This is a list of special function eponyms in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, to cover the theory of

special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...

, the differential equations they satisfy, named

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

s of the theory (but not intended to include every mathematical

eponym An eponym is a noun after which or for which someone or something is, or is believed to be, named. Adjectives derived from the word ''eponym'' include ''eponymous'' and ''eponymic''. Eponyms are commonly used for time periods, places, innovati ...

). Named

symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...

s, and other special polynomials, are included.

A

* Niels Abel: Abel polynomials - Abelian function - Abel–Gontscharoff interpolating polynomial * Sir George Biddell Airy:

Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ...

* Waleed Al-Salam (1926–1996): Al-Salam polynomial - Al Salam–Carlitz polynomial - Al Salam–Chihara polynomial * C. T. Anger: Anger–Weber function * Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral *

Paul Émile Appell :''M. P. Appell is the same person: it stands for Monsieur Paul Appell''. Paul Émile Appell (27 September 1855 in Strasbourg – 24 October 1930 in Paris) was a French mathematician and Rector of the University of Paris. Appell polynomials and ...

(1855–1930): Appell hypergeometric series, Appell polynomial, Generalized Appell polynomials * Richard Askey: Askey–Wilson polynomial, Askey–Wilson function (with James A. Wilson)

B

* Ernest William Barnes: Barnes G-function * E. T. Bell: Bell polynomials ** Bender–Dunne polynomial *

Jacob Bernoulli Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...

Bernoulli polynomial In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur ...

Friedrich Bessel Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesy, geodesist. He was the first astronomer who determined reliable values for the distance from the Sun to another star by th ...

Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...

Bessel–Clifford function In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel func ...

* H. Blasius: Blasius functions * R. P. Boas, R. C. Buck: Boas–Buck polynomial ** Böhmer integral * Erland Samuel Bring: Bring radical * de Bruijn function * Buchstab function * Burchnall, Chaundy: Burchnall–Chaundy polynomial

C

Leonard Carlitz Leonard Carlitz (December 26, 1907 – September 17, 1999) was an American mathematician. Carlitz supervised 44 doctorates at Duke University and published over 770 papers. Chronology * 1907 Born Philadelphia, PA, USA * 1927 BA, University ...

: Carlitz polynomial *

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...

, Capelli: Cayley–Capelli operator ** Celine's polynomial ** Charlier polynomial *

Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...

Chebyshev polynomials The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: ...

* Elwin Bruno Christoffel, Darboux: Christoffel–Darboux relation *

Cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th prim ...

D

* H. G. Dawson: Dawson function * Charles F. Dunkl: Dunkl operator, Jacobi–Dunkl operator, Dunkl–Cherednik operator ** Dickman–de Bruijn function

E

* Engel:

Engel expansion The Engel expansion of a positive real number ''x'' is the unique non-decreasing sequence of positive integers (a_1,a_2,a_3,\dots) such that :x=\frac+\frac+\frac+\cdots = \frac\!\left(1 + \frac\!\left(1 + \frac\left(1+\cdots\right)\right)\right) ...

* Erdélyi Artúr: Erdelyi–Kober operator *

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

: Euler polynomial, Eulerian integral, Euler hypergeometric integral

F

*V. N. Faddeeva: Faddeeva function (also known as the complex error function; see

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...

)

G

* C. F. Gauss: Gaussian polynomial,

Gaussian distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...

Hypergeometric function In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...

₂''F''₁, etc. * Leopold Bernhard Gegenbauer: Gegenbauer polynomials ** Gottlieb polynomial ** Gould polynomial * Christoph Gudermann:

Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwe ...

H

*Wolfgang Hahn: Hahn polynomial, (with H. Exton) Hahn–Exton Bessel function * Philip Hall: Hall polynomial, Hall–Littlewood polynomial *

Hermann Hankel Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix. Biography Hankel was born on ...

: Hankel function * Heine: Heine functions *

Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite p ...

Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...

* Karl L. W. M. Heun (1859 – 1929): Heun's equation * J. Horn: Horn hypergeometric series *

Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, mathematical analysis, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a ...

: Hurwitz zeta-function *

₂''F''₁

J

* Henry Jack (1917–1978) Dundee: Jack polynomial * F. H. Jackson: Jackson derivative Jackson integral * Carl Gustav Jakob Jacobi: Jacobi polynomial,

Jacobi theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube do ...

