Concept

# Hypergeometric function

Summary
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by and . There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic. The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by . Studies in the nineteenth century included those of , and the fundamental characterisation by of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities. The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list). The hypergeometric function is defined for z < 1 by the power series It is undefined (or infinite) if c equals a non-positive integer. Here (q)n is the (rising) Pochhammer symbol, which is defined by: The series terminates if either a or b is a nonpositive integer, in which case the function reduces to a polynomial: For complex arguments z with ≥ 1 it can be analytically continued along any path in the complex plane that avoids the branch points 1 and infinity.