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Fungrim entry: 70111e

1F1 ⁣(a,b,z)=k=0(a)kΓ ⁣(b+k)zkk!\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\Gamma\!\left(b + k\right)} \frac{{z}^{k}}{k !}
Assumptions:aCandbCandzCa \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
TeX:
\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\Gamma\!\left(b + k\right)} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Hypergeometric1F1Regularized1F1 ⁣(a,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) Regularized Kummer confluent hypergeometric function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("70111e"),
    Formula(Equal(Hypergeometric1F1Regularized(a, b, z), Sum(Mul(Div(RisingFactorial(a, k), GammaFunction(Add(b, k))), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, Infinity)))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), 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.

2019-08-21 11:44:15.926409 UTC