Hypergeometric

Hypergeometric refers to a broad class of special functions and distributions built from hypergeometric series and differential equations. The most common examples are the Gauss hypergeometric function 2F1 and the generalized hypergeometric functions pFq. These functions arise in many areas of mathematics and physics and include numerous classical functions as special cases.

The Gauss hypergeometric function 2F1(a,b;c;z) is defined by the series, for |z| < 1, 2F1(a,b;c;z) = sum_{n=0}^∞ (a)_n (b)_n / (c)_n * z^n / n!, where (q)_n is the rising factorial. Generalized hypergeometric functions pFq extend this form with p upper parameters a1,...,ap and q lower parameters b1,...,bq: pFq(a1,...,ap; b1,...,bq; z) = sum_{n=0}^∞ (a1)_n ... (ap)_n / (b1)_n ... (bq)_n * z^n / n!. These functions encompass many elementary and special functions and admit analytic continuations beyond their initial radius of convergence.

The hypergeometric differential equation z(1 − z) y'' + [c − (a + b + 1) z] y' − a b y = 0 has, in general, two linearly independent solutions that can be expressed in terms of hypergeometric functions. In particular, solutions include 2F1(a,b;c;z) and z^{1−c} 2F1(a−c+1, b−c+1; 2−c; z).

Hypergeometric distribution describes sampling without replacement. If a population of size N contains K successes and n items are drawn, the probability of observing 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).

Historically, the term derives from work by Euler and Gauss, with hypergeometric functions later appearing across analysis, number theory, and physics. The concept also connects to many areas through the study of differential equations, special functions, and combinatorial identities.