Hypertranscendental function

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 $\mathbb{Z}$ (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.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}}, [https://books.google.com/books?id=fNontpCu9kQC&pg=PA390 p. 390]{{harvtxt|Morduhaĭ-Boltovskoĭ|1949}}

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.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) [https://www.ams.org/journals/tran/1913-014-03/S0002-9947-1913-1500949-2/S0002-9947-1913-1500949-2.pdf 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}}

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

Examples