Hypertranscendental number

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 ${\displaystyle \mathbb {Z} [r]}$ and with algebraic initial conditions.

The term was introduced by D. D. Morduhai-Boltovskoi in "Hypertranscendental numbers and hypertranscendental functions" (1949).^[1]

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 ${\displaystyle e}$ is transcendental but not hypertranscendental, as it can be generated from the solution to the differential equation ${\displaystyle y'=y}$.

Any hypertranscendental number is also a transcendental number.

See also

• Hypertranscendental function