Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C{{su|p=(α)|b=n}}(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x²)^α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.

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 {{harv|Stein|Weiss|1971|loc=§IV.2}}:

:: $\frac{1}{(1-2xt+t^2)^\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x) t^n \qquad (0 \leq |x| < 1, |t| \leq 1, \alpha > 0)$

The polynomials satisfy the recurrence relation {{harv|Suetin|2001}}:

:: $\begin{align}
C_0^{(\alpha)}(x) & = 1 \\
C_1^{(\alpha)}(x) & = 2 \alpha x \\
(n+1) C_{n+1}^{(\alpha)}(x) & = 2(n+\alpha) x C_{n}^{(\alpha)}(x) - (n+2\alpha-1)C_{n-1}^{(\alpha)}(x).
\end{align}$

Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation {{harv|Suetin|2001}}:

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

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

:When α = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the Chebyshev polynomials 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 in certain cases where the series is in fact finite:

:: $C_n^{(\alpha)}(z)=\frac{(2\alpha)_n}{n!}
\,_2F_1\left(-n,2\alpha+n;\alpha+\frac{1}{2};\frac{1-z}{2}\right).$

:(Abramowitz & Stegun [https://personal.math.ubc.ca/~cbm/aands/page_561.htm p. 561]). Here (2α)_n is the rising factorial. Explicitly,

:: $C_n^{(\alpha)}(z)=\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)k!(n-2k)!}(2z)^{n-2k}.$

:From this it is also easy to obtain the value at unit argument:

:: $C_n^{(\alpha)}(1)=\frac{\Gamma(2\alpha+n)}{\Gamma(2\alpha)n!}.$

They are special cases of the Jacobi polynomials {{harv|Suetin|2001}}:

:: $C_n^{(\alpha)}(x) = \frac{(2\alpha)_n}{(\alpha+\frac{1}{2})_{n}}P_n^{(\alpha-1/2,\alpha-1/2)}(x).$

:in which $(\theta)_n$ represents the rising factorial of $\theta$ .

:One therefore also has the Rodrigues formula

:: $C_n^{(\alpha)}(x) = \frac{(-1)^n}{2^n n!}\frac{\Gamma(\alpha+\frac{1}{2})\Gamma(n+2\alpha)}{\Gamma(2\alpha)\Gamma(\alpha+n+\frac{1}{2})}(1-x^2)^{-\alpha+1/2}\frac{d^n}{dx^n}\left[(1-x^2)^{n+\alpha-1/2}\right].$

An alternative normalization sets $C_n^{(\alpha)}(1)=1$ . Assuming this alternative normalization, the derivatives of Gegenbauer are expressed in terms of Gegenbauer:{{Cite journal |last=Doha |first=E. H. |date=1991-01-01 |title=The coefficients of differentiated expansions and derivatives of ultraspherical polynomials |url=https://www.sciencedirect.com/science/article/pii/089812219190089M |journal=Computers & Mathematics with Applications |volume=21 |issue=2 |pages=115–122 |doi=10.1016/0898-1221(91)90089-M |issn=0898-1221}}

$\begin{aligned}
\frac{d^q}{dx^q}C_{q+2 j+1}^{(\alpha)}(x)=\frac{2^q(q+2 j+1)!}{(q-1)!\Gamma(q+2 j+2 \alpha+1)} & \sum_{i=0}^j \frac{(2 i+\alpha+1) \Gamma(2 i+2 \alpha+1)}{(2 i+1)!(j-i)!} \\
& \times \frac{\Gamma(q+j+i+\alpha+1)}{\Gamma(j+i+\alpha+2)}(q+j-i-1)!C_{2 i+1}^{(\alpha)}(x)
\end{aligned}$

Orthogonality and normalization

For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_774.htm p. 774])

: $w(z) = \left(1-z^2\right)^{\alpha-\frac{1}{2}}.$

To wit, for n ≠ m,

: $\int_{-1}^1 C_n^{(\alpha)}(x)C_m^{(\alpha)}(x)(1-x^2)^{\alpha-\frac{1}{2}}\,dx = 0.$

They are normalized by

: $\int_{-1}^1 \left[C_n^{(\alpha)}(x)\right]^2(1-x^2)^{\alpha-\frac{1}{2}}\,dx = \frac{\pi 2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)[\Gamma(\alpha)]^2}.$

Applications

The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potential theory and harmonic analysis. The Newtonian potential in Rⁿ has the expansion, valid with α = (n − 2)/2,

: $\frac{1}$

\mathbf{x}-\mathbf{y}|^{n-2}} = \sum_{k=0}^\infty \frac{|\mathbf{x}|^k}{|\mathbf{y}|^{k+n-2}}C_k^{(\alpha)}(\frac{\mathbf{x}\cdot \mathbf{y}}{|\mathbf{x}

\mathbf{y}

When n = 3, this gives the Legendre polynomial expansion of the gravitational potential. Similar expressions are available for the expansion of the Poisson kernel in a ball {{harv|Stein|Weiss|1971}}.

It follows that the quantities $C^{((n-2)/2)}_k(\mathbf{x}\cdot\mathbf{y})$ are spherical harmonics, when regarded as a function of x only. They are, in fact, exactly the zonal spherical harmonics, up to a normalizing constant.

Gegenbauer polynomials also appear in the theory of positive-definite functions.

The Askey–Gasper inequality reads

: $\sum_{j=0}^n\frac{C_j^\alpha(x)}{{2\alpha+j-1\choose j}}\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).$

In spectral methods for solving differential equations, if a function is expanded in the basis of Chebyshev polynomials and its derivative is represented in a Gegenbauer/ultraspherical basis, then the derivative operator becomes a diagonal matrix, leading to fast banded matrix methods for large problems.{{cite journal | last1 = Olver | first1 = Sheehan | last2 = Townsend | first2 = Alex | title = A Fast and Well-Conditioned Spectral Method | journal = SIAM Review | date = January 2013 | volume = 55 | issue = 3 | pages = 462–489 | issn = 0036-1445 | eissn = 1095-7200 | doi = 10.1137/120865458 | arxiv = 1202.1347 }}

Other properties

Dirichlet–Mehler-type integral representation:{{Cite web |title=DLMF: §18.10 Integral Representations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.10 |access-date=2025-03-18 |website=dlmf.nist.gov}} $\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma\left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\,\mathrm{d}\phi,$ Laplace-type $\begin{aligned}
\frac{P_n^{(\alpha, \alpha)}(\cos \theta)}{P_n^{(\alpha, \alpha)}(1)} & =\frac{C_n^{\left(\alpha+\frac{1}{2}\right)}(\cos \theta)}{C_n^{\left(\alpha+\frac{1}{2}\right)}(1)} \\
& =\frac{\Gamma(\alpha+1)}{\pi^{\frac{1}{2}} \Gamma\left(\alpha+\frac{1}{2}\right)} \int_0^\pi(\cos \theta+i \sin \theta \cos \phi)^n(\sin \phi)^{2 \alpha} \mathrm{~d} \phi
\end{aligned}$

References

{{Abramowitz_Stegun_ref|22|773}}*{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
{{citation|last1=Stein|first1=Elias|authorlink1=Elias Stein|first2=Guido|last2=Weiss|authorlink2=Guido Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=978-0-691-08078-9|location=Princeton, N.J.|url-access=registration|url=https://archive.org/details/introductiontofo0000stei}}.
{{springer|title=Ultraspherical polynomials|id=U/u095030|first=P.K.|last=Suetin}}.

;Specific

Category:Orthogonal polynomials

Category:Special hypergeometric functions