Jacobi polynomials P_n^(α,β)(x): orthogonal on [-1,1] with weight (1−x)^α(1+x)^β. The 'mother' of all classical orthogonal polynomials on finite intervals. Special cases: Legendre (0,0), Chebyshev T (−½,−½), Chebyshev U (½,½), Gegenbauer (α=β).
How to Use
Enter n, α, β, x:
- Value: P_n^(α,β)(x)
- Presets: Known families
- Sequence: n=0..N
Family Hierarchy
Jacobi(α,β) → Gegenbauer(α=β) → Legendre(0,0) & Chebyshev(−½,−½). Each specialization adds symmetry. Jacobi is the most general with TWO parameters controlling the weight at each endpoint.
Applications
- Spectral methods with non-uniform weights
- Random matrix theory (Jacobi ensemble)
- Representation theory of SU(2)
- Approximation theory on [-1,1]
Step-by-Step Instructions
- 1Enter n, α, β.
- 2Enter x.
- 3Compute value.
- 4Try presets.
- 5View sequence.