# Fungrim entry: 2c1df7

$R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\lambda z_{1}, \lambda z_{2}, \ldots, \lambda z_{n}\right]\right) = {\lambda}^{-a} R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)$
Assumptions:$a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(b_{k} \in \mathbb{R} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > a > 0 \;\mathbin{\operatorname{and}}\; \lambda \in \left(0, \infty\right)$
TeX:
R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\lambda z_{1}, \lambda z_{2}, \ldots, \lambda z_{n}\right]\right) = {\lambda}^{-a} R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(b_{k} \in \mathbb{R} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > a > 0 \;\mathbin{\operatorname{and}}\; \lambda \in \left(0, \infty\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
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
Sum$\sum_{n} f(n)$ Sum
OpenInterval$\left(a, b\right)$ Open interval
Source code for this entry:
Entry(ID("2c1df7"),
Formula(Equal(CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 1, n)), List(Mul(lamda, z_(k)), For(k, 1, n))), Mul(Pow(lamda, Neg(a)), CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 1, n)), List(z_(k), For(k, 1, n)))))),
Variables(a, b_, z_, n, lamda),
Assumptions(And(Element(a, RR), Element(n, ZZGreaterEqual(1)), All(Element(b_(k), RR), ForElement(k, Range(1, n))), All(Element(z_(k), SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), ForElement(k, Range(1, n))), Greater(Sum(b_(k), For(k, 1, n)), a, 0), Element(lamda, OpenInterval(0, Infinity)))))

## 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