# Integrals from functions generated by increasing factorial powers

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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.