Horn hypergeometric series

Summary

A Horn hypergeometric series is a double power series $\sum_{m, n} A_{m, n} x^m y^n$ with a special kind of coefficients. Moving one step in either index multiplies the coefficient by a rational function of $(m, n)$. This is the two-variable form of the defining property of the Gauss hypergeometric series, whose consecutive coefficients have a rational ratio. Horn classified the convergent series of this type of order two. His list has 34 essentially distinct series, including the four Appell series and their confluent forms.