HYPERGEOMETRIC FORMS OF SOME FUNCTIONS INVOLVING ARCSINE (x) USING DIFFERENTIAL EQUATION APPROACH

In this paper, by changing the independent and dependent variables in the suitable ordinary differential equations of second and third order and comparing the resulting ordinary differential equations with standard ordinary hypergeometric differential equations of Gauss and Clausen, we obtain the hypergeometric forms of following functions:

\begin{equation*}

\frac{\sin^{-1}(x)}{\sqrt{(1-x^2)}},~~~[{\sin^{-1}(x)}]^{2}~~~and~~~{\sin^{-1}(x)}.

\end{equation*}