General Mathematics ,Vol. 16, Nr. 1 (2008), 19–23
Fractional Derivate of Riemann-Liouville via Laguerre
Polynomials1
M.A. Acevedo M, J. López-Bonilla, M. Sánchez-Meraz
Abstract
We show that well known properties of Laguerre polynomials permit to motivate the definition of Riemann-Liouville for the fractional
derivative.
2000 Mathematics Subject Classification: 33C45, 26A33
Key words: Laguerre polynomials, fractional derivative.
1
Introduction
The associated Laguerre polynomials are given by [1]:
(1)
(2)
therefore
(3)
dk
Ln+k (x), k = 0, 1, 2, . . .
dxk
Ã
!
n
X
n
+
k
xr
=
(−1)r
,
r!
n
−
r
r=0
Lkn (x) = (−1)k
Ln (y) = L0n (y) =
n
X
r=0
1
(−1)r
Ã
n
r
!
Received 1 August 2007
Accepted for publication (in revised form) 5 December 2007
19
yr
,
r!
20
M.A. Acevedo M, J. López-Bonnila, M. Sánchez-Meraz
and thus
1
L0 (y) = 1, L1 (y) = 1 − y, L2 (y) = (2 − 4y + y 2 ), . . .
2
In this work we accept that (1) is valid for k = −1, −2, . . . , with two
implications:
1. It permits to obtain an expression for Ln (x) in terms of the Lkm (x);
2. It motivates the fractional derivative of Riemann-Liouville [2-4], wich are
studied in Section 2 and 3, respectively.
(4)
2
Ln(x) in terms of their associated polynomials
dN
to
We put k = −N = −1, −2, . . . in (1) and we apply the operator
dxN
deduce that:
dN
Ln−N (x) = (−1)N n L−N
(5)
(x), n ≥ N,
dx n
but using (2) it is easy to show the property [5]:
p−q
Lp−q
q (y) = (−1)
(6)
then
(7)
N
L−N
n (x) = (−1)
p! q−p q−p
y Lp (y),
q!
(n − N )! N N
x Ln−N (x),
n!
and thus (5) implies:
Ln−N (x) =
N
(n − N )! X
=
n!
m=0
(8)
(n − N )! dN N N
(x Ln−N ),
n!
dxN
Ã
!
dN −m N dm N
x · m Ln−N ,
dxN −m
dx
Ã
!
N
N
N !(n − N )! X
xm N +m
=
Ln−N −m
(−1)m
n!
m!
m
m=0
N
m
Fractional Derivate of Riemann-Liouville via Laguerre Polynomials
21
where we have employed the known relation ([1]):
a!
dc a
db z a
a−b
=
z
,
L = (−1)c La+c
b−c .
dz b
(a − b)!
dxc b
(9)
N +m
In (8) the upper limit of Σ may be (n − M ) because Ln−N
−m = 0 if m >
(n − N ), then finnaly (8) adopts the form:
à !
Ã
!
r
X
n
n
−
r
xm n−r+m
(10)
(x)
Lr (x) =
L
(−1)m
m! r−m
r
m
m=0
which is not common in the literature. The expansion (10) permits to write
Lr (x) in terms of their associated polynomials.
3
Fractional derivative of Riemann-Liouville
From (1) and (7) it is clear that:
(n − N )! N N
d−N
x Ln−N (x).
Ln−N (x) =
−N
dx
n!
(11)
On the other hand, the definition (2) and the recurence relation ([1]):
Lp−1
= Lpq − Lpq−1 =
q
(12)
d p−1
(L
− Lp−1
q+1 )
dx q
imply the known integral property ([1]):
Z x
Z x
(13)
Lm (t)Ln (x − t)dt =
Lm+n (t)dt = Lm+n (x) − Lm+n+1 (x)
0
0
If in (13) we put n = 0, 1, 2, . . . , and use (4) and the recurrence expression
([1]):
(14)
xLp+1
= (p + q + 1)Lqp − (q + 1)Lpq+1 ,
q
then it is immediate that:
Z
0
x
Ln−1 (t)dt =
x 1
L
n n−1
22
M.A. Acevedo M, J. López-Bonnila, M. Sánchez-Meraz
Z
x
(x − t)Ln−2 (t)dt =
0
(15)
1
(N − 1)!
Z
x2
L2
n(n − 1) n−2
.. ..
. .
x
(x − t)N −1 Ln−N (t)dt =
0
(n − N )! N N
x Ln−N (x)
n!
that in union of (11) implies the interesting relation:
(16)
d−N
1
Ln−N (x) =
−N
dx
(N − 1)!
Z
x
(x − t)N −1 Ln−N (t)dt,
0
which is a strong motivation for the fractional derivative of Riemann-Liouville
([2-4]):
Z x
1
f (t)
dq
(17)
f (x) =
dt, q < 0
q
dx
Γ(−q) 0 (x − t)1+q
for the case q = −N = −1, −2, . . .
The generalization of (11) and (16) is given by [1]:
1
d−β α α
(x)]
=
[x
L
m
dx−β
Γ(β)
Z
x
(x − t)β−1 tα Lαm (t)dt ,
0
(18)
=
Γ(α + m + 1)
xα+β Lα+β(x)
,
m
Γ(α + β + m + 1)
for α > −1 and β > 0.
References
[1] M. Abramowitz and I.A. Stegun, Handbook of mathematical function,
John Wiley & Sons (1972) Chap. 22
[2] K.B. Olham and J. Spanier, The fractional calclus, Academic Press
(1974) Chap. 1
Fractional Derivate of Riemann-Liouville via Laguerre Polynomials
23
[3] R. Hilfer, Applicatiion of fractional calculus in Physics, World Scientific
(1999)
[4] I. Podlubny, Fractional differential equations, Academic Press(1999)
[5] J.D.Talman, Special functions: A Group theoretic approach, W.A. Benjamin, Inc. (1968) Chap. 13
Sección de Estudios de Posgrado e Investigación
Escuela Superior de ingenieria Mecánica y Eléctrica
Instituto Politécnico Nacional
Edif. Z, Acc. 3-3er piso, Col. Lindavista, C.P. 07738 México, D.F.
E-mail: lopezbjl@hotmail.com, mmeraz@ipn.mx