A power series is called differentially algebraic if it satisfies an algebraic differential equation. This class includes many of the classical functions of mathematical physics and corresponds to series which can be generated by general-purpose analog computers. A contrary function which satisfies no algebraic differential equation is called transcendentally transcendental. This paper is a rich account of many of the known transcendentally transcendental functions. These include \(\Gamma\) (x), \(\zeta\) (s) and, in a certain sense, ‘most’ entire functions. This topic has intriguing connections with transcendental number theory, centred on Mahler’s work on functions such as \(\sum z^{2^ n}\) and \(\sum [n\omega]z^ n\). The paper concludes with a sally in this direction:
Outrageous Conjecture. Let \(\epsilon_ n=0,1\) for \(n=0,1,2,..\). Then \(\sum \epsilon_ nz^ n\) is a differentially algebraic function if and only if \(\sum \epsilon_ n2^{-n}\) is an algebraic number.