Generalizations of fractional q-Leibniz formulae and applications
© Mansour; licensee Springer 2013
Received: 5 September 2012
Accepted: 24 January 2013
Published: 11 February 2013
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
This paper is organized as follows. In Section 2, we mention in brief some known fractional and q-fractional Leibniz formulae. In Section 3, we generalize the fractional q-Leibniz formula of the Riemann-Liouville fractional q-derivative introduced by Agarwal in . Finally, in Section 4, we extend the fractional q-Leibniz formula introduced by Purohit  for the q-Weyl derivatives of nonnegative integral orders to any real order.
2 Fractional and q-fractional Leibniz formulas
formally. An analytic proof of (16) is introduced in  where the following theorem is introduced.
Then (16) holds for and .
Proof See . □
3 A generalization for Agarwal’s fractional q-Leibniz formula
In this section we introduce a q-analogue of the fractional Leibniz formula (15) when . The case of the derived fractional q-Leibniz formula is the fractional q-Leibniz formula (17) introduced by Agarwal .
where , and .
Remark 3.2 It is worthwhile to notice that if we set in (21), we obtain Agarwal’s fractional Leibniz rule (17) with less restrictive conditions on the functions and . Actually, the special case of Theorem 3.1 is an extension of the result given by Manocha and Sharma in .
and the theorem follows. □
Equating (33) and (34) gives (30). □
Combining (39) and (40), we obtain (35). □
4 A q-extension of the Leibniz rule via Weyl-type of a q-derivative operator
In this section, we prove that the q-expansion in (18) can be derived for any . The proof we introduce is completely different from the one introduced by Purohit for nonnegative integer values of α. We start with characterizing a sufficient class of functions for which exists for some α.
Proposition 4.2 If , then exists for any function f defined on . If and , then exists.
In the following, we define a sufficient class of functions for which exists for all α.
It is clear that if , then for all α. The spaces and are q-analogues of the spaces of fairly good functions and good functions, respectively, introduced by Lighthill [, p.15], see also [, Chapter VII].
where is a polynomial of degree n and a is a constant such that for all .
The identity in (41) leads to the following result.
Simple manipulations give (43). □
Proof The proof is easy and is omitted. □
for all and for all . If , then (45) may not hold for all α on ℝ but only for α in a subdomain of ℝ.
Combining this latter identity with (50) yields the theorem. □
Equating (54) and (55) gives (51). □
Combining (60) and (61), we obtain (56). □
This research is supported by NPST Program of King Saud University; project number 10-MAT1293-02.
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