Generalizations of fractional q -Leibniz formulae and applications
- PDF / 224,141 Bytes
- 16 Pages / 595.28 x 793.7 pts Page_size
- 36 Downloads / 167 Views
RESEARCH
Open Access
Generalizations of fractional q-Leibniz formulae and applications Zeinab SI Mansour* *
Correspondence: [email protected] Department of Mathematics, Faculty of Science, King Saud University, Riyadh, Saudi Arabia
Abstract In this paper we generalize the fractional q-Leibniz formula introduced by Agarwal in (Ganita 27(1-2):25-32, 1976) for the Riemann-Liouville fractional q-derivative. This extension is a q-version of a fractional Leibniz formula introduced by Osler in (SIAM J. Appl. Math. 18(3):658-674, 1970). We also introduce a generalization of the fractional q-Leibniz formula introduced by Purohit for the Weyl fractional q-difference operator in (Kyungpook Math. J. 50(4):473-482, 2010). Applications are included.
1 q-notions and notations Let q be a positive number, < q < . In the following, we follow the notations and notions of q-hypergeometric functions, the q-gamma function q (x), Jackson q-exponential functions Eq (x), and the q-shifted factorial as in [, ]. By a q-geometric set A, we mean a set that satisfies if x ∈ A, then qx ∈ A. Let f be a function defined on a q-geometric set A. The q-difference operator is defined by Dq f (x) :=
f (x) – f (qx) , x – qx
x = .
()
The nth q-derivative, Dnq f , can be represented by its values at the points {qj x, j = , , . . . , n} through the identity
Dnq f (x) = (–)n ( – q)–n x–n q–n(n–)/
n (–) qr(r–)/ f xqn–r r r=
n
r
()
q
for every x in A\{}. After some straightforward manipulations, formula () can be written as Dnq f (x) = ( – q)–n x–n
n r=
qr
(q–n ; q)r r f xq (q; q)r
for x ∈ A \ {}.
()
Moreover, formula () can be inverted through the relation
f xq
n
k n = (–) ( – q)k xk q() Dkq f (x). k k= n
k
()
q
© 2013 Mansour; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Mansour Advances in Difference Equations 2013, 2013:29 http://www.advancesindifferenceequations.com/content/2013/1/29
Page 2 of 16
Formulas () and () are well known and follow easily by induction. Jackson [] introduced an integral denoted by
b
f (x) dq x a
as a right inverse of the q-derivative. It is defined by
b
b
f (t) dq t :=
a
f (t) dq t –
a
f (t) dq t,
a, b ∈ C,
()
where
x
f (t) dq t := ( – q)
∞
xqn f xqn ,
x ∈ C,
()
n=
provided that the series at the right-hand side of () converges at x = a and b. In [], Hahn defined the q-integration for a function f over [, ∞) and [x, ∞), x > , by
∞
∞
f (t) dq t = ( – q)
qn f qn ,
n=–∞ ∞
f (t) dq t = ( – q)
∞
x
xq–n ( – q)f xq
–n
() ,
n=
respectively, provided that the series converges absolutely. Al-Salam [] defined a fractional q-integral operator Kq–α by
Kq–α φ(x) :=
q– α(α–) q (α)
∞
x
t α– (x/t; q)α– φ t
Data Loading...
