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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 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

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

where 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
Hypertranscendental functions

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

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

All algebraic functions, in particular polynomials.

Hypertranscendental number

Notes

D. D. Mordykhai-Boltovskoi, "On hypertranscendence of the function ξ(x, s)", Izv. Politekh. Inst. Warsaw 2:1-16 (1914), cited in Anatoly A. Karatsuba, S. M. Voronin, The Riemann Zeta-Function, 1992, ISBN 3-11-013170-6, p. 390
Morduhaĭ-Boltovskoĭ (1949)
Eliakim H. Moore, "Concerning Transcendentally Transcendental Functions", Mathematische Annalen 48:1-2:49-74 (1896) doi:10.1007/BF01446334
R. D. Carmichael, "On Transcendentally Transcendental Functions", Transactions of the American Mathematical Society 14:3:311-319 (July 1913) full text JSTOR 1988599 doi:10.1090/S0002-9947-1913-1500949-2

Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", The American Mathematical Monthly 96:777-788 (November 1989) JSTOR 2324840

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 (N.S.) (in Russian), 64: 21–24, MR 0028347