A hypertranscendental function or transcendentally transcendental function is a transcendental

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...

which is not the solution of an

algebraic differential equation In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to in ...

with coefficients in Z (the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s) and with algebraic

initial condition In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...

History

The term 'transcendentally transcendental' was introduced by

E. H. Moore Eliakim Hastings Moore (; January 26, 1862 – December 30, 1932), usually cited as E. H. Moore or E. Hastings Moore, was an American mathematician. Life Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, di ...

in 1896; the term 'hypertranscendental' was introduced by D. D. Morduhai-Boltovskoi in 1914.

Definition

One standard definition (there are slight variants) defines solutions of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s of the form :

F\left(x, y, y', \cdots, y^ \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 In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

is in this category.Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", ''The American Mathematical Monthly'' 96:777-788 (November 1989) Hypertranscendental functions usually arise as the solutions to

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

s, for example the

Examples

Hypertranscendental functions

* The zeta functions of

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

s, in particular, the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

* The

(''cf.'' Hölder's theorem)

Transcendental but not hypertranscendental functions

* The

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...

, and the

trigonometric Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

and

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

functions. * The

generalized hypergeometric function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...

s, including special cases such as

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

s (except some special cases which are algebraic).

Non-transcendental (algebraic) functions

* All

algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additio ...

s, in particular

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

Notes

References

* Loxton,J.H., Poorten,A.J. van der,
A class of hypertranscendental functions
,

Aequationes Mathematicae ''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topics ...

, 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 Analytic functions Mathematical analysis Types of functions