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:

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

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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