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

Confluent hypergeometric functions