Appell series

Builds on Pochhammer symbol Hypergeometric series Gamma function

Definition

Let $a, a', b, b', c, c'$ be complex numbers, with every lower parameter $c, c'$ outside $\set{0, -1, -2, \dots}$ . Write $(x)_n$ for the Pochhammer symbol. The four Appell series are the double power series in two complex variables $x, y$ :

$F_1(a; b, b'; c; x, y) = \sum_{m, n \ge 0} \frac{(a)_{m + n} (b)_m (b')_n}{(c)_{m + n}\, m!\, n!} x^m y^n$ , convergent for $\abs{x} < 1$ and $\abs{y} < 1$ ;
$F_2(a; b, b'; c, c'; x, y) = \sum_{m, n \ge 0} \frac{(a)_{m + n} (b)_m (b')_n}{(c)_m (c')_n\, m!\, n!} x^m y^n$ , convergent for $\abs{x} + \abs{y} < 1$ ;
$F_3(a, a'; b, b'; c; x, y) = \sum_{m, n \ge 0} \frac{(a)_m (a')_n (b)_m (b')_n}{(c)_{m + n}\, m!\, n!} x^m y^n$ , convergent for $\abs{x} < 1$ and $\abs{y} < 1$ ;
$F_4(a; b; c, c'; x, y) = \sum_{m, n \ge 0} \frac{(a)_{m + n} (b)_{m + n}}{(c)_m (c')_n\, m!\, n!} x^m y^n$ , convergent for $\sqrt{\abs{x}} + \sqrt{\abs{y}} < 1$ .

Remarks

Setting $y = 0$ leaves the terms with $n = 0$ , and each series becomes a Gauss hypergeometric series in $x$ ; for instance $F_1(a; b, b'; c; x, 0) = {}_2F_1(a, b; c; x)$ . Example: with $b = c$ and $b' = c'$ , the series $F_2$ becomes $\sum_{m, n} \frac{(a)_{m + n}}{m!\, n!} x^m y^n = \sum_{N \ge 0} \frac{(a)_N}{N!} (x + y)^N = (1 - x - y)^{-a}$ , by the binomial theorem; this converges when $\abs{x} + \abs{y} < 1$ . Non-example: for $a = 1$ the same series at $x = y = \frac12$ is $\sum_N 1$ , which diverges; the point lies on the boundary $\abs{x} + \abs{y} = 1$ of the region of convergence. For $\operatorname{Re} c > \operatorname{Re} a > 0$ , with $\Gamma$ the gamma function, $F_1$ has Picard's integral representation $F_1(a; b, b'; c; x, y) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c - a)} \int_0^1 t^{a - 1} (1 - t)^{c - a - 1} (1 - xt)^{-b} (1 - yt)^{-b'}\, dt.$

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Tag	`GFJ20FBYF8`
URI	`m2://GFJ20FBYF8`
Kind	definition
Writing	W0 stub
MSC	33C65
Defines	Appell series
Uses	Pochhammer symbol, Hypergeometric series, Gamma function

Appell series

Definition

Remarks

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