Hypertranscendental function

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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.

Contents 1 Examples 1.1 Known hypertranscendental functions 1.2 Functions which are not hypertranscendental 2 See also 3 References

[edit] Examples

[edit] Known hypertranscendental functions

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

[edit] Functions which are not hypertranscendental

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

[edit] See also

• Hypertranscendental number

[edit] References

• Loxton,J.H., Poorten,A.J. van der, "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.