Gegenbauer polynomials

Short description: Polynomial sequence

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α)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.

## Contents

* 1 Characterizations
* 2 Orthogonality and normalization
* 3 Applications
* 4 Other properties
* 5 Asymptotics
* 6 See also
* 7 References
* 7.1 Specific

## Characterizations

* 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

* Gegenbauer polynomials with α=1

* Gegenbauer polynomials with α=2

* Gegenbauer polynomials with α=3

* 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:[1]

1(1−2xt+t2)α=∑n=0∞Cn(α)(x)tn(0≤|x|<1,|t|≤1,α>0)

* The polynomials satisfy the recurrence relation:[2]

C0(α)(x)=1C1(α)(x)=2αx(n+1)Cn+1(α)(x)=2(n+α)xCn(α)(x)−(n+2α−1)Cn−1(α)(x).

* Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation:[2]

(1−x2)y″−(2α+1)xy′+n(n+2α)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.[3]

* They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:

Cn(α)(z)=(2α)nn!2F1(−n,2α+n;α+12;1−z2).

[4] Here (2α)n is the rising factorial. Explicitly,

Cn(α)(z)=∑k=0⌊n/2⌋(−1)kΓ(n−k+α)Γ(α)k!(n−2k)!(2z)n−2k.
From this it is also easy to obtain the value at unit argument:

Cn(α)(1)=Γ(2α+n)Γ(2α)n!.

* They are special cases of the Jacobi polynomials:[2]

Cn(α)(x)=(2α)n(α+12)nPn(α−1/2,α−1/2)(x).
in which (θ)n represents the rising factorial of θ.
One therefore also has the Rodrigues formula

Cn(α)(x)=(−1)n2nn!Γ(α+12)Γ(n+2α)Γ(2α)Γ(α+n+12)(1−x2)−α+1/2dndxn[(1−x2)n+α−1/2].

* An alternative normalization sets Cn(α)(1)=1. Assuming this alternative normalization, the derivatives of Gegenbauer are expressed in terms of Gegenbauer:[5]

dqdxqCq+2j+1(α)(x)=2q(q+2j+1)!(q−1)!Γ(q+2j+2α+1)∑i=0j(2i+α+1)Γ(2i+2α+1)(2i+1)!(j−i)!×Γ(q+j+i+α+1)Γ(j+i+α+2)(q+j−i−1)!C2i+1(α)(x)

## Orthogonality and normalization

For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function[6]

w(z)=(1−z2)α−12.

To wit, for n ≠ m,

∫−11Cn(α)(x)Cm(α)(x)(1−x2)α−12dx=0.

They are normalized by

∫−11[Cn(α)(x)]2(1−x2)α−12dx=π21−2αΓ(n+2α)n!(n+α)[Γ(α)]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,

1|𝐱−𝐲|n−2=∑k=0∞|𝐱|k|𝐲|k+n−2Ck(α)(𝐱⋅𝐲|𝐱||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.[7]

It follows that the quantities Ck((n−2)/2)(𝐱⋅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

∑j=0nCjα(x)(2α+j−1j)≥0(x≥−1,α≥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.[8]

## Other properties

Dirichlet–Mehler-type integral representation:[9]Pn(α,α)(cos⁡θ)Pn(α,α)(1)=Cn(α+12)(cos⁡θ)Cn(α+12)(1)=2α+12Γ(α+1)π12Γ(α+12)(sin⁡θ)−2α∫0θcos⁡((n+α+12)ϕ)(cos⁡ϕ−cos⁡θ)−α+12dϕ,Laplace-type integral representationPn(α,α)(cos⁡θ)Pn(α,α)(1)=Cn(α+12)(cos⁡θ)Cn(α+12)(1)=Γ(α+1)π12Γ(α+12)∫0π(cos⁡θ+isin⁡θcos⁡ϕ)n(sin⁡ϕ)2αdϕAddition formula:[10]

Cnλ(cos⁡θ1cos⁡θ2+sin⁡θ1sin⁡θ2cos⁡ϕ)=∑k=0nan,kλ(sin⁡θ1)kCn−kλ+k(cos⁡θ1)(sin⁡θ2)kCn−kλ+k(cos⁡θ2)⋅Ckλ−1/2(cos⁡ϕ),an,kλ constants

## Asymptotics

Given fixed λ∈(0,1),M∈{1,2,…},δ∈(0,π/2), uniformly for all θ∈[δ,π−δ], for n->∞,[11][12]Cn(λ)(cos⁡θ)=22λΓ(λ+12)π12Γ(λ+1)(2λ)n(λ+1)n(∑m=0M−1(λ)m(1−λ)mm!(n+λ+1)mcos⁡θn,m(2sin⁡θ)m+λ+RM(θ))

where (⋅)m is the Pochhammer symbol, andθn,m=(n+m+λ)θ−12(m+λ)πThe remainder RM=O(1nM) has an explicit upper bound:|RM(θ)|≤(2/π)sin⁡(λπ)Γ(n+2λ)Γ(λ)Γ(M+λ)Γ(M−λ+1)M!Γ(n+M+λ+1)max⁡(|cos⁡θ|−1,2sin⁡θ)(2sin⁡θ)M+λwhere Γ is the Gamma function.

Other asymptotic formulas can be obtained as special cases of asymptotic formulas for the more general Jacobi polynomials.

## See also

* Rogers polynomials, the q-analogue of Gegenbauer polynomials
* Chebyshev polynomials
* Romanovski polynomials

## References

* Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18
* Szegő, G. (1975). Orthogonal Polynomials. Colloquium Publications. XXIII (4th ed.). Providence, RI: American Mathematical Society.

### Specific

1. ↑ (Stein Weiss)
2. ↑ 2.0 2.1 2.2 Hazewinkel, Michiel, ed. (2001), "Ultraspherical polynomials", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=U/u095030
3. ↑ Arfken, Weber, and Harris (2013) "Mathematical Methods for Physicists", 7th edition; ch. 18.4
4. ↑ Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 773. LCCN 65-12253. ISBN 978-0-486-61272-0. http://www.math.sfu.ca/~cbm/aands/page_773.htm.
5. ↑ Doha, E. H. (1991-01-01). "The coefficients of differentiated expansions and derivatives of ultraspherical polynomials". Computers & Mathematics with Applications 21 (2): 115–122. doi:10.1016/0898-1221(91)90089-M. ISSN 0898-1221. https://www.sciencedirect.com/science/article/pii/089812219190089M.
6. ↑ (Abramowitz Stegun)
7. ↑ Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9, https://archive.org/details/introductiontofo0000stei
8. ↑ Olver, Sheehan; Townsend, Alex (January 2013). "A Fast and Well-Conditioned Spectral Method". SIAM Review 55 (3): 462–489. doi:10.1137/120865458. ISSN 0036-1445.
9. ↑ "DLMF: §18.10 Integral Representations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials". https://dlmf.nist.gov/18.10.
10. ↑ Koornwinder, Tom (September 1973). "The Addition Formula for Jacobi Polynomials and Spherical Harmonics" (in en). SIAM Journal on Applied Mathematics 25 (2): 236–246. doi:10.1137/0125027. ISSN 0036-1399. http://epubs.siam.org/doi/10.1137/0125027.
11. ↑ (Szegő 1975)
12. ↑ "DLMF: §18.15 Asymptotic Approximations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials". https://dlmf.nist.gov/18.15.

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