In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.

A hypergeometric series is formally defined as a power series

in which the ratio of successive coefficients is a rational function of n. That is,

where A(n) and B(n) are polynomials in n.

For example, in the case of the series for the exponential function,

we have:

So this satisfies the definition with A(n) = 1 and B(n) = n + 1.

It is customary to factor out the leading term, so β₀ is assumed to be 1. The polynomials can be factored into linear factors of the form (a_j + n) and (b_k + n) respectively, where the a_j and b_k are complex numbers.