A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in ${\displaystyle \mathbb {Z} }$ (the integers) and with algebraic initial conditions.

History

The term 'transcendentally transcendental' was introduced by E. H. Moore in 1896; the term 'hypertranscendental' was introduced by D. D. Morduhai-Boltovskoi in 1914.^[1][2]

Definition

One standard definition (there are slight variants) defines solutions of differential equations of the form

${\displaystyle F\left(x,y,y',\cdots ,y^{(n)}\right)=0}$,

where ${\displaystyle F}$ is a polynomial with constant coefficients, as algebraically transcendental or differentially algebraic. Transcendental functions which are not algebraically transcendental are transcendentally transcendental. Hölder's theorem shows that the gamma function is in this category.^[3][4][5]

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

Examples

See also

• Hypertranscendental number

Notes

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.
• Morduhaĭ-Boltovskoĭ, D. (1949), "On hypertranscendental functions and hypertranscendental numbers", Doklady Akademii Nauk SSSR, New Series (in Russian), 64: 21–24, MR 0028347