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Fungrim entry: 65693e

2F1 ⁣(a,b,n,z)=(a)n+1(b)n+1zn+1(n+1)!2F1 ⁣(a+n+1,b+n+1,n+2,z)\,{}_2{\textbf F}_1\!\left(a, b, -n, z\right) = \frac{\left(a\right)_{n + 1} \left(b\right)_{n + 1} {z}^{n + 1}}{\left(n + 1\right)!} \,{}_2F_1\!\left(a + n + 1, b + n + 1, n + 2, z\right)
Assumptions:aCandbCandnZ0andzC{1}a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{1\right\}
TeX:
\,{}_2{\textbf F}_1\!\left(a, b, -n, z\right) = \frac{\left(a\right)_{n + 1} \left(b\right)_{n + 1} {z}^{n + 1}}{\left(n + 1\right)!} \,{}_2F_1\!\left(a + n + 1, b + n + 1, n + 2, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{1\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
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Powab{a}^{b} Power
Factorialn!n ! Factorial
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("65693e"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, Neg(n), z), Mul(Div(Mul(Mul(RisingFactorial(a, Add(n, 1)), RisingFactorial(b, Add(n, 1))), Pow(z, Add(n, 1))), Factorial(Add(n, 1))), Hypergeometric2F1(Add(Add(a, n), 1), Add(Add(b, n), 1), Add(n, 2), z)))),
    Variables(a, b, n, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(1))))))

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

2019-08-19 14:38:23.809000 UTC