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Fungrim entry: dec042

1F1 ⁣(n,b,z)=k=0n(n)k(b)kzkk!\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}
Assumptions:nZ0  and  bC  and  not(b{0,1,}  and  b>n)  and  zCn \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{not} \left(b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b > -n\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\;  \operatorname{not} \left(b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b > -n\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
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
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(Hypergeometric1F1(Neg(n), b, z), Sum(Mul(Div(RisingFactorial(Neg(n), k), RisingFactorial(b, k)), Div(Pow(z, k), Factorial(k))), For(k, 0, n)))),
    Variables(n, b, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(b, CC), Not(And(Element(b, ZZLessEqual(0)), Greater(b, Neg(n)))), 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