Abstract

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.

Keywords and Phrases

The Gauss' Hypergeometric function, The Maclaurin's series expansion, The Taylor's series expansion, The Leibniz theorem, The Chebyshev polynomials, The Euler's linear transformation.

A.M.S. subject classification

33C05, 34A35, 41A58, 33B10.

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