---
tag: 2P0V6X8GZ3
kind: definition
name: Horn hypergeometric series
msc: [33C65, 33C70]
defines: X24AGT6TK9
uses: [428M453CR2, 033WGN00BF]
sources:
  - url: https://en.wikipedia.org/wiki/Horn_function
---

## Definition

Let $(A_{m, n})_{m, n \ge 0}$ be complex numbers with $A_{0, 0} = 1$.
Consider the [[formal power series|033WGN00BF]] $\sum_{m, n \ge 0} A_{m, n} x^m y^n$.
It is a *Horn hypergeometric series* if there are rational functions $f$ and $g$ of two variables such that for all $m, n \ge 0$:

1. $A_{m + 1, n} = f(m, n)\, A_{m, n}$;
2. $A_{m, n + 1} = g(m, n)\, A_{m, n}$.

Write $f = P / Q$ and $g = R / S$ with polynomials $P, Q, R, S$ in lowest terms.
The *order* of the series is the largest of the degrees of $P, Q, R, S$.

## Remarks

The two conditions mean that the ratios of neighbouring coefficients are rational in $(m, n)$, as for the Gauss [[hypergeometric series|428M453CR2]] in one variable.
Example: Appell's $F_1(a; b, b'; c; x, y)$ has $A_{m, n} = \frac{(a)_{m + n} (b)_m (b')_n}{(c)_{m + n}\, m!\, n!}$.
Then $f(m, n) = \frac{(a + m + n)(b + m)}{(c + m + n)(m + 1)}$ and $g(m, n) = \frac{(a + m + n)(b' + n)}{(c + m + n)(n + 1)}$, both rational, so $F_1$ is a Horn series of order $2$.
Non-example: $A_{m, n} = 2^{m^2}$ gives $A_{m + 1, n} / A_{m, n} = 2^{2m + 1}$, which is not a rational function of $m$, so the first clause fails.
Horn's list contains $34$ essentially distinct convergent series of order $2$: $14$ complete ones, among them the four Appell series, and $20$ confluent ones.