
is the Riemann-Liouville derivative of
[17].
This convolutlon exists if,
for example,
f and g are in C [0,oo), the space of infinitely differentlable
complex functions on [0,).
The following properties of K
K(2 p/)
g(t/p) /2
I
1
(t/p)
f0
-/2
X
will be used:
t
--I exp(-px-)
t
0 x-i exp(-px-)
Re() > -i, Re(p) > 0, Re(t) > 0.
dx
(5.1)
dx,
(5.2)
J. CONLAN AND E. L. KOII
148
-in t+0(1),
r()+0[t min(l’)],
2(pt)K(2 p/)
(1,-.la)
t
2 4 -2
e
p {l+o(ll
1
pf) }
> 0
(pt)la+O(1),
]_!
{d (pt)-+V/2Kv(2
0
-
t0
-1 < la < 0
(6.1)
i
)},
t
(6.2)
+=,
KI(2 p/).
-p(pt)
(7)
In order that the Meijer transform (3) converges, it is
sufficient for f(t)
and
f(t)
to be locally Lebesgue integrable on (0, oo)
< Ce 2y/t (t
/
> 0 and for f(t) to remain
) for
bounded in the neighbourhood of the origin for-i < V < 0.
The
integral then converges absolutely within the parabolic region
Re/
> y.
This is clear from the asymptotic behaviors
and (6.2)
Indeed,
l 0f(t)(pt)2Kv(2/)dtl
<
suplf(t
0<t<
+
(6.1)
_<
01f(t)l(pt)2K(2 p/)
dt
P)dt + Tlf(t) l(pt)2Kp(2 p/)dt
IE(pt)2K(2
0
fTlf(t) (pt)2Kp(2 p) dt,
for some 0 <
< T < m.
(8)
The first integral on the right hand side of (8) exists because
of (6.1); the second exists because of the local integrabillty
of f(t) and the continuity of
(pt)K
(2/pt); and the last
MEIJER TRANSFORMATION
integral exists because of (6.2) provided
149
Re/p > y.
We state this
result in
THEOREM i.
y such that
f(t)Loc(0,),
If
< Ce 2Y/ as t
If(t)
then (3) converges absolutely in
if there are constants C and
oo, and if lira f(t)
t/0 +
Re/
f(0 +) <
,
> y for all g(-l,).
Furthermore, the integral (3) as a function of p is analytic in
the region of convergence.
The proof of the analyticity is standard and is omitted.
When a function f(t) satisfies the hypothesis of theorem i, we
shall write, for brevity, fEHypl.
continuous derivative on
THEOREM 2.
k(Bf)
[0, ) and f’Hypl, then fgHypl.
2
If fEC [0,=) and f’EHypl, then
p(kf)
pf(0 + ).
PROOF.
k(B f)
+i
=2p,,
r( +)
Clearly, if a function f has
2p
-P-+I
2
r( + I)
_]!
t
0
-
d tl+ddt f(t) ]t2K(2/)dt
{t 2 K (2 p)t +1 d__f
dt
0
df
d(t K(2 p/)dt}.
f 0 (t +i Ff)
The limit terms vanish because f’Hypl.
another integration by parts to yield
We now use (7) and
150
k(Bf)
_
J. CONLAN AND E. L. KOH
2E2 ;
2p
r (+)
3
+!
df
2
0
KIa+I ( 2 p) d t
3
{-J0m+ [f(t)t
pk (f)(p)-pf(0
+i
+).
2
1
Kla+l(2 p)]+p210ft2Kla(2 pCr)dt }
QED
This result immediately generalizes to the next theorem by
induction.
THEOREM 3.
k (B
If
kf)
U
fec2k[0,=)
P
k
and
2k-l)eHypl, then"
k
Z p J B k-J f(O).
kla[f]
U
J=l
Note that this theorem is the integral transform version
of Lemma i of
B
[8].
The operational calculus for the operator
is now effected through this formula.
