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 [math]\displaystyle{ \mathbb{Z}[r] }[/math] 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 [math]\displaystyle{ e }[/math] is transcendental but not hypertranscendental, as it can be generated from the solution to the differential equation [math]\displaystyle{ y' = y }[/math].

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)