The bilateral hypergeometric series pHp is defined by where is the rising factorial or Pochhammer symbol.
Suppose that none of the variables a or b are integers, so that all the terms of the series are finite and non-zero.
When |z|=1, the series converges if The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of several variables whose singularities are branch points at z = 0 and z=1 and simple poles at ai = −1, −2,... and bi = 0, 1, 2, ...
Similarly the terms with n negative converge for |z| >1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, so can also be continued to a multivalued function with these points as branch points.
The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of z other than 0 and 1, and satisfies a linear differential equation in z similar to the hypergeometric differential equation.