Module special_functions 

Special functions (Gap analysis §3.5) — beyond the Gamma/erf/incomplete family already in crate::solvers::statistics::distributions::special.

• orthogonal — Legendre, Chebyshev (T/U), Hermite, Laguerre polynomials by their three-term recurrences.
• bessel — Bessel J/Y and modified I/K (integer order): convergent series for the order-0 building blocks, the Wronskian for order 1, then the standard recurrences.
• airy — Airy Ai/Bi via their Maclaurin series.
• [zeta] — Riemann ζ(s) for real s > 1 via Euler–Maclaurin acceleration.

Domain-restricted functions fail closed (Option/None) rather than return a fabricated value (e.g. Y_n/K_n require x > 0; ζ requires s > 1). The series methods are accurate for moderate arguments; the convergence regime is documented per function.

Re-exports

pub use airy::airy_ai;

pub use airy::airy_bi;

pub use bessel::bessel_i;

pub use bessel::bessel_j;

pub use bessel::bessel_k;

pub use bessel::bessel_y;

pub use orthogonal::chebyshev_t;

pub use orthogonal::chebyshev_u;

pub use orthogonal::hermite;

pub use orthogonal::laguerre;

pub use orthogonal::legendre;

pub use zeta::zeta;

Modules

airy

Airy functions Ai(x) and Bi(x) via their Maclaurin series. Accurate for moderate |x| (the series converge for all x but lose digits for large argument).

bessel

Bessel functions of integer order: J_n, Y_n (first/second kind) and the modified I_n, K_n. The order-0 functions come from their convergent power series (with the log + harmonic-number terms for the second kinds); order 1 from the Wronskian relations; higher orders from the standard upward recurrences. Accurate for moderate |x| (the series converge for all x but lose digits for large argument — documented).

orthogonal

Classical orthogonal polynomials by their three-term recurrences. Each evaluates P_n(x) in O(n) with no allocation.

zeta

Riemann zeta function ζ(s) for real s > 1, via Euler–Maclaurin acceleration: sum the first N−1 terms directly, then add the integral tail and Bernoulli corrections. None for s ≤ 1 (the series there needs analytic continuation — out of this function’s honest domain).