Integrals from functions generated by increasing factorial powers

Mathematical models of various natural and industrial processes often lead to problems, exact solutions of which it is impossible to obtain by means of well-known classical methods. This is the reason for further development of function theory and numerical analysis. Enlargement “library” of non-elementary functions leads to the enlargement of tasks that can be solved in closed form. That’s why the introducing of new non-elementary functions and studying their properties are actual tasks. Further studying of the new non-elementary functions is prospective and very useful for different branches of science.

The classical transcendental functions $\operatorname{cos} x$, $\operatorname{sin} x$ is given by the corresponding power series with factorials, which can be written as the falling factorial power $n^{\underline{n}}$ (i.e. usual funtorials). Replacing the falling factorial powers by the corresponding rising factorial powers $n^{\overline{n}}$, we get the new non-elementary real functions Sin($x$), Cos($x$).

In general, duality of rising and falling factorial powers is a common feature in the combinatorial analysis. In other words, if a problem leads to some combinatorial identity constructed with the help of falling factorial powers, then there is often a dual combinatorial problem, which leads to a dual combinatorial identity involving rising factorial powers.

In this paper we consider new integral functions $\widetilde{S}(x) = \int_{0}^{x} \operatorname{Sin}(t)dt$, $\widetilde{C}(x) = \int_{0}^{x}\operatorname{Cos}(t)dt$.
We sketch graphs of these functions and find some of their basic properties. In particular, we established the relationship of these functions with the generalized hypergeometric function $_2F_3(a_1, a_2; b_1, b_2, b_3; z)$. We also showed that functions $\widetilde{S}(x)$, $\widetilde{C}(x)$ are solutions of linear ordinary differential equations of four order with variables coefficients.