CAOP

Plot

Jacobi Polynomials

Definition

The Jacobi polynomials are defined as

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

Differential Equation

Recurrence Equation

Parameters

Variables

\(n\)

\(x\)

Parameters

\(\alpha\) \(\alpha > -1\)

\(\beta\) \(\beta > -1\)

factor (use Maple-style input)

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