Jacobi Polynomial Calculator
Enter the degree n, parameters α and β, and a point x to evaluate the Jacobi polynomial P_n^(α,β)(x) with a three-term recurrence, plus its derivative, a per-degree value table, and a plot.
Input
Evaluate the orthogonal Jacobi polynomial P_n^(α,β)(x) with a three-term recurrence. Enter the degree n, the parameters α and β, and a point x.
Integer of 0 or more
Real number greater than -1
Real number greater than -1
Usually in the range -1 to 1
Result
Value of P_4^(α=1,β=1)
-0.7421875
at x = 0.5
Degree n
4
Parameter α
1
Parameter β
1
Derivative of P_4
-2.1875
Graph of P_4
The curve over the interval -1 to 1. The orange dot marks the entered x.
Value of P_k at each degree k
Values computed step by step by the recurrence from degree 0 to n.
| Degree k | Value of P_k |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 0.1875 |
| 3 | -0.625 |
| 4 | -0.7421875 |
How it works
- The Jacobi polynomials P_n^(α,β)(x) are orthogonal on the interval from -1 to 1 with respect to the weight (1-x)^α (1+x)^β. This tool evaluates them step by step with a three-term recurrence.
- The starting values are P_0(x) = 1 and P_1(x) = (1/2)[(α-β) + (α+β+2)x]. Higher degrees follow the standard recurrence c1 P_n = (c2 x + c3) P_(n-1) - c4 P_(n-2), raising the degree by one each step.
- For convergence and orthogonality the parameters are assumed to satisfy α greater than -1 and β greater than -1.
- The derivative is computed from the relation d/dx P_n^(α,β)(x) = (1/2)(n+α+β+1) P_(n-1)^(α+1,β+1)(x).
- When α = β = 0 the Jacobi polynomial reduces to the Legendre polynomial P_n(x).
- When α = β it is proportional to the Gegenbauer (ultraspherical) polynomial; α = β = -1/2 gives a multiple of the Chebyshev polynomial of the first kind and α = β = 1/2 the second kind.
- At the endpoints the values are P_n^(α,β)(1) = binomial C(n+α, n) and P_n^(α,β)(-1) = (-1)^n C(n+β, n).
- Very large degrees or large α and β can cause floating-point overflow or rounding error. If a result is not finite, narrow the input range.
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Jacobi Polynomial Calculator