chebyshev_polynomial_u

→Tensor

chebyshev_polynomial_u(x: Tensor, n: int)

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Chebyshev polynomial of the second kind, $U_n(x)$ .

The U polynomials are orthogonal on $[-1, 1]$ with weight $\sqrt{1 - x^2}$ . They appear naturally as the derivative of the T polynomials (up to a normalisation) and in Gauss-Chebyshev quadrature of the second kind.

Parameters

xTensor

Argument tensor; any floating-point dtype.

nint

Non-negative polynomial degree.

Returns

Tensor

$U_n(x)$ element-wise, same shape and dtype as x.

Notes

Defined by $U_n(\cos\theta) = \sin((n+1)\theta)/\sin\theta$ and obeying

U_0(x) = 1, \quad U_1(x) = 2x, \quad U_{k+1}(x) = 2x\, U_k(x) - U_{k-1}(x).

Linked to T by $T_n'(x) = n\, U_{n-1}(x)$ . Raises ValueError for n < 0.

Examples

>>> import lucid
>>> from lucid.special import chebyshev_polynomial_u
>>> chebyshev_polynomial_u(lucid.tensor([-1.0, 0.0, 0.5, 1.0]), n=3)
tensor([-4., -0., -1., 4.])