/**
 * Polygamma functions and classical orthogonal polynomials.
 *
 * `polygamma(n, x)` is the n-th derivative of the digamma function,
 * ψ^(n)(x) = d^n/dx^n ψ(x). It is computed by shifting the argument up via
 * the standard recurrence
 *
 *   ψ^(n)(x) = ψ^(n)(x + m) + (-1)^(n+1) n! * sum_{k=0}^{m-1} 1/(x+k)^(n+1)
 *
 * until x + m is large enough (>= SHIFT_THRESHOLD) for the asymptotic
 * (Bernoulli) expansion (DLMF 5.15.8) to converge to machine precision:
 *
 *   psi^(n)(X) ~ (-1)^(n-1) * [ (n-1)!/X^n + n!/(2 X^(n+1))
 *                 + sum_j B_{2j} * (2j+n-1)! / ((2j)! * X^(2j+n)) ]
 *
 * Since (-1)^(n+1) = (-1)^(n-1), both pieces share the same overall sign.
 *
 * `jacobiP` and `gegenbauerC` are evaluated with their standard stable
 * three-term recurrences (DLMF 18.9.2 and 18.9.1 respectively), matching the
 * pattern already used for `chebyshevT` / `hermiteH` / `laguerreL` /
 * `legendreP`.
 *
 * @packageDocumentation
 */
/**
 * Polygamma function ψ^(n)(x): the n-th derivative of the digamma function.
 *
 * `polygamma(0, x)` delegates to `digamma(x)`. For `n >= 1`, x is shifted up
 * via the standard recurrence until it is large enough for the Bernoulli
 * asymptotic expansion to converge.
 *
 * @param n - Derivative order (nonnegative integer)
 * @param x - Argument (must not be a nonpositive integer, where ψ^(n) has poles)
 * @returns ψ^(n)(x)
 *
 * @example
 * polygamma(1, 2) // ~0.6449340668 (trigamma(2))
 * polygamma(2, 1) // ~-2.4041138063
 */
export declare function polygamma(n: number, x: number): number;
/**
 * Trigamma function ψ'(x) = ψ^(1)(x): the first derivative of the digamma
 * function. Equivalent to `polygamma(1, x)`.
 *
 * @param x - Argument
 * @returns ψ'(x)
 *
 * @example
 * trigamma(2) // ~0.6449340668 (= zeta(2) - 1 = pi^2/6 - 1)
 */
export declare function trigamma(x: number): number;
/**
 * Jacobi polynomial P_n^(alpha,beta)(x), evaluated via the standard
 * three-term recurrence (DLMF 18.9.2).
 *
 * @param n - Degree (nonnegative integer)
 * @param alpha - Parameter alpha (> -1)
 * @param beta - Parameter beta (> -1)
 * @param x - Evaluation point
 * @returns P_n^(alpha,beta)(x)
 *
 * @example
 * jacobiP(2, 1, 1, 0.5) // 0.1875
 */
export declare function jacobiP(n: number, alpha: number, beta: number, x: number): number;
/**
 * Gegenbauer (ultraspherical) polynomial C_n^(alpha)(x), evaluated via the
 * standard three-term recurrence (DLMF 18.9.1).
 *
 * @param n - Degree (nonnegative integer)
 * @param alpha - Parameter alpha (> -1/2, alpha != 0)
 * @param x - Evaluation point
 * @returns C_n^(alpha)(x)
 *
 * @example
 * gegenbauerC(2, 1, 1) // 3 (C_2^(1)(x) = 4x^2 - 1)
 */
export declare function gegenbauerC(n: number, alpha: number, x: number): number;
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