Fungrim entry: b81ca0

R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\underbrace{z, \ldots, z}_{n \text{ times}}\right]\right) = {z}^{-a}

Assumptions:

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left(b_{k} \in \mathbb{R} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > a > 0 \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right]

TeX:

R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\underbrace{z, \ldots, z}_{n \text{ times}}\right]\right) = {z}^{-a}

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left(b_{k} \in \mathbb{R} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > a > 0 \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonHypergeometricR`	$R_{-a}\!\left(b, z\right)$	Carlson multivariate hypergeometric function
`Pow`	${a}^{b}$	Power
`RR`	$\mathbb{R}$	Real numbers
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`Sum`	$\sum_{n} f(n)$	Sum
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("b81ca0"),
    Formula(Equal(CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 1, n)), List(Repeat(z, n))), Pow(z, Neg(a)))),
    Variables(a, b_, n, z),
    Assumptions(And(Element(a, RR), All(Element(b_(k), RR), ForElement(k, Range(1, n))), Greater(Sum(b_(k), For(k, 1, n)), a, 0), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))

Topics using this entry

Series representations of Carlson symmetric elliptic integrals