Fungrim entry: a1f7ea

R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \frac{1}{\mathrm{B}\!\left(a, c\right)} \int_{0}^{\infty} {t}^{c - 1} \prod_{k=1}^{n} {\left(t + z_{k}\right)}^{-b_{k}} \, dt\; \text{ where } c = -a + \sum_{j=1}^{n} b_{j}

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

TeX:

R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \frac{1}{\mathrm{B}\!\left(a, c\right)} \int_{0}^{\infty} {t}^{c - 1} \prod_{k=1}^{n} {\left(t + z_{k}\right)}^{-b_{k}} \, dt\; \text{ where } c = -a + \sum_{j=1}^{n} b_{j}

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

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
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Product`	$\prod_{n} f(n)$	Product
`Infinity`	$\infty$	Positive infinity
`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
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("a1f7ea"),
    Formula(Equal(CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 1, n)), List(z_(k), For(k, 1, n))), Where(Mul(Div(1, BetaFunction(a, c)), Integral(Mul(Pow(t, Sub(c, 1)), Product(Pow(Add(t, z_(k)), Neg(b_(k))), For(k, 1, n))), For(t, 0, Infinity))), Def(c, Add(Neg(a), Sum(b_(j), For(j, 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), SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), ForElement(k, Range(1, n))), Greater(Sum(b_(k), For(k, 1, n)), a, 0))))

Topics using this entry

Series representations of Carlson symmetric elliptic integrals