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Fungrim entry: 651a4a

2F1 ⁣(a,b,c,z)=(1z)cab2F1 ⁣(ca,cb,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{c - a - b} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c, z\right)
Assumptions:aC  and  bC  and  cC  and  zC  and  z1a \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 \ne 1
TeX:
\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{c - a - b} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c, z\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 \ne 1
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
Source code for this entry:
Entry(ID("651a4a"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Mul(Pow(Sub(1, z), Sub(Sub(c, a), b)), Hypergeometric2F1Regularized(Sub(c, a), Sub(c, b), c, z)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotEqual(z, 1))))

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2020-04-08 16:14:44.404316 UTC