Editing Classical orthogonal polynomials (section)

=== Gegenbauer polynomials ===
When one sets the parameters <math>\alpha</math> and <math>\beta</math> in the Jacobi polynomials equal to each other, one obtains the '''Gegenbauer''' or '''ultraspherical''' polynomials.  They are written <math>C_n^{(\alpha)}</math>, and defined as

:<math>C_n^{(\alpha)}(x) = \frac{\Gamma(2\alpha\!+\!n)\,\Gamma(\alpha\!+\!1/2)}{\Gamma(2\alpha) \,\Gamma(\alpha\!+\!n\!+\!1/2)}\! \  P_n^{(\alpha-1/2, \alpha-1/2)}(x).</math>

We have <math>Q(x) = 1-x^2</math> and
<math>L(x) = -(2\alpha+1)\, x</math>.
The parameter <math>\alpha</math> is required to be greater than&nbsp;&minus;1/2.

(Incidentally, the standardization given in the table below would make no sense for ''α'' = 0 and ''n'' ≠ 0, because it would set the polynomials to zero.  In that case, the accepted standardization sets <math>C_n^{(0)}(1) = \frac{2}{n}</math> instead of the value given in the table.)

Ignoring the above considerations, the parameter <math>\alpha</math> is closely related to the derivatives of <math>C_n^{(\alpha)}</math>:

:<math>C_n^{(\alpha+1)}(x) = \frac{1}{2\alpha}\! \  \frac{d}{dx}C_{n+1}^{(\alpha)}(x)</math>

or, more generally:

:<math>C_n^{(\alpha+m)}(x) = \frac{\Gamma(\alpha)}{2^m\Gamma(\alpha+m)}\! \  C_{n+m}^{(\alpha)[m]}(x).</math>

All the other classical Jacobi-like polynomials (Legendre, etc.) are special cases of the Gegenbauer polynomials, obtained by choosing a value of <math>\alpha</math> and choosing a standardization.

For further details, see [[Gegenbauer polynomials]].