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]

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.

Hypertranscendental functions edit

• The zeta functions of algebraic number fields, in particular, the Riemann zeta function
• The gamma function (cf. Hölder's theorem)

Transcendental but not hypertranscendental functions edit

• The exponential function, logarithm, and the trigonometric and hyperbolic functions.
• The generalized hypergeometric functions, including special cases such as Bessel functions (except some special cases which are algebraic).

Non-transcendental (algebraic) functions edit

• All algebraic functions, in particular polynomials.

• Hypertranscendental number

• 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