Gegenbauer polynomials

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

• 
  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):

$\frac{1}{(1-2xt+t^2{)}^{\alpha }}=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\alpha )}(x)t^n.$

• The polynomials satisfy the recurrence relation (Suetin 2001):

$\begin{array}{rl}{C}_0^{\alpha }(x)& =1\\ C_1^{\alpha }(x)& =2\alpha x\\ C_n^{\alpha }(x)& =\frac{1}{n}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{array}$

• Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):

$(1-x^2)y^{″}-(2\alpha +1)x{y}^{\prime }+n(n+2\alpha )y=0.$

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.

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

$C_n^{(\alpha )}(z)=\frac{(2\alpha {)}_n}{n!}{}_2{F}_1(-n,2\alpha +n;\alpha +\frac{1}{2};\frac{1-z}{2}).$

(Abramowitz & Stegun p. 561). Here (2α)_n is the rising factorial. Explicitly,

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

• They are special cases of the Jacobi polynomials (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

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

Orthogonality and normalization

For a fixed α, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun p. 774)

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

To wit, for n ≠ m,

$\underset{-1}{\overset{1}{\int }}{C}_n^{(\alpha )}(x)C_m^{(\alpha )}(x)(1-x^2{)}^{\alpha -\frac{1}{2}}dx=0.$

They are normalized by

$\underset{-1}{\overset{1}{\int }}{[C_n^{(\alpha )}(x)]}^2(1-x^2{)}^{\alpha -\frac{1}{2}}dx=\frac{\pi 2^{1-2\alpha }\mathrm{\Gamma }(n+2\alpha )}{n!(n+\alpha )[\mathrm{\Gamma }(\alpha ){]}^2}.$

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,

$\frac{1}{|\mathbf{x}-\mathbf{y}{|}^{n-2}}=\sum _{k=0}^{\mathrm{\infty }}\frac{|\mathbf{x}{|}^k}{|\mathbf{y}{|}^{k+n-2}}{C}_k^{(\alpha )}(\mathbf{x}\cdot \mathbf{y}).$

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 (Stein & Weiss 1971).

It follows that the quantities $C_k^{((n-2)/2)}(\mathbf{x}\cdot \mathbf{y})$ 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

$\sum _{j=0}^n\frac{C_j^{\alpha }(x)}{(\genfrac{}{}{0ex}{}{2\alpha +j-1}{j})}\ge 0(x\ge -1,\alpha \ge 1/4).$

See also

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

References

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "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. 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.

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