To solve the initial
value problem
Q(B )f(t)
g(t)
CO, Btaf(0)
f(0)
C,
...,
B k-1 (o)
c k-1
(9)
where Q(z) is a polynomial, we transform (9) into
Q(p)k f
P(p) + k g
where P(p) is a polynomial of degree less than or equal to that
of Q(p).
Therefore
k f
la
P(P)
Q(p)
4
1
Q(p’---’
(klag)(P)
and f(t) is retrieved by means of an inversion formula and
MEIJER RANSFORMAION
151
possibly a convolution theorem.
The following inversion theorem is obtained from
Theorem [18]
and y
/
through a simple change of variables, vlz. x
/
2/.
THEOREM 4.
not less than
be a complex number whose real part is
Let
-.
1
Assume that in
Re/
>
70 _>
where q >
23--Re+2.
k(f)
f(t)
Then for real c >
70
O, F(p) is an
IF(p>
analytic function and is bounded according to
F(p)
MeiJer’s
and for
<
Mlpl -q
Re > c,
where
r(+l)t
2 i
2
I
F(p)p
--12 I
(2 p/)dp.
(i0)
Re/p=c
The following lemma will be used in proving a convolution
theorem for k
LEMMA.
Letting D
denote the Weyl derivative of order
we have
Do {(z/t)2K (2/{)} (-z/t)K2(2/)"
(d/dt)kwk-{f(t) },
PROOF. By definition, D {f(t) }
where k-i <
< k, and where
WU{f(t)}
[-I/F(9)]
I t f(s)(t-s)V-lds.
Since
K
we have
2/)
i
/2
exp{-(zy+t/y)
}y--idy
(ii)
J. CONIMkN AND E. L. KOH
152
W k-lJ { t -ti
12Kl
2) }
(-z-/2/2r(k_l)}
j"
(t_s)
{-z-/2/2r(k-l)}
f0
exp (-.y)
k-]-I
ds
foeXp{- (zy+s/y)}y
dy
y-l-I d Y f exp(s/y)
t
(t_s)k-p-lds
yA, ds
On putting s-t
ydA and using the deflnlton of the gamma
function, this becomes
(-1)k-u{z-UI212} foeXp{-(zy+t/y)}y k_2_ld y.
Differentiating k times with respect to t, we get
=
D {t
]/
2K
( 2 ’) )
(-l)-g{z-P/212}
and using (11), with
THEOREM 5.
C [0o) and
kg(f,g)
0exp{-(zy+tly)}y-
replaced by 2, completes the proof.
(Convolution theorem)
f(n)
and
g(n)
PROOF.
(kf)(kg)
ReCap
>
7f+78
and
for pc(-l,=).
We use the two-dlmenslonal Laplace convolution
theorem (pp. 26-29, [5])
Fi(p,q)
If f and g belong to
satisfy HypI for every n, then
converges absolutely in
k(fwg)
dy,
L(f i)
Let
fofo -px-qyf i(x,y)dxdy,
e
i
1,2
MEIJER TRANSFORMATION
15 3
2 [x-,y-]dd.
(12)
and
x
Soloyf
If F
i
converges absolutely, then so does F*(p,q)
F*(p,q)
Fl(p,q)F2(p,q).
In (12), let f l(x,y)
Then f*(x,y)
Y
xPf(xy)
and
(13)
xPg(xy).
f2(x,y)
1
t:(xy) where
2]/
[(t-w)(1-v)] dvdw,
fotflwP(t-w)Pf(wv)g
o
i[f*]
-
i(f*) and
and
_fofo e -px-qy (xy)y -2p dxdy
-I
f O (t)f 0 e -qy-pty y- 2p-I dydt.
(14)
Similarly,
/" [xf (xY)
fO f(t)dt
f 0 exp(-py-qty -i )y U-I dy
2f0 f(t)P-(pqt)2Kv(2r/pqt)dt,
on using the integral representation (5.2).
