{{Short description|Mathematics analytic function}} {{Use American English|date = January 2019}} 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 <math>\mathbb{Z}</math> (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.<ref>D. D. Mordykhai-Boltovskoi, "On hypertranscendence of the function ξ(x, s)", ''Izv. Politekh. Inst. Warsaw''<!-- what is the full name of this journal? presumably something like Izwiestija Politechniki Instituti Warszawskiej, but (a) I don't know Polish (b) I haven't been able to find a journal of that name or similar--> '''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]</ref><ref>{{harvtxt|Morduhaĭ-Boltovskoĭ|1949}}</ref>
==Definition==
One standard definition (there are slight variants) defines solutions of differential equations of the form :<math>F\left(x, y, y', \cdots, y^{(n)} \right) = 0</math>, where <math>F</math> 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.<ref>Eliakim H. Moore, "Concerning Transcendentally Transcendental Functions", ''Mathematische Annalen'' '''48''':1-2:49-74 (1896) {{doi|10.1007/BF01446334}}</ref><ref>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}}</ref><ref>Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", ''The American Mathematical Monthly'' '''96''':777-788 (November 1989) {{JSTOR|2324840}}</ref>
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.
==See also==
==Notes== {{Reflist}}
==References== * Loxton, J.H., Poorten, A.J. van der, "[http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN356261603_0016 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. * {{citation|mr=0028347|last=Morduhaĭ-Boltovskoĭ|first=D.|title=On hypertranscendental functions and hypertranscendental numbers|language=Russian|journal=Doklady Akademii Nauk SSSR |series=New Series|volume=64|year=1949|pages= 21–24}}
Category:Analytic functions Category:Mathematical analysis Category:Types of functions Category:Ordinary differential equations