K

* Joseph Marie Kampe de Feriet (1893–1982): Kampe de Feriet hypergeometric series *

David Kazhdan David Kazhdan (), born Dmitry Aleksandrovich Kazhdan (), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 MacArthur Fellow. Biography Kazhdan was born on 20 June 1946 in Moscow, USSR. His father ...

, George Lusztig:

Kazhdan–Lusztig polynomial In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial P_(q) is a member of a family of integral polynomials introduced by . They are indexed by pairs of elements ''y'', ''w'' of a Coxeter group ''W'', which can in parti ...

Lord Kelvin William Thomson, 1st Baron Kelvin (26 June 182417 December 1907), was a British mathematician, Mathematical physics, mathematical physicist and engineer. Born in Belfast, he was the Professor of Natural Philosophy (Glasgow), professor of Natur ...

: Kelvin function ** Kibble–Slepian formula * Kirchhoff: Kirchhoff polynomial * Tom H. Koornwinder: Koornwinder polynomial ** Kostka polynomial, Kostka–Foulkes polynomial * Mikhail Kravchuk: Kravchuk polynomial

L

* Edmond Laguerre:

Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: xy'' + (1 - x)y' + ny = 0,\ y = y(x) which is a second-order linear differential equation. Thi ...

Johann Heinrich Lambert Johann Heinrich Lambert (; ; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, at that time allied to the Switzerland, Swiss Confederacy, who made important contributions to the subjects of mathematics, phys ...

Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...

Gabriel Lamé Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...

: Lamé polynomial * G. Lauricella Lauricella-Saran: Lauricella hypergeometric series *

Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transforma ...

Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...

* Eugen Cornelius Joseph von Lommel (1837–1899), physicist: Lommel polynomial,

Lommel function Lommel () is a Municipalities of Belgium, municipality and City status in Belgium, city in the Belgium, Belgian province of Limburg (Belgium), Limburg. Lying in the Campine, Kempen, it has about 34,000 inhabitants and is part of the arrondissement ...

, Lommel–Weber function

M

* Ian G. Macdonald:

Macdonald polynomial In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald orig ...

, Macdonald–Kostka polynomial, Macdonald spherical function ** Mahler polynomial ** Maitland function * Émile Léonard Mathieu: Mathieu function * F. G. Mehler, student of Dirichlet (Ferdinand): Mehler's formula, Mehler–Fock formula, Mehler–Heine formula, Mehler functions ** Meijer G-function * Josef Meixner: Meixner polynomial, Meixner-Pollaczek polynomial * Mittag-Leffler: Mittag-Leffler polynomials ** Mott polynomial

P

* Paul Painlevé: Painlevé transcendents * Poisson–Charlier polynomial * Pollaczek polynomial

R

* Giulio Racah: Racah polynomial *

Jacopo Riccati Jacopo Francesco Riccati (28 May 1676 – 15 April 1754) was a Venetian mathematician and jurist from Venice. He is best known for having studied the equation that bears his name. Education Riccati was educated first at the Jesuit school for th ...

: Riccati–Bessel function *

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

: Riemann zeta-function * Olinde Rodrigues: Rodrigues formula * Leonard James Rogers: Rogers–Askey–Ismail polynomial, Rogers–Ramanujan identity, Rogers–Szegő polynomials

S

* Schubert polynomial *

Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...

: Schur polynomial *

Atle Selberg Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded ...

: Selberg integral ** Sheffer polynomial ** Slater's identities *

Thomas Joannes Stieltjes Thomas Joannes Stieltjes ( , ; 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics ...

: Stieltjes polynomial, Stieltjes–Wigert polynomials ** Strömgren function * Hermann Struve: Struve function

T

* Francesco Tricomi: Tricomi–Carlitz polynomial

W

* Wall polynomial * Wangerein: Wangerein functions * Weber function *

Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...

Weierstrass function In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiab ...

* Louis Weisner: Weisner's method * E. T. Whittaker: Whittaker function * Wilson polynomial {{anchor, Y

Z

Frits Zernike Frits Zernike (; 16 July 1888 – 10 March 1966) was a Dutch physicist who received the Nobel Prize in Physics in 1953 for his invention of the phase-contrast microscope. Early life and education Frederick "Frits" Zernike was born on 16 July ...

: Zernike polynomials Eponyms of special functions Eponymous functions