A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in
(the integers) and with algebraic initial conditions.
The term 'transcendentally transcendental' was introduced by E. H. Moore in 1896; the term 'hypertranscendental' was introduced by D. D. Morduhai-Boltovskoi in 1914.
[1][2] One standard definition (there are slight variants) defines solutions of differential equations of the form where
Hölder's theorem shows that the gamma function is in this category.