hypergeometriseen

Hypergeometric describes a family of mathematical objects linked to hypergeometric functions and hypergeometric distributions. In mathematics, the generalized hypergeometric function pFq is defined by a power series, pFq(a1,...,ap; b1,...,bq; z) = sum_{n=0}^∞ [(a1)_n ... (ap)_n / (b1)_n ... (bq)_n] z^n / n!, where (q)_n denotes the rising factorial. The case p=2, q=1 yields the Gauss hypergeometric function 2F1(a,b;c;z). Hypergeometric functions satisfy the linear differential equation z(1−z) f'' + [c − (a+b+1)z] f' − a b f = 0 and can be analytically continued beyond |z|<1. They encode many special functions and appear in algebra, number theory, combinatorics, and physics. Generalized hypergeometric functions with p and q arbitrary extend this framework and include many classical functions as particular cases. Special values at z=1 are governed by summation formulas such as Gauss's theorem.

Hypergeometric distributions model sampling without replacement. If a population contains K successes in N items, and n items are drawn without replacement, the probability of exactly k successes is [C(K,k) C(N−K,n−k)] / C(N,n). The mean is nK/N and the variance is n(K/N)(1−K/N) (N−n)/(N−1).

Hypergeometric functions also underpin many families of orthogonal polynomials and appear in quantum mechanics, statistics, and computational mathematics. The term hypergeometric thus denotes both a family of special functions and their probabilistic counterpart, linked by combinatorial and analytic structures.