Special functions

2020 Mathematics Subject Classification: Primary: 33-XX [MSN][ZBL]

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, special functions are the special functions of mathematical physics, which arise when solving partial differential equations by the method of separation of variables (cf. Separation of variables, method of).

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 (cf. Hypergeometric function; Confluent hypergeometric function); Bessel functions; Legendre functions; parabolic cylinder functions (cf. Parabolic cylinder function); integral sine and integral cosine functions; incomplete gamma- and beta-functions (cf. Incomplete gamma-function; Incomplete beta-function); probability integrals (cf. Probability integral); various classes of orthogonal polynomials in one or several variables; elliptic functions and integrals (cf. Elliptic function; Elliptic integral); Lamé functions (cf. Lamé function) and Mathieu functions; the Riemann zeta-function; automorphic functions (cf. Automorphic function); and some special functions of a discrete argument.

The theory of special functions is connected with the representation of groups (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 tables of values of special functions and also tables of integrals and series.

Comments

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 recent survey [Ko], and [ViKl], vols. 2–3, for more details.

References

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[GrRy]	I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products", Acad. Press (1980) (Translated from Russian) MR0582453
[Ho]	E.W. Hobson, "The theory of spherical and ellipsoidal harmonics", Cambridge Univ. Press (1931) MR0064922 MR1522948 Zbl 0004.21001 JFM Zbl 57.0405.06
[JaEmLö]	E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen", Teubner (1966)
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[KrFr]	A. Krazer, W. Franz, "Transzendente Funktionen", Akademie Verlag (1960)
[Le]	N.N. Lebedev, "Special functions and their applications", Prentice-Hall (1965) (Translated from Russian) MR0174795 Zbl 0131.07002
[Mi]	W. Miller jr., "Lie theory and special functions", Acad. Press (1968) MR0264140 Zbl 0174.10502
[NiUf]	A.F. Nikiforov, V.B. Ufarov, "Special functions of mathematical physics", Birkhäuser (1988) (Translated from Russian) MR0922041 Zbl 0624.33001
[PrBrMa]	A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, "Integrals and series. Elementary functions", Gordon & Breach (1986) (Translated from Russian) MR0888165 MR0874987 MR0874986 Zbl 0511.00044
[PrBrMa2]	A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, "Integrals and series. Special functions", Gordon & Breach (1986) (Translated from Russian) MR0888165 MR0874987 MR0874986 Zbl 0626.00033 Zbl 0733.00004
[PrBrMa3]	A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, "Integrals and series. Additional chapters", Gordon & Breach (1987) (Translated from Russian) MR1162980 MR1162979 MR1054647
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[Vi]	N.Ya. Vilenkin, "Special functions and the theory of group representations", Amer. Math. Soc. (1968) (Translated from Russian) MR0229863 Zbl 0172.18404
[ViKl]	N.Ya. Vilenkin, A.U. Klimyk, "Representations of Lie groups, special functions and integral transforms", 1–3, Kluwer (1991-) (Translated from Russian)
[Wa]	G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) MR1349110 Zbl 0849.33001
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