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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
The Jacobi polynomials $P_n^{(\alpha,\beta)}$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (3) such that $\deg P_n^{(\alpha,\beta)}=n$ and $P_n^{(\alpha,\beta)}(1) = \binom{n+\alpha}{n}$. Due to (4) and (2) we restrict to $\alpha < \beta$.
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).
(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]])
}