# 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

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