(15)
Set p
1 in
(14) and (15) and invoke (13); we have
f0(e)def0
exp
(-qy-ty-l) y
"2 (It+l)
21.t-i dy
q
2
(kpf) (k
g).
It remains to show that the left hand side of (16) is equal
r 2 (+) k (f,g). Indeed, letting
to
denote the Riemann2
U
q
R:k
Llouvllle integral of order
(16)
J. CO AND E. L. KOtl
154
i
k(f,g)
k[F(+l)
q
r(+)
k
B t-DB(t)
(17)
[t-D
(18)
2q2
Kta (2 q/-) t -V (d) kRk_ v (t) dt
(qt)
r2 (]+1) f 0
(-1)k2q 2 f
F(k_) F2 (+i) 0
k
{(q/t) ]/2 K1a(2 )}dt
(19)
t
f0E(s)
(t-s)k-V-lds
2
r2 (]+1)
0
i;
(20)
(t)D]a{ (q/t) ]/2 K(2 q/)}dt
(21)
2q 2
;(t) (q/t)K2(2 q-)dt
F2 (+i) f 0
q
r
2
(+)
f o (t)dt f o exp(-qy-t/y)y
(22)
(23)
dy
which proves our assertion concerning the left hand side of
(16).
Equation (17)
Equation (18)
for
f(n)
follows from theorem 2 since
g(n)eHypl.
and
[16], we have f,g
as t
/
0
+.
follows from the definition of convolution.
Thus
Furthermore, from a theorem of Ritt
0(i)
lim
t/0+
t
Dt-DE(t)eHypl
-D
(t)
O(t
(t)
O.
2+I)
t-D(t)
Equation (20)
follows from
(19) by the definition of the Riemann-Liouville integral,
and integrating by parts k-times.
at t
0 and t
by (6.1),
O(t)
Rk_
The integrated terms vanish
(6.2), and the fact that in the
definition of (t), the functions f and g satisfy the hypotheses
of theorem 5.
Equation (22)
is due to the preceding lemma.
That
MEIJER TRANSFORMATION
k (f,g) converges absolutely follows from the absolute convergence
of (13).
3.
This completes the proof.
SOME OPERATIONAL FORMULAS.
10f(x)K
Let F(y)
2
If we set y
f (/{)
k
U
(xy)l/2dx
and x
=/,
’
(24)
we get
22/3 P 3/4+la / 2
t-i/4-/2
r(+)
F ( 2 ,/’’)
(25)
From Erdlyi [7] p. 137 (16),
28+U a 8 y U+I/2 F(8++I) (y2+a2)-8-U-I
[xS+U-I/2j 8(ax)]
Rell]-l,
Re >
(26)
lmpl
Rey >
By (25)
aS/2 p U +i
k[tS/2Js(2/)]
Re/
>
[t(v
(27)
[’ (1+1)
ImKI.
v-u, this becomes, for Rev > Reu,
8
Letting
k
1" (8+1+!,)
(p+a) 8+1+1
)/2 J
Since
v- (2/-{)1
K(.)
K
[xS-+I/2js(ax)]^
a(V-U)/2 p
(28)
(p+a) v+l
(.), we have from (26), for
28-aSy +I/2F(8_U+I)
2
(y +a 2’ -+8+I
Re/
>
llmal
(29)
From (25)
k
[tS/2-J 8 (2/)1
aS/2..(8-u..+l)P
F (u+l) (p+a) 8-+i
(30)
156
J. CO AND E. L. KOH
Setting
k
U
t
(v-p) / 2
If v
Jv+1.4
,I’ (y+l) a (v+u) / 2
2/-{)
r
0 in
k [r(p+l)
(at)-/2J
(2V)]
Letting a /-a, and using
k
U
[r(+l)(at)-/2I p (2/)]
Equation (31)
k
P
(32)
p+a
-
l(z)
J
e
(33)
p-a
can be written as
[r(p+l)
tv (at) -(v+u)/2 Jv+u(2)]
F(v+l)
p
(p+a)
v+l
(34)
/
e -wi’2 J (iz), this gives
Again letting a /-a, and I (z)
k
(31)
v+l
1.1+i ) p+a )
v
[F(+l)t
(at) -(v+)/2 Iv+(2/)]
r(v+l)
p
v+l"
(p_a)
(35)
These expressions are useful in inverting rational functions.
