/**
 * Numerical special functions for exact, seekable gamma/beta sampling.
 *
 * gamma and beta have no closed-form inverse CDF, but their CDFs — the
 * regularized lower incomplete gamma `P(a,x)` and regularized incomplete beta
 * `I_x(a,b)` — are computable, and inverting them numerically (bisection) gives
 * an EXACT inverse-CDF sampler that consumes exactly ONE uniform. That keeps
 * gamma/beta deterministic and seekable like every other distribution — no
 * rejection sampling (variable draws), no approximation.
 *
 * Algorithms follow the standard series / continued-fraction forms (Numerical
 * Recipes). Hand-rolled — no dependency.
 */
/** Natural log of the gamma function (Lanczos approximation). */
export declare function lgamma(z: number): number;
/** Regularized lower incomplete gamma `P(a,x)` — the CDF of gamma(a, 1) at x. */
export declare function gammaP(a: number, x: number): number;
/** Regularized incomplete beta `I_x(a,b)` — the CDF of beta(a,b) at x. */
export declare function betaI(x: number, a: number, b: number): number;
/** Inverse of `gammaP(a, ·)`: the x ≥ 0 with `P(a,x) = u`, for u in (0,1). */
export declare function gammaPInv(a: number, u: number): number;
/** Inverse of `betaI(·, a, b)`: the x in (0,1) with `I_x(a,b) = u`. */
export declare function betaIInv(a: number, b: number, u: number): number;
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