This holds unless a nonpositive integer value of at least one of its parameters ap reduces the hypergeometric function to a finite polynomial, in which case the gamma prefactor of either G-function vanishes and the parameter sets of the G-functions violate the requirement "a""k" − "b""j" ≠ 1, 2, 3, ... for "k" = 1, 2, ..., "n" and "j" = 1, 2, ..., "m" from the definition above. Apart from this restriction, the relationship is valid whenever the generalized hypergeometric series "p""F""q"("z") converges, i. e. for any finite "z" when "p" ≤ "q", and for |"z"| < 1 when "p" = "q" + 1. In the latter case, the relation with the G-function automatically provides the analytic continuation of "p""F""q"("z") to |"z"| ≥ 1 with a branch cut from 1 to ∞ along the real axis. Finally, the relation furnishes a natural extension of the definition of the hypergeometric function to orders "p" > "q" + 1. By means of the G-function we can thus solve the generalized hypergeometric differential equation for "p" > "q" + 1 as well.