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Fungrim entry: ad8db2

2F1 ⁣(a,b,c,z)=k=0(a)k(b)k(c)kzkk!\,{}_2F_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}
Assumptions:aCandbCandcC{0,1,}andzCand(z<1ora{0,1,}orb{0,1,})a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left|z\right| < 1 \,\mathbin{\operatorname{or}}\, a \in \{0, -1, \ldots\} \,\mathbin{\operatorname{or}}\, b \in \{0, -1, \ldots\}\right)
TeX:
\,{}_2F_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left|z\right| < 1 \,\mathbin{\operatorname{or}}\, a \in \{0, -1, \ldots\} \,\mathbin{\operatorname{or}}\, b \in \{0, -1, \ldots\}\right)
Definitions:
Fungrim symbol Notation Short description
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("ad8db2"),
    Formula(Equal(Hypergeometric2F1(a, b, c, z), Sum(Mul(Div(Mul(RisingFactorial(a, k), RisingFactorial(b, k)), RisingFactorial(c, k)), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, Infinity)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, CC), Or(Less(Abs(z), 1), Element(a, ZZLessEqual(0)), Element(b, ZZLessEqual(0))))))

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

2019-09-15 14:14:26.267625 UTC