A complex number is said to be hypertranscendental if it is not the value at an algebraic point of a function which is the solution of an algebraic differential equation with coefficients in Z [ r ] {\displaystyle \mathbb {Z} [r]} and with algebraic initial conditions. The term was introduced by D. D. Morduhai-Boltovskoi in "Hypertranscendental numbers and hypertranscendental functions" (1949). The term is related to transcendental numbers, which are numbers which are not a solution of a non-zero polynomial equation with rational coefficients. The number e {\displaystyle e} is transcendental but not hypertranscendental, as it can be generated from the solution to the differential equation y ′ = y {\displaystyle y'=y} . Any hypertranscendental number is also a transcendental number.
See also Hypertranscendental function
References
Hiroshi Umemura, "On a class of numbers generated by differential equations related with algebraic groups", Nagoya Math. Journal. Volume 133 (1994), 1-55. (Downloadable from ProjectEuclid)