Taylor theory in quantum calculus: a general approach

Let the function beta be strictly increasing and continuous on an interval I subset of R. The beta-difference operator is defined by D-beta f (t) = (f(beta(t)) - f(t)/(beta(t) - t), where t not equal beta(t), and D-beta f (t) = f ' t(t) when t = s(0) is a fixed point of the function beta. This quantum operator is a generalization of q-Jackson, Hahn, power and other quantum operators. As a convenience of the beta-function: beta(t) turns into the probability distribution function with the probability measure 1, and the sample space R, in the case of its conditions are relaxed to be increasing and continuous from the right, that is, lim(t ->infinity) beta(t) = 1 and lim(t ->infinity) beta(t) = 0, and also by using the Lebesgue-Stieltjes measure of the interval [a, b] to be beta(b) - beta(a). In this paper, we investigate a beta-Taylor's formula associated with the operator D-beta when the function beta has a unique fixed point s(0) is an element of I, which may allow for more flexible and accurate approximations of functions. An estimation of its remainder is given. Additionally, the beta-power series is defined. Furthermore, as application, the beta-expansion form of some fundamental functions is introduced. Finally, we find the unique solution of the beta-shifting problem.