Fungrim home page

Fungrim entry: a1f7ea

Ra ⁣([b1,b2,,bn],[z1,z2,,zn])=1B ⁣(a,c)0tc1k=1n(t+zk)bkdt   where c=a+j=1nbjR_{-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:aR  and  nZ1  and  (bkR   for all k{1,2,,n})  and  (zkC(,0]   for all k{1,2,,n})  and  k=1nbk>a>0a \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
CarlsonHypergeometricRRa ⁣(b,z)R_{-a}\!\left(b, z\right) Carlson multivariate hypergeometric function
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Productnf(n)\prod_{n} f(n) Product
Infinity\infty Positive infinity
Sumnf(n)\sum_{n} f(n) Sum
RRR\mathbb{R} Real numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC