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 Z ["r"] and with algebraic initial conditions.

The term was introduced by Mordukhai-Boltovski 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" is transcendental but not hypertranscendental, as it can be generated from the solution to the differential equation $y\text{'}\; =\; y$.

Any hypertranscendental number is also a transcendental number.

ee also

* Hypertranscendental function

References

* Mordukhai-Boltovski, "Hypertranscendental numbers and hypertranscendental functions", Dokl. Akad. Nauk. SSSR, 64 (1949)

* Hiroshi Umemura, "On a class of numbers generated by differential equations related with algebraic groups", Nagoya Math. Journal. Volume 133 (1994), 1-55. ( [http://projecteuclid.org/euclid.nmj/1118779837 Downloadable] from [http://projecteuclid.org ProjectEuclid] )