As an application, consider the problem of solving
(B2+3B
+2)(t)
(o)
o’
(B) (0)
f(t),
l"
One gets
(p
2
+3p+2)k()
2
0 p +(3O+l)P+k U f
whence
k ()
Therefore
(0+1) P
p+2
+
(20+1) P
p+l
+
1
(
p
p
+ 2(p+2)
)k f.
p+l
15 7
MEIJER TRANSFORMATION
-(o+)r(+)
+(20+i) F (+i) t
1
(]+1) (2t)
+{I
ACKNOWLEDGMENT.
-/2J (2v) +f (t)
-]/2
J](2 2/)-r(]+l)t
J
(2/{)}*f(t).
This research was partly supported by the
National Research Council of Canada under grant number A7184.
The work of the second author was conducted while he was on
Sabbatical Leave at the Technischen Hochschule Darmstadt,
Germany.
He wishes to thank Professor E. Meister for the dis-
cussions on this paper.
REFERENCES
i.
Dimovski, I.H.
An Explicit Expression for the Convolution
of the Meijer Transformation, Comptes Rendus de
l’Academie Bulgare des Sciences, Tome 26, N
I0, 1973,
1293-1296.
2.
Dimovski, I.H.
On a Bessel-type Integral Transformation due
to N. Obrechkoff, Comptes Rendus de l’Academie Bulga.r.e
des Sciences, Tome 27, N
3.
Dimovski, I.H.
i, 1974, 23-26.
Foundations of Operational Calculi for the
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4.
Bularicae
51-63.
Ditkin, V.A. Operational Calculus Theory, Dokl. Akad. Nauk.
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5.
Ditkin, V.A., and A.P. Prudnikov.
Operational Calculus in
Two Variables, Pergamon Press, New York, 1962.
6.
Ditkin, V.A., ’nd A.P. Prudnikov.
Integral Transforms and
J. CONLAN AND E. L. KOH
158
Operational Calculus,
7.
Erdlyi, A., -t. al.
Persamon Press,
New York, 1965.
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Koh, E.L., A
Proc. of the Conf. on Differential Equations at Dundee,
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KrHtzel, E.
Eine Verallgemeinerung der Laplace und
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Transformation, Wiss. Z. Univ. Jena, Math. Naturw. Reihe,
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i0.
KrHtzel, E.
Die Faltung fur
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11.
KrHtzel, E. Bemerkungen zur MeiJer Transf. und Andwendungen,
Math. Nachr. 30 (1965) 327-334.
KrHtzel, E.
Differentiationssatze
=er Transformation
und
Differentialgleichungen nach dem Operator
1
1
-v
n
t
i---d_
d-d
t
)
n-I v-l- 2-n
t
Math. Nachr. 35 (1967)
105-
114.
Meller, N.A.
On an Operational Calculus for the Operator
u+l d
d--’
Vichislitelnaya Matematika 6 (1960)
161-168.
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Meller, N.A.
Some Applications of Operational Calculus to
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Mikusinski, J.G.
Operational Calculus, Pergamon Press,
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Rift, J.F.
On the Differentiability of Asymptotic Series,
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MEIJER TRANSFORMATION
Bull. Amer. Math. Soc. 24 (1918) 225-227.
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Ross, B. (ed).
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Zemanian, A.H.
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KEY WORDS AND PHRASES. Intzgral transforms, Mijr transformation, opnal
alclu and ractional iegral operator.
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
44A15, 44A35.