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Fungrim entry: 306ef7

2F1 ⁣(a,b,c,z)=k=0(a)k(b)kΓ ⁣(c+k)zkk!\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\Gamma\!\left(c + k\right)} \frac{{z}^{k}}{k !}
Assumptions:aC  and  bC  and  cC  and  zC  and  (z<1  or  a{0,1,}  or  b{0,1,}  or  c{0,1,})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}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; c \in \{0, -1, \ldots\}\right)
\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\Gamma\!\left(c + k\right)} \frac{{z}^{k}}{k !}

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}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; c \in \{0, -1, \ldots\}\right)
Fungrim symbol Notation Short description
Hypergeometric2F1Regularized2F1 ⁣(a,b,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) Regularized Gauss hypergeometric function
Sumnf(n)\sum_{n} f(n) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
GammaΓ(z)\Gamma(z) Gamma function
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Sum(Mul(Div(Mul(RisingFactorial(a, k), RisingFactorial(b, k)), Gamma(Add(c, k))), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), Or(Less(Abs(z), 1), Element(a, ZZLessEqual(0)), Element(b, ZZLessEqual(0)), Element(c, 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.

2021-03-15 19:12:00.328586 UTC