New Gelfond-Type Transcendental Numbers

Abstract

It is well known that value at a non-zero algebraic number of each of the functions $e^x,\ln x,\sin x,\cos x,\tan x,\csc x,\sec x,\cot x,\sinh x,$ $\cosh x,$ $\tanh x,$ and $\coth x$ is transcendental number (see Theorem 9.11 of \cite{N}). In the work, we show that for any one of the above mentioned functions, $f(x)$, and for a polynomial $g(x)$ with rational coefficients the zero, if any, of the equation $f(x)=g(x)$ is a transcendental number. We also show that if $f(x)$ and $g(x)$ are polynomials with rational coefficients, then a zero of the equation $e^{f(x)}=g(x)$ is a transcendental number. Finally we show that the existence of an abelian group whose non-zero elements are transcendental numbers.

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