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Jacobi polynomials

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Polynomials

$\alpha$

$\beta$

$n$

$P_n^{(\alpha,\beta)}(x)$

INPUT{polynomials.yaml} (not shown in preview)

Definition

Parameters

$\alpha$

— real number ($\alpha > -1$)

$\beta$

— real number ($\beta > -1$)

$n$

— integer ($n \geq 0$)

Formulas

(1)

$\sum_{n=0}^\infty P_n^{(\alpha,\beta)}(x)t^n = \frac{2^{\alpha+\beta}}{R(1-t+R)^\alpha (1+t+R)^\beta}$, where $R = (1-2zt+t)^{1/2}$, defined via the standard branch of the square root (generating function).

(2)

$P_n^{(\alpha,\beta)}(-x) = (-1)^n P_n^{(\beta,\alpha)}(x)$ (symmetry).

Comments

(3)

The $P_n^{(\alpha,\beta)}$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) (1-x)^{\alpha}(1+x)^{\beta} dx$.

(4)

The polynomials with $\alpha=\beta$ are called Gegenbauer polynomials (with parameter $\alpha+1/2$).

(5)

The polynomials with $\alpha=\beta=\pm 1/2$ are called Chebyshev polynomials of the second kind and Chebyshev polynomials of the first kind, respectively.

(6)

The polynomials with $\alpha=\beta=0$ are called Legendre polynomials.

Programs

(P1)

Sage

```
polynomials = {(alpha,beta):
{n: jacobi_P(n,alpha,beta,x) for n in [0..10]}
for alpha, beta in cartesian_product([[0..2],[0..2]])
}
```

Links

Data properties

Entries are of type: rational polynomial

Table is complete: no