Hypergeometric series

In mathematics, the term hypergeometric series, first used by John Wallis (Arithmetica Infinitorum, 1655), means a series such that the ratio of two successive terms is a simple function of the index, such as:

• Appell series, a 2-variable generalization of hypergeometric series
• Basic hypergeometric series where the ratio of terms is a periodic function of the index
• Bilateral hypergeometric series _pH_p are similar to generalized hypergeometric series, but summed over all integers
• Binomial series ₁F₀
• Confluent hypergeometric series ₁F₁(a;c;z)
• Elliptic hypergeometric series where the ratio of terms is an elliptic function of the index
• Euler hypergeometric integral, an integral representation of ₂F₁
• Gaussian hypergeometric series ₂F₁(a,b;c;z)
• General hypergeometric function introduced by Gelfand.
• Generalized hypergeometric series _pF_q where the ratio of terms is a rational function of the index
• Geometric series, where the ratio of terms is a constant
• Horn function, 34 distinct convergent hypergeometric series of order two
• Humbert series 7 hypergeometric functions of 2 variables
• Hypergeometric differential equation, a second-order linear ordinary differential equation
• Hypergeometric distribution, a discrete probability distribution
• Hypergeometric function of a matrix argument, the multivariate generalization of the hypergeometric series
• Kampé de Fériet function, hypergeometric series of two variables
• Lauricella hypergeometric series, hypergeometric series of three variables
• Modular hypergeometric series, a terminating form of the elliptic hypergeometric series
• Theta hypergeometric series A special sort of elliptic hypergeometric series