$R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[0, z_{2}, z_{3}, \ldots, z_{n}\right]\right) = \frac{\mathrm{B}\!\left(a, c - b_{1}\right)}{\mathrm{B}\!\left(a, c\right)} R_{-a}\!\left(\left[b_{2}, b_{3}, \ldots, b_{n}\right], \left[z_{2}, z_{3}, \ldots, z_{n}\right]\right)\; \text{ where } c = -a + \sum_{k=1}^{n} b_{k}$
Assumptions:$a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(b_{k} \in \mathbb{R} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\text{ for all } k \in \{2, 3, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > 0 \;\mathbin{\operatorname{and}}\; \sum_{k=2}^{n} b_{k} > a$
References:
• https://dlmf.nist.gov/19.16
TeX:
R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[0, z_{2}, z_{3}, \ldots, z_{n}\right]\right) = \frac{\mathrm{B}\!\left(a, c - b_{1}\right)}{\mathrm{B}\!\left(a, c\right)} R_{-a}\!\left(\left[b_{2}, b_{3}, \ldots, b_{n}\right], \left[z_{2}, z_{3}, \ldots, z_{n}\right]\right)\; \text{ where } c = -a + \sum_{k=1}^{n} b_{k}

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} \;\text{ for all } k \in \{2, 3, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > 0 \;\mathbin{\operatorname{and}}\; \sum_{k=2}^{n} b_{k} > a
Definitions:
Fungrim symbol Notation Short description
CarlsonHypergeometricR$R_{-a}\!\left(b, z\right)$ Carlson multivariate hypergeometric function
BetaFunction$\mathrm{B}\!\left(a, b\right)$ Beta function
Sum$\sum_{n} f(n)$ Sum
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
Source code for this entry:
Entry(ID("cbcad9"),
Formula(Equal(CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 1, n)), List(0, Step(z_(k), For(k, 2, n)))), Where(Mul(Div(BetaFunction(a, Sub(c, b_(1))), BetaFunction(a, c)), CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 2, n)), List(z_(k), For(k, 2, n)))), Def(c, Add(Neg(a), Sum(b_(k), For(k, 1, n))))))),
Variables(a, b_, z_, n),
Assumptions(And(Element(a, RR), Element(n, ZZGreaterEqual(1)), All(Element(b_(k), RR), ForElement(k, Range(1, n))), All(Element(z_(k), CC), ForElement(k, Range(2, n))), Greater(Sum(b_(k), For(k, 1, n)), 0), Greater(Sum(b_(k), For(k, 2, n)), a))),
References("https://dlmf.nist.gov/19.16"))

## 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