In this way a new type of fractional calculus was established.
Also, in [7], a generalized q-Taylor formula in fractional q-calculus is established and used in deriving certain q-generating functions for the basic hyper-geometric functions.
A calculus based on the operator (D_{beta}) and its inverse is established.
A general theory of operators on vector-valued white noise functionals is established in line with Hida′s white noise calculus.
Relevant theory about fractional q-difference calculus has been established [16], such as q-analogues of integral and difference fractional operators properties as Mittag-Leffler function [17], q-Laplace transform, q-Taylor's formula [18, 19], just to mention some.
The relevant theory of fractional q-difference calculus has been established [15], such as q-analogues of the fractional integral and the difference operators properties as Mitlagel Leffler function [16], q-Laplace transform, q-Taylor's formula [17, 18], just to mention some.
Certain fundamental results of calculus have already been established in the AB model: Laplace transforms [7], integration by parts [27], the product rule and chain rule [26], etc.
1). 2. There are some rare earlier examples for the calculation of chances, mainly for games of chance, but no calculus for chances was established; see Bellhouse (2000), Hald (2003, Chap. 4), Schneider (1988).
An incomplete calculus indicates that there are entailment relations that cannot be established within the calculus.
Full composition formulas are established for the calculus, which contains the resolvents of APS-type problems.
Moreover, the Euler Lagrange equations together with the natural boundary conditions associated to the total potential energy functional are established with variational calculus and they coincide with analogous relations already obtained by means of mechanical considerations.
