Gegenbauer polynomials
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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.
Characterizations
[edit]- 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 (Stein & Weiss 1971, §IV.2):
- The polynomials satisfy the recurrence relation (Suetin 2001):
- Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):
- 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.[1]
- They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:
- (Abramowitz & Stegun p. 561). Here (2α)n is the rising factorial. Explicitly,
- From this it is also easy to obtain the value at unit argument:
- They are special cases of the Jacobi polynomials (Suetin 2001):
- in which represents the rising factorial of .
- One therefore also has the Rodrigues formula
Orthogonality and normalization
[edit]For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun p. 774)
To wit, for n ≠ m,
They are normalized by
Applications
[edit]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,
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 (Stein & Weiss 1971).
It follows that the quantities 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
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.[2]
See also
[edit]- Rogers polynomials, the q-analogue of Gegenbauer polynomials
- Chebyshev polynomials
- Romanovski polynomials
References
[edit]- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 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. p. 773. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.*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.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
- Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
- Suetin, P.K. (2001) [1994], "Ultraspherical polynomials", Encyclopedia of Mathematics, EMS Press.
- Specific