chebyshev_polynomial_w

→Tensor

chebyshev_polynomial_w(x: Tensor, n: int)

source

Chebyshev polynomial of the fourth kind, $W_n(x)$ .

The W polynomials are orthogonal on $[-1, 1]$ with weight $\sqrt{(1 - x)/(1 + x)}$ — the mirror image of the V family — and partner with V under the substitution $x \mapsto -x$ .

Parameters

xTensor

Argument tensor; any floating-point dtype.

nint

Non-negative polynomial degree.

Returns

Tensor

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

Notes

Half-angle definition

W_n(\cos\theta) = \frac{\sin((n + \tfrac{1}{2})\theta)} {\sin(\theta/2)},

with the recurrence

W_0(x) = 1, \quad W_1(x) = 2x + 1, \quad W_{k+1}(x) = 2x\, W_k(x) - W_{k-1}(x).

Symmetry relation: $W_n(-x) = (-1)^n V_n(x)$ . Raises ValueError for n < 0.

Examples

>>> import lucid
>>> from lucid.special import chebyshev_polynomial_w
>>> chebyshev_polynomial_w(lucid.tensor([-1.0, 0.0, 1.0]), n=2)
Tensor([1.0000, -1.0000, 3.0000])