Fungrim entry: 504717

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-b} \,{}_2{\textbf F}_1\!\left(c - a, 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)}^{-b} \,{}_2{\textbf F}_1\!\left(c - a, 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("504717"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Mul(Pow(Sub(1, z), Neg(b)), Hypergeometric2F1Regularized(Sub(c, a), 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

Gauss hypergeometric function