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Fungrim entry: 13f252

Ra ⁣([β,,βn times],[z1,z2,,zn])=AaRa ⁣([β,,βn times],[z1A,z2A,,znA])   where A=1nk=1nzkR_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[\frac{z_{1}}{A}, \frac{z_{2}}{A}, \ldots, \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}
Assumptions:aR  and  β(0,)  and  nZ1  and  (zkC  and  1nzkj=1nzj<1   for all k{1,2,,n})  and  (Re ⁣(zk)>0   for all k{1,2,,n})a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{n z_{k}}{\sum_{j=1}^{n} z_{j}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}\!\left(z_{k}\right) > 0 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)
TeX:
R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[\frac{z_{1}}{A}, \frac{z_{2}}{A}, \ldots, \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{n z_{k}}{\sum_{j=1}^{n} z_{j}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}\!\left(z_{k}\right) > 0 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonHypergeometricRRa ⁣(b,z)R_{-a}\!\left(b, z\right) Carlson multivariate hypergeometric function
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
RRR\mathbb{R} Real numbers
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("13f252"),
    Formula(Where(Equal(CarlsonHypergeometricR(Neg(a), List(Repeat(beta, n)), List(z_(k), For(k, 1, n))), Mul(Pow(A, Neg(a)), CarlsonHypergeometricR(Neg(a), List(Repeat(beta, n)), List(Div(z_(k), A), For(k, 1, n))))), Def(A, Mul(Div(1, n), Sum(z_(k), For(k, 1, n)))))),
    Variables(a, beta, z_, n),
    Assumptions(And(Element(a, RR), Element(beta, OpenInterval(0, Infinity)), Element(n, ZZGreaterEqual(1)), All(And(Element(z_(k), CC), Less(Abs(Sub(1, Div(Mul(n, z_(k)), Sum(z_(j), For(j, 1, n))))), 1)), ForElement(k, Range(1, n))), All(Greater(Re(z_(k)), 0), ForElement(k, Range(1, n))))))

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2020-08-27 09:56:25.682319 UTC