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

• 

  Gegenbauer polynomials

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):

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

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

${\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{{n-1}}^{\alpha }(x)-(n+2\alpha -2)C_{{n-2}}^{\alpha }(x)].\end{aligned}}$

• Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (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.

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

$C_{n}^{{(\alpha )}}(z)={\frac {(2\alpha )_{n}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right).$

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

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

• 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

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

Orthogonality and normalization

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

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

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,

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

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 $C_{{n,k}}^{{((n-2)/2)}}({\mathbf {x}}\cdot {\mathbf {y}})$ 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

$\sum _{{j=0}}^{n}{\frac {C_{j}^{\alpha }(x)}{{2\alpha +j-1 \choose j}}}\geq 0\qquad (x\geq -1,\,\alpha \geq 1/4).$

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., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253.
• Bayin, S.S. (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 5.
• 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., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, 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), "Ultraspherical polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.