Fungrim entry: cbcad9

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

Series representations of Carlson symmetric elliptic integrals