Tessero\SpecialSeq

legendreP(int $n, float $x, int $diffN = 0): array

Legendre polynomial P_n(x) and its derivatives up to order $diffN (0, 1 or 2).

legendrePAll(int $n, float $x, int $diffN = 0): array

P_k(x) and derivatives for every degree k = 0..n.

lqn(int $n, float $x): array

Legendre function of the second kind Q_k(x) and its derivative for k = 0..n (|x| < 1).

riccatiJn(int $n, float $x): array

Riccati-Bessel function of the first kind psi_k(x) = x * j_k(x) and its derivative, k = 0..n.

riccatiYn(int $n, float $x): array

Riccati-Bessel function of the second kind chi_k(x) = -x * y_k(x) and its derivative, k = 0..n.

assocLegendreP(int $n, int $m, float $x, int $diffN = 0, bool $norm = false): array

Associated Legendre polynomial (Ferrers function) P_n^m(x) and its x-derivatives up to $diffN (0-2), with the Condon-Shortley phase and branch_cut=2 (|x| < 1). If $norm, apply the normalisation factor sqrt((2n+1)(n-m)! / (2(n+m)!)).

assocLegendrePAll(int $n, int $m, float $x, int $diffN = 0, bool $norm = false): array

P_k^j(x) and derivatives for every degree k = 0..n and order j, laid out like scipy's assoc_legendre_p_all: shape [$diffN + 1][n + 1][2m + 1] with the order axis wrapped (index 0..m = orders 0..m, then m+1..2m = orders -m..-1).

sphLegendreP(int $n, int $m, float $theta, int $diffN = 0): array

Spherical (fully normalised) Legendre function Y_n^m(theta) without the azimuthal factor, and its theta-derivatives up to $diffN (0-2).

sphLegendrePAll(int $n, int $m, float $theta, int $diffN = 0): array

sph_legendre_p for every degree/order, laid out like scipy's sph_legendre_p_all.

sphHarmYAll(int $n, int $m, float $theta, float $phi): array

Spherical harmonics Y_n^m(theta, phi) for every degree 0..n and order, laid out like scipy's sph_harm_y_all: Y = sph_legendre_p(n, m, theta) * exp(i*m*phi). Each complex value is returned as a [re, im] pair; the result is [n + 1][2m + 1][2].

lqmn(int $m, int $n, float $x): array

Legendre function of the second kind Q_n^m(x) and its x-derivative, for all orders 0..m and degrees 0..n (|x| < 1). Returns [q, dq], each shaped [m + 1][n + 1] like scipy's lqmn.

stirling2(array|int $n, array|int $k): array|float

Stirling number(s) of the second kind via the floating-point dynamic program (matching scipy's exact=False for n <= 50). Scalars return a float; arrays broadcast element-wise.