# Fungrim entry: b25089

$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-a} \,{}_2{\textbf F}_1\!\left(a, c - b, c, \frac{z}{z - 1}\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)$
TeX:
\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-a} \,{}_2{\textbf F}_1\!\left(a, c - b, c, \frac{z}{z - 1}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
Hypergeometric2F1Regularized$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$ Regularized Gauss hypergeometric function
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("b25089"),
Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Mul(Pow(Sub(1, z), Neg(a)), Hypergeometric2F1Regularized(a, Sub(c, b), c, Div(z, Sub(z, 1)))))),
Variables(a, b, c, z),
Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, ClosedOpenInterval(1, Infinity)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC