CAOP

Plot

Gegenbauer Polynomials

Definition

The Gegenbauer polynomials are defined as

\[ \begin{align} C_n^{(\alpha)}(x) &= \frac{1}{\Gamma(\alpha)}\,\sum_{k=0}^{[n/2]} \frac{(-1)^k\,\Gamma(\alpha+n-k)}{k!\,(n-2k)!}\,(2x)^{n-2k}\\ &= {n+\alpha-1 \choose n} (2x)^n {}_2F_1 \left(\left. {-n/2, -n/2+\frac{1}{2} \atop -n-\alpha+1} \; \right| \frac{1}{x^2} \right) \end{align} \]

Differential Equation

Recurrence Equation

Parameters

Variables

\(n\)

\(x\)

Parameters

\(\alpha\) \(\)

factor (use Maple-style input)

   hypergeometric term in \(n\) and hyperexponential term in \(x\) required