mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomial ...

on the interval minus;1,1with respect to the

weight function A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...

(1 − ''x''²)^''α''–1/2. They generalize

Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...

and

Chebyshev polynomials The Chebyshev polynomials are two sequences of 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: The Chebyshe ...

, and are special cases of

Jacobi polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of Classical orthogonal polynomials, classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta ...

. They are named after

Leopold Gegenbauer Leopold Bernhard Gegenbauer (2 February 1849, Asperhofen – 3 June 1903, Gießhübl) was an Austrian mathematician remembered best as an algebraist. Gegenbauer polynomials are named after him. Leopold Gegenbauer was the son of a doctor. He ...

Characterizations

File:Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg, Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D File:Mplwp gegenbauer Cn05a1.svg, Gegenbauer polynomials with ''α''=1 File:Mplwp gegenbauer Cn05a2.svg, Gegenbauer polynomials with ''α''=2 File:Mplwp gegenbauer Cn05a3.svg, Gegenbauer polynomials with ''α''=3 File:Gegenbauer polynomials.gif, An animation showing the polynomials on the ''xα''-plane for the first 4 values of ''n''. A variety of characterizations of the Gegenbauer polynomials are available. * The polynomials can be defined in terms of their

generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...

: ::

\frac=\sum_^\infty C_n^(x) t^n \qquad (0 \leq , x,  < 1, , t,  \leq 1, \alpha > 0)

* The polynomials satisfy the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

: ::

\begin
C_0^(x) & = 1 \\
C_1^(x) & = 2 \alpha x \\
(n+1) C_^(x) & = 2(n+\alpha) x C_^(x) - (n+2\alpha-1)C_^(x).
\end

* Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation : ::

(1-x^)y''-(2\alpha+1)xy'+n(n+2\alpha)y=0.\,

:When ''α'' = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the

. :When ''α'' = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the

of the second kind.Arfken, Weber, and Harris (2013) "Mathematical Methods for Physicists", 7th edition; ch. 18.4 * They are given as

Gaussian hypergeometric series Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

in certain cases where the series is in fact finite: ::

C_n^(z)=\frac
\,_2F_1\left(-n,2\alpha+n;\alpha+\frac;\frac\right).

:(Abramowitz & Stegu
p. 561
. Here (2α)_''n'' is the

rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...

. Explicitly, ::

C_n^(z)=\sum_^ (-1)^k\frac(2z)^.

* They are special cases of the

: ::

C_n^(x) = \fracP_n^(x).

:in which

(\theta)_n

represents the

\theta

. :One therefore also has the

Rodrigues formula In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out ...

C_n^(x) = \frac\frac(1-x^2)^\frac\left 1-x^2)^\right

Orthogonality and normalization

For a fixed ''α'', the polynomials are orthogonal on minus;1, 1with respect to the weighting function (Abramowitz & Stegu
p. 774
:

w(z) = \left(1-z^2\right)^.

To wit, for ''n'' ≠ ''m'', :

\int_^1 C_n^(x)C_m^(x)(1-x^2)^\,dx = 0.

They are normalized by :

2(1-x^2)^\,dx = \frac.

Applications

The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...

and

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

. The

Newtonian potential In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object ...

in R^''n'' has the expansion, valid with α = (''n'' − 2)/2, :

\frac = \sum_^\infty \fracC_k^(\frac).

When ''n'' = 3, this gives the Legendre polynomial expansion of the

gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric po ...

. Similar expressions are available for the expansion of the

Poisson kernel In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...

in a ball . It follows that the quantities

C^_k(\mathbf\cdot\mathbf)

are

spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...

, when regarded as a function of x only. They are, in fact, exactly the

zonal spherical harmonic In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion ...

s, up to a normalizing constant. Gegenbauer polynomials also appear in the theory of

Positive-definite function In mathematics, a positive-definite function is, depending on the context, either of two types of function. Most common usage A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such th ...

s. The

Askey–Gasper inequality In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture. Statement It states that if \beta\geq 0, \alpha+\beta\geq -2, and -1\leq x\leq 1 then :\sum_^n \fr ...

reads :

\sum_^n\frac\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).

spectral methods Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain " basis functio ...

for solving differential equations, if a function is expanded in the basis of

and its derivative is represented in a Gegenbauer/ultraspherical basis, then the derivative operator becomes a

diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...

, leading to fast banded matrix methods for large problems.

References

* * * . * {{springer, title=Ultraspherical polynomials, id=U/u095030, first=P.K., last=Suetin. ;Specific Orthogonal polynomials Special hypergeometric functions

Characterizations

Orthogonality and normalization

Applications

See also

References