Gegenbauer polynomials explained

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(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

A variety of characterizations of the Gegenbauer polynomials are available.

• The polynomials can be defined in terms of their generating function :

(1)/(( 1-2 xt + t ^{2)} ^{\alpha}) = \sum ^{inf ty} _{n =0} C ^{(\alpha)} _{n} (x) t ^{n} ( 0 \leq |x| < 1 , |t| \leq 1 , \alpha > 0 )

• The polynomials satisfy the recurrence relation :

\begin{align} C ^{(\alpha)} _{0} (x) & = 1 \\ C ^{(\alpha)} _{1} (x) & = 2 \alpha x \\ (n +1 ) C ^{(\alpha)} _{n +1} (x) & = 2 (n + \alpha) x C ^{(\alpha)} _{n} (x) - (n +2 \alpha -1 )C ^{(\alpha)} _{n -1} (x). \end{align}

• Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation :

( 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.^[1]

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

C ^{(\alpha)} _{n} (z) = (( 2 \alpha) _{n})/(n!) _{2F} ; _{1 \left( - n, 2 \alpha + n;\alpha + (1)/(2)} (1- z)/(2) \right).

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

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

From this it is also easy to obtain the value at unit argument:

C ^{(\alpha)} _{n} ( 1 ) = (\Gamma( 2 \alpha + n))/(\Gamma( 2 \alpha)n!) .

• They are special cases of the Jacobi polynomials :

C ^{(\alpha)} _{n} (x) = (( 2 \alpha) _{n})/((\alpha + (1)/() _{n} 2)) P ^{(\alpha -1 / 2 ,\alpha -1 / 2 )} _{n} (x).

in which

(\theta) _{n} represents the rising factorial of \theta .

One therefore also has the Rodrigues formula

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

Orthogonality and normalization

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

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

To wit, for n ≠ m,

\int ^{1} _{-1} C ^{(\alpha)} _{n} (x)C ^{(\alpha)} _{m} (x)( 1- x ^{2)} ^{\alpha - (1)/(2)} dx = 0 .

They are normalized by

\int ^{1} _{-1} \left[C ^{(\alpha)} _{n} (x)\right] ^{2(1-x} ^{2)} ^{\alpha - (1)/(2)} dx = (\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,

(1)/(| x - y | ^{n-2}) = \sum ^{inf ty} _{k =0} (| x | ^{k})/(| y | ^{k+n-2}) C ^{(\alpha)} _{k} ( (x ⋅ y)/(| x || 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 .

It follows that the quantities

C ^{((n -2 )/ 2 )} _{k( x ⋅ 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 ^{n (C ^{\alpha(x)} _{j})/({ 2 \alpha + j -1 \choose j)} _{j =0} }\ge 0\qquad (x\ge-1,\, \alpha\ge 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.^[2]

See also

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

References

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Specific

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