In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratio
of two terms is a rational function of n. The definition of the generalized hypergeometric series is similar, except that the terms with negative n must vanish; the bilateral series will in general have infinite numbers of non-zero terms for both positive and negative n.
The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function defined for most rational functions. There are several summation formulas giving its values for special values where it does converge.