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Gegenbauer polynomials

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Polynomial sequence

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C
n(x) are orthogonal polynomials on the interval with respect to the weight function (1 − x). 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 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=1
  • Gegenbauer polynomials with α=2 Gegenbauer polynomials with α=2
  • Gegenbauer polynomials with α=3 Gegenbauer polynomials with α=3
  • An animation showing the polynomials on the xα-plane for the first 4 values of n. 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.

1 ( 1 − 2 x t + t 2 ) α = ∑ n = 0 ∞ C n ( α ) ( x ) t n ( 0 ≤ | x | < 1 , | t | ≤ 1 , α > 0 ) {\displaystyle {\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)}
C 0 ( α ) ( x ) = 1 C 1 ( α ) ( x ) = 2 α x ( n + 1 ) C n + 1 ( α ) ( x ) = 2 ( n + α ) x C n ( α ) ( x ) − ( n + 2 α − 1 ) C n − 1 ( α ) ( x ) . {\displaystyle {\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 α + 1 ) x y ′ + n ( n + 2 α ) y = 0. {\displaystyle (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.
C n ( α ) ( z ) = ( 2 α ) n n ! 2 F 1 ( − n , 2 α + n ; α + 1 2 ; 1 − z 2 ) . {\displaystyle 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 ( α ) ( z ) = ∑ k = 0 ⌊ n / 2 ⌋ ( − 1 ) k Γ ( n − k + α ) Γ ( α ) k ! ( n − 2 k ) ! ( 2 z ) n − 2 k . {\displaystyle 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 ( α ) ( 1 ) = Γ ( 2 α + n ) Γ ( 2 α ) n ! . {\displaystyle C_{n}^{(\alpha )}(1)={\frac {\Gamma (2\alpha +n)}{\Gamma (2\alpha )n!}}.}
C n ( α ) ( x ) = ( 2 α ) n ( α + 1 2 ) n P n ( α − 1 / 2 , α − 1 / 2 ) ( x ) . {\displaystyle C_{n}^{(\alpha )}(x)={\frac {(2\alpha )_{n}}{(\alpha +{\frac {1}{2}})_{n}}}P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).}
in which ( θ ) n {\displaystyle (\theta )_{n}} represents the rising factorial of θ {\displaystyle \theta } .
One therefore also has the Rodrigues formula
C n ( α ) ( x ) = ( − 1 ) n 2 n n ! Γ ( α + 1 2 ) Γ ( n + 2 α ) Γ ( 2 α ) Γ ( α + n + 1 2 ) ( 1 − x 2 ) − α + 1 / 2 d n d x n [ ( 1 − x 2 ) n + α − 1 / 2 ] . {\displaystyle 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.}

Orthogonality and normalization

For a fixed α > -1/2, the polynomials are orthogonal on with respect to the weighting function (Abramowitz & Stegun p. 774)

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

To wit, for n ≠ m,

∫ − 1 1 C n ( α ) ( x ) C m ( α ) ( x ) ( 1 − x 2 ) α − 1 2 d x = 0. {\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx=0.}

They are normalized by

∫ − 1 1 [ C n ( α ) ( x ) ] 2 ( 1 − x 2 ) α − 1 2 d x = π 2 1 − 2 α Γ ( n + 2 α ) n ! ( n + α ) [ Γ ( α ) ] 2 . {\displaystyle \int _{-1}^{1}\left^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\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 R has the expansion, valid with α = (n − 2)/2,

1 | x − y | n − 2 = ∑ k = 0 ∞ | x | k | y | k + n − 2 C k ( α ) ( x ⋅ y | x | | y | ) . {\displaystyle {\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 ) ( x ⋅ y ) {\displaystyle 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

∑ j = 0 n C j α ( x ) ( 2 α + j − 1 j ) ≥ 0 ( x ≥ − 1 , α ≥ 1 / 4 ) . {\displaystyle \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.

See also

References

Specific
  1. Arfken, Weber, and Harris (2013) "Mathematical Methods for Physicists", 7th edition; ch. 18.4
  2. Olver, Sheehan; Townsend, Alex (January 2013). "A Fast and Well-Conditioned Spectral Method". SIAM Review. 55 (3): 462–489. arXiv:1202.1347. doi:10.1137/120865458. eISSN 1095-7200. ISSN 0036-1445.
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