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Fungrim entry: d1b3b5

U ⁣(a,b,z)=k=0n1(a)k(ab+1)kk!(z)k+Rn ⁣(a,b,z)U^{*}\!\left(a, b, z\right) = \sum_{k=0}^{n - 1} \frac{\left(a\right)_{k} \left(a - b + 1\right)_{k}}{k ! {\left(-z\right)}^{k}} + R_{n}\!\left(a,b,z\right)
Assumptions:aCandbCandzCandz0andnZ0a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}
TeX:
U^{*}\!\left(a, b, z\right) = \sum_{k=0}^{n - 1} \frac{\left(a\right)_{k} \left(a - b + 1\right)_{k}}{k ! {\left(-z\right)}^{k}} + R_{n}\!\left(a,b,z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
Sumnf(n)\sum_{n} f(n) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Factorialn!n ! Factorial
Powab{a}^{b} Power
HypergeometricUStarRemainderRn ⁣(a,b,z)R_{n}\!\left(a,b,z\right) Error term in asymptotic expansion of Tricomi confluent hypergeometric function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("d1b3b5"),
    Formula(Equal(HypergeometricUStar(a, b, z), Add(Sum(Div(Mul(RisingFactorial(a, k), RisingFactorial(Add(Sub(a, b), 1), k)), Mul(Factorial(k), Pow(Neg(z), k))), For(k, 0, Sub(n, 1))), HypergeometricUStarRemainder(n, a, b, z)))),
    Variables(a, b, z, n),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0), Element(n, ZZGreaterEqual(0)))))

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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