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

• 
  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):

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

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

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

• 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.

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:

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

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

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

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 (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:

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

Orthogonality and normalization

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

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

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.

Other properties

Dirichlet–Mehler-type integral representation:$${\frac {P_{n}^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_{n}^{(\alpha ,\alpha )}\left(1\right)}}={\frac {C_{n}^{(\alpha +{\frac {1}{2}})}\left(\cos \theta \right)}{C_{n}^{(\alpha +{\frac {1}{2}})}\left(1\right)}}={\frac {2^{\alpha +{\frac {1}{2}}}\Gamma \left(\alpha +1\right)}{{\pi }^{\frac {1}{2}}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}(\sin \theta )^{-2\alpha }\int _{0}^{\theta }{\frac {\cos \left((n+\alpha +{\tfrac {1}{2}})\phi \right)}{(\cos \phi -\cos \theta )^{-\alpha +{\frac {1}{2}}}}}\,\mathrm {d} \phi ,$$Laplace-type integral representation$${\begin{aligned}{\frac {P_{n}^{(\alpha ,\alpha )}(\cos \theta )}{P_{n}^{(\alpha ,\alpha )}(1)}}&={\frac {C_{n}^{\left(\alpha +{\frac {1}{2}}\right)}(\cos \theta )}{C_{n}^{\left(\alpha +{\frac {1}{2}}\right)}(1)}}\\&={\frac {\Gamma (\alpha +1)}{\pi ^{\frac {1}{2}}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\pi }(\cos \theta +i\sin \theta \cos \phi )^{n}(\sin \phi )^{2\alpha }\mathrm {~d} \phi \end{aligned}}$$Addition formula:

$${\begin{aligned}&C_{n}^{\lambda }\left(\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}\cos \phi \right)\\&\quad =\sum _{k=0}^{n}a_{n,k}^{\lambda }\left(\sin \theta _{1}\right)^{k}C_{n-k}^{\lambda +k}\left(\cos \theta _{1}\right)\left(\sin \theta _{2}\right)^{k}C_{n-k}^{\lambda +k}\left(\cos \theta _{2}\right)\\&\quad \cdot C_{k}^{\lambda -1/2}(\cos \phi ),\quad a_{n,k}^{\lambda }{\text{ constants }}\end{aligned}}$$

Asymptotics

Given fixed $\lambda \in (0,1),M\in \{1,2,\dots \},\delta \in (0,\pi /2)$, uniformly for all $\theta \in [\delta ,\pi -\delta ]$, for $n\to \infty$,$$C_{n}^{(\lambda )}\left(\cos \theta \right)={\frac {2^{2\lambda }\Gamma \left(\lambda +{\frac {1}{2}}\right)}{{\pi }^{\frac {1}{2}}\Gamma \left(\lambda +1\right)}}{\frac {\left(2\lambda \right)_{n}}{\left(\lambda +1\right)_{n}}}\left(\sum _{m=0}^{M-1}{\dfrac {{\left(\lambda \right)_{m}}{\left(1-\lambda \right)_{m}}}{m!\,{\left(n+\lambda +1\right)_{m}}}}{\dfrac {\cos \theta _{n,m}}{(2\sin \theta )^{m+\lambda }}}+R_{M}(\theta )\right)$$

where $(\cdot )_{m}$ is the Pochhammer symbol, and$$\theta _{n,m}=(n+m+\lambda )\theta -{\tfrac {1}{2}}(m+\lambda )\pi$$The remainder $R_{M}=O\left({\frac {1}{n^{M}}}\right)$ has an explicit upper bound:$$|R_{M}(\theta )|\leq (2/\pi )\sin(\lambda \pi ){\frac {\Gamma (n+2\lambda )}{\Gamma (\lambda )}}{\frac {\Gamma (M+\lambda )\Gamma (M-\lambda +1)}{M!\Gamma (n+M+\lambda +1)}}{\frac {\max \left(|\cos \theta |^{-1},2\sin \theta \right)}{(2\sin \theta )^{M+\lambda }}}$$where $\Gamma$ 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

• 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.
• Szegő, G. (1975). Orthogonal Polynomials. Colloquium Publications. Vol. XXIII (4th ed.). Providence, RI: American Mathematical Society.
• Suetin, P.K. (2001) [1994], "Ultraspherical polynomials", Encyclopedia of Mathematics, EMS Press.

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