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Fungrim entry: a61f01

1F1 ⁣(a,b,z)=k=0(a)k(b)kzkk!\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}
Assumptions:aCandbC{0,1,}andzCa \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
TeX:
\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Hypergeometric1F11F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric 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:
Entry(ID("a61f01"),
    Formula(Equal(Hypergeometric1F1(a, b, z), Sum(Mul(Div(RisingFactorial(a, k), RisingFactorial(b, k)), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, 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.

2019-10-05 13:11:19.856591 UTC