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, 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].

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

References

[Er]	A. Erdélyi et al. (ed.), Higher transcendental functions, 1–3, McGraw-Hill (1953–1955) MR0058756 MR0066496 Zbl 0051.30303 Zbl 0052.29502 Zbl 0064.06302
[Fe]	W. Feller, "An introduction to probability theory and its applications", 1, third ed. (1968), 2, second ed. (1971), Wiley MR0228020 MR0270403 Zbl 0155.23101 Zbl 0219.60003
[GrRy]	I.S. Gradshteyn, I.M. Ryzhik, "Table of integrals, series and products", Elsevier/Acad. Press, seventh ed. (2007) (Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger) MR2360010 Zbl 1208.65001
[Ho]	E.W. Hobson, "The theory of spherical and ellipsoidal harmonics", Cambridge Univ. Press (1931) MR0064922 MR1522948 Zbl 0004.21001 JFM 57.0405.06
[Ko]	H.T. Koelink, "Askey-Wilson polynomials and the quantum SU(2) group: survey and applications" Acta Appl. Math., 44 (1996) pp. 295–352 MR1407326Zbl 0865.33013
[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
[Olv]	F.W.J. Olver et al., "NIST handbook of mathematical functions", Cambridge Univ. Press (2010) MR2723248 Zbl 1198.00002; online version: NIST Digital Library of Mathematical Functions (DLMF)
[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
[SrKa]	H.M. Srivastava, B.R.K. Kashyap, "Special functions in queuing theory", Acad. Press (1982) MR0657766 Zbl 0492.60089
[Sz]	G. Szegö, "Orthogonal polynomials", Amer. Math. Soc. (1975) MR0310533
[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
[Wa2]	A. Wawrzyńczyk, "Group representations and special functions", Reidel (1984) MR0750113 Zbl 0545.43001
[WhWa]	E.T. Whittaker, G.N. Watson, "A course of modern analysis", Cambridge Univ. Press (1952) pp. Chapt. 6 MR1424469 Zbl 0951.30002