/**
 * Prolate spheroidal wave functions — angular characteristic values
 * `λ_mn(c)` and the angular function `S_mn(c, η)` via the associated-Legendre
 * expansion (Flammer / Hodge). The pentadiagonal recurrence in the expansion
 * coefficients reduces to a symmetric tridiagonal eigenproblem on each
 * parity chain (even/odd `n−m`); we reuse matrix `eig`.
 *
 * As `c → 0`, `λ_mn → n(n+1)` and `S_mn(c, η) → P_n^m(η)` (Ferrers).
 * Oracle: `c = 0` closed form + finite-difference residual of the angular
 * ODE; scipy's `pro_cv` / `pro_ang1` when available in the gap suite.
 *
 * @packageDocumentation
 */
type f64 = number;
/**
 * Ferrers associated Legendre `P_n^m(x)` on `[-1, 1]` via the standard
 * recurrence (stable for the modest `n` used by the expansion).
 */
export declare function ferrersP(n: number, m: number, x: f64): f64;
/**
 * Angular characteristic value `λ_mn(c)` of the prolate spheroidal equation.
 * `λ_mn(0) = n(n+1)`.
 */
export declare function spheroidalLambda(m: number, n: number, c: f64): f64;
/** Alias matching the DLMF `λ_mn(c)` notation. */
export declare const spheroidalCharacteristic: typeof spheroidalLambda;
/**
 * Prolate angular spheroidal function `S_mn(c, η)` on `η ∈ [-1, 1]`.
 * `S_mn(0, η) = P_n^m(η)`.
 */
export declare function spheroidalAngular(m: number, n: number, c: f64, eta: f64): f64;
/**
 * Prolate radial function of the first kind `R_mn^{(1)}(c, ξ)` for `ξ ≥ 1`,
 * obtained from the angular function by the joining relation
 * `R_mn^{(1)}(c, ξ) ∝ S_mn(c, ξ)` continued off `[-1,1]` via the same
 * associated-Legendre series (analytic continuation of Ferrers to `ξ > 1`
 * through the same recurrence — valid for the modest `n` used here).
 *
 * Scaled so `R_mn^{(1)}(c, ξ) ∼ j_n(c ξ)` as `ξ → ∞` is NOT enforced (that
 * needs the spherical-Bessel joining factor). This export is the angular
 * series evaluated at `ξ`, which satisfies the radial ODE in `ξ` after the
 * standard `η ↔ ξ` substitution and is the quantity the characteristic-value
 * tests pin.
 */
export declare function spheroidalRadial(m: number, n: number, c: f64, xi: f64): f64;
export {};
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