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Fungrim entry: 504717

2F1 ⁣(a,b,c,z)=(1z)b2F1 ⁣(ca,b,c,zz1)\,{}_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:aC  and  bC  and  cC  and  zC  and  z[1,)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
Hypergeometric2F1Regularized2F1 ⁣(a,b,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) Regularized Gauss hypergeometric function
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ClosedOpenInterval[a,b)\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)))))

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2021-03-15 19:12:00.328586 UTC