Fractional and operational calculus with generalized fractional ...

Integral Transforms and Special Functions 801
2. Properties of the generalized fractional derivative operator and relationships with the
Mittag–Leffler functions
In this section, we derive several continuity properties of the generalized fractional derivative
operator D α,β
a+ . Each of the following results (Lemma 1 as well as Theorems 1 and 2) are easily
derivable by suitably specializing the corresponding general results proven recently by Srivastava
and Tomovski [24].
Lemma 1 [24]
The following fractional derivative formula holds true:
(
D α,β [
a+ (t − a)
ν−1 ]) (x) = Ɣ (ν)
Ɣ (ν − α) (x − a)ν−α−1 (2.1)
( )
x>a; 0 a; 0 0 .
Theorem 2 [24]
ϕ ∈ L(a, b):
The following relationship holds true for any Lebesgue integrable function
D α,β
a+
(
E
λ
µ,ν,ω;a+ ϕ ) = E λ µ,ν−α,ω;a+ ϕ (2.3)
(
x > a (a = a); 0 0
)
.
In addition to the space L(a, b) given by (1.13), we shall need the weighted L p -space
X p c (a, b) (
c ∈ R; 1 ≦ p ≦ ∞
)
,
which consists of those complex-valued Lebesgue integrable functions f on (a, b) for which
with
‖f ‖ X
p
c
< ∞,
(∫ b ∣
‖f ‖ p Xc = ∣t c f (t) ∣ ) 1/p p dt ( )
1 ≦ p a) in R, by
W α,p
a+ (a, b) = { f : f ∈ L p (a, b) and D α a+ f ∈ Lp (a, b) (0 0
)}
.
See also the notational convention mentioned in connection with (1.13).