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Chebyshev polynomials of the first kind $T_n$
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Polynomials
$n$
$T_n(x)$
INPUT{polynomials.yaml} (not shown in preview)
Definition
The Chebyshev polynomials of the first kind $T_n$, $n\geq 0$, can be defined via $T_n(\cos \alpha) = \cos(n\alpha)$.
Parameters
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$T_0(x) = 1$, $T_1(x) = x$, and $T_{n+1}(x) = 2x T_n(x)-T_{n-1}(x)$ for $n\geq 1$ (recurrence).
(2)
$\sum_{n=0}^\infty T_n(x)t^n = \frac{1-tx}{1-2tx+t^2}$ (generating function).
The $T_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x)(1-x^2)^{-1/2} dx$.
polynomials = {n: chebyshev_T(n,x) for n in [0..100]}