Gaussian hypergeometric series | Kummer's series

A generalized hypergeometric function _p F_q(a_1, ..., a_p ;b_1, ..., b_q ;x) is a function which can be defined in the form of 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)) x.

Appell hypergeometric function | Barnes' lemma | Bradley's theorem | Cayley's hypergeometric function theorem | Clausen formula | closed form | confluent hypergeometric function of the first kind | confluent hypergeometric function of the second kind | confluent hypergeometric limit function | contiguous function | Darling's products | generalized hypergeometric function | Gosper's algorithm | hypergeometric identity | hypergeometric series | Jacobi polynomial | Kummer's formulas | Kummer's quadratic transformation | Kummer's relation | Orr's theorem | Pfaff transformation | q-hypergeometric function | Ramanujan's hypergeometric identity | Saalschützian | Sister Celine's method | Zeilberger's algorithm