Hypertranscendental function

A hypertranscendental function is a function which is not the solution of an algebraic differential equation with coefficients in Z (the integers) and with algebraic initial conditions.

The term was introduced by Mordukhai-Boltovski in "Hypertranscendental numbers and hypertranscendental functions" (1949).

Hypertranscendental functions usually arise as the solutions to functional equations, for example the Gamma function.

Examples

Known hypertranscendental functions

* The zeta functions of algebraic number fields, in particular, the Riemann zeta function
* The Gamma function

Functions which are not hypertranscendental

* Any polynomial with algebraic coefficients
* The exponential function and the logarithm
* The sine, cosine and tangent trigonometric functions

ee also

*Hypertranscendental number

References

* Loxton,J.H., Poorten,A.J. van der, " [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN356261603_0016 A class of hypertranscendental functions] ", Aequationes Mathematicae, Periodical volume 16

* Mahler,K., "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585.