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Fungrim entry: 4c41ad

0F1 ⁣(a,z)=k=01(a)kzkk!\,{}_0F_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\left(a\right)_{k}} \frac{{z}^{k}}{k !}
Assumptions:aC{0,1,}  and  zCa \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
\,{}_0F_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\left(a\right)_{k}} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol Notation Short description
Hypergeometric0F10F1 ⁣(a,z)\,{}_0F_1\!\left(a, z\right) Confluent hypergeometric limit function
Sumnf(n)\sum_{n} f(n) 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
Source code for this entry:
    Formula(Equal(Hypergeometric0F1(a, z), Sum(Mul(Div(1, RisingFactorial(a, k)), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
    Variables(a, z),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(z, CC))))

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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