In the broad sense, a set of several classes of functions that arise in the solution of both theoretical and applied problems in various branches of mathematics.
In the narrow sense, the special functions of mathematical physics, which arise when solving partial differential equations by the method of separation of variables.
Special functions can be defined by means of power series, generating functions, infinite products, repeated differentiation, integral representations, differential, difference, integral, and functional equations, trigonometric series, or other series in orthogonal functions.
The most important classes of special functions are the following: the gamma function and the beta function; hypergeometric functions and confluent hypergeometric functions; Bessel functions; Legendre functions; parabolic cylinder functions; integral sine and integral cosine functions; incomplete gamma functions and incomplete beta functions; probability integrals; various classes of orthogonal polynomials in one or several variables; elliptic functions and elliptic integrals; Lamé functions and Mathieu functions; the Riemann zeta function; automorphic functions; and some special functions of a discrete argument.
The theory of special functions is connected with group representations (cf. representation theory), with methods of integral representations based on the generalization of the Rodrigues formula for classical orthogonal polynomials, and with methods in probability theory.
There are software libraries for computation of special functions, and also tables of integrals, series and other formulas for special functions. See in particular [Olv].
Given a Lie group $G$ and a (matrix) representation $\rho$ of it, one can regard the matrix coefficients of $\rho$ as functions on $G$. Many special functions can be seen as arising essentially in this way, and this point of view "explains" many of the special properties of special functions, e.g. various orthogonality relations. Cf. [Vi], [Mi], [Wa2], and the encyclopaedic treatment [ViKl], vol. 1, for more details.
Many special functions have so-called $q$-analogues, $q$-special functions. That means, roughly, that it is possible to insert a parameter $q$ to obtain a family of special functions in such a way that many of the characteristic properties of special functions are retained. These $q$-special functions correspond to quantum groups in the same way that special functions relate to Lie groups. Cf. the survey [Ko], and [ViKl], vols. 2–3, for more details.
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Special functions. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Special_functions&oldid=26594