The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written c_(k + 1)/c_k = (P(k))/(Q(k)) = ((k + a_1)(k + a_2)...(k + a_p))/((k + b_1)(k + b_2)...(k + b_q)(k + 1)). (The factor of k + 1 in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written sum_(k = 0)^∞ c_k x^k | = | _p F_q[a_1, a_2, ..., a_p b_1, b_2, ..., b_q;x] | = | sum_(k = 0)^∞ ((a_1)_k (a_2)_k ...(a_p)_k)/((b_1)_k (b_2)_k ...(b_q)_k) x^k/(k!), where (a)_k is the Pochhammer symbol or rising factorial (a)_k congruent (Γ(a + k))/(Γ(a)) = a(a + 1)...(a + k - 1).

Carlson's theorem | Clausen formula | confluent hypergeometric function of the first kind | confluent hypergeometric limit function | Dixon's theorem | Dougall-Ramanujan identity | Dougall's theorem | generalized hypergeometric differential equation | Gosper's algorithm | hypergeometric function | hypergeometric identity | hypergeometric series | Jackson's identity | Kampé de Fériet function | k-balanced | Kummer's theorem | Lauricella functions | nearly-poised | q-hypergeometric function | Ramanujan's hypergeometric identity | Saalschützian | Saalschütz's theorem | Sister Celine's method | Slater's formula | Thomae's theorem | Watson's theorem | well-poised | Whipple's identity | Whipple's transformation | Wilf-Zeilberger pair | Zeilberger's algorithm