Gegenbauer polynomials

Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. They are named for Leopold Gegenbauer (1849-1903). They are obtained from hypergeometric series in cases where the series is in fact finite:

:C_n^{(alpha)}(z)=frac{(2alpha)^{underline{n}{n!},_2F_1left(-n,2alpha+n;alpha+frac{1}{2};frac{1-z}{2} ight)

where underline{n} is the falling factorial. (Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_561.htm p561] )

Gegenbauer polynomials appear from solving the Gegenbauer differential equation:

:(1-x^{2})y"-(2n+3)xy'+{alpha}y=0

They are closely related to ultraspherical polynomials and can be viewed as an extension of the Legendre polynomials, since they can be obtained from the generating function:

:frac{1}{(1-2xt+t^{2})^{alpha=sum_{n=0}^{infty}C_n^{(alpha)}(x) t^{n}

They are orthogonal with respect to the weighting function (Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_774.htm p774] ):: w(z) = left(1-z^2 ight)^{alpha-frac{1}{2

References

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