HYPERGEOMETRIC FORMS OF SOME COMPOSITE FUNCTIONS CONTAINING ARCCOSINE($x$) USING MACLAURIN'S EXPANSION

In this article, we have derived the hypergeometric forms of some composite functions containing, arccosine$(x)$ and arccosh$(x)$ like:

$~\exp{(b\cos^{-1}x)},$ $~\frac{\exp{(b\cos^{-1}x)}}{\sqrt{(1-x^2)}},$ $~\frac{\cos^{-1}x}{\sqrt{(1-x^2)}},$ $~ \frac{\sin~(b\cos^{-1}x)}{\sqrt{(1-x^2)}},$ $~

\exp{(a\cosh^{-1}x)},$ $~\frac{\exp{(a\cosh^{-1}x)}}{\sqrt{(x^2-1)}},$ $~\frac{\cosh^{-1}x}{\sqrt{(x^2-1)}}$ and $\frac{\sin~(a\cosh^{-1}x)}{\sqrt{(x^2-1)}}$

by using the Leibniz theorem for successive differentiation, the Maclaurin's series expansion, the Taylor's series expansion and the Euler's linear transformation, as the proof of the hypergeometric forms of the above functions is not available in the literature. Some applications of the functions are also obtained in the form of the Chebyshev polynomials and the Chebyshev functions.