chebyshev_polynomial_v

→Tensor

chebyshev_polynomial_v(x: Tensor, n: int)

source

Chebyshev polynomial of the third kind, $V_n(x)$ .

The V polynomials are orthogonal on $[-1, 1]$ with weight $\sqrt{(1 + x)/(1 - x)}$ and arise in problems with one Dirichlet and one Neumann boundary condition along the interval.

Parameters

xTensor

Argument tensor; any floating-point dtype.

nint

Non-negative polynomial degree.

Returns

Tensor

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

Notes

Half-angle definition

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

with the recurrence

V_0(x) = 1, \quad V_1(x) = 2x - 1, \quad V_{k+1}(x) = 2x\, V_k(x) - V_{k-1}(x).

Raises ValueError for n < 0.

Examples

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