# Fungrim entry: d1b3b5

$U^{*}\!\left(a, b, z\right) = \sum_{k=0}^{n - 1} \frac{\left(a\right)_{k} \left(a - b + 1\right)_{k}}{k ! {\left(-z\right)}^{k}} + R_{n}\!\left(a,b,z\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
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
HypergeometricUStar$U^{*}\!\left(a, b, z\right)$ Scaled Tricomi confluent hypergeometric function
Sum$\sum_{n} f(n)$ Sum
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Factorial$n !$ Factorial
Pow${a}^{b}$ Power
HypergeometricUStarRemainder$R_{n}\!\left(a,b,z\right)$ Error term in asymptotic expansion of Tricomi confluent hypergeometric function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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), NotEqual(z, 0), Element(n, ZZGreaterEqual(0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC