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Gegenbauer polynomials
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Polynomials
$\alpha$
$n$
$C_n^{\alpha}(x)$
INPUT{polynomials.yaml} (not shown in preview)
Definition
The Gegenbauer polynomials $C_n^{\alpha}$, $n\geq 0$, can be defined via the recurrence formula (1).
Parameters
$\alpha$
—   real number
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$C_0^{\alpha}(x) = 1$, $C_1^{\alpha}(x) = 2\alpha x$, and $(n+1)C_{n+1}^{\alpha}(x) = 2x(n+\alpha) C_n^{\alpha}(x) - (n+2\alpha-1) C_{n-1}^{\alpha}(x)$
for $n\geq 1$ (recurrence formula).
(2)
$\sum_{n=0}^\infty C_n^{\alpha}(x)t^n = \frac{1}{(1-2tx+t^2)^\alpha}$ (generating function).
(3)
The $C_n^{\alpha}$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) (1-x^2)^{\alpha-1/2} dx$.
Programs
(P1)
Sage
polynomials = {alpha:
{n: gegenbauer(n,alpha,x) for n in [0..10]}
for alpha in [-1/2,0,1/2,1]
}