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Fungrim entry: 4cb707

Ra ⁣([b1,b2,,bn],[z1,z2,,zn])=N=0(a)N(c)NTN ⁣([b1,b2,,bn],[1z1,1z2,,1zn])   where c=k=1nbkR_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[1 - z_{1}, 1 - z_{2}, \ldots, 1 - z_{n}\right]\right)\; \text{ where } c = \sum_{k=1}^{n} b_{k}
Assumptions:aR  and  nZ1  and  (bkR   for all k{1,2,,n})  and  (zkC  and  1zk<1   for all k{1,2,,n})  and  k=1nbk>0  and  (Re ⁣(zk)>0   for all k{1,2,,n})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} \;\mathbin{\operatorname{and}}\; \left|1 - z_{k}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > 0 \;\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[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[1 - z_{1}, 1 - z_{2}, \ldots, 1 - z_{n}\right]\right)\; \text{ where } c = \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} \;\mathbin{\operatorname{and}}\; \left|1 - z_{k}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \sum_{k=1}^{n} b_{k} > 0 \;\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
Sumnf(n)\sum_{n} f(n) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
CarlsonHypergeometricTTN ⁣(b,z)T_{N}\!\left(b, z\right) Term in expansion of Carlson multivariate hypergeometric function
Infinity\infty Positive infinity
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
Absz\left|z\right| Absolute value
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("4cb707"),
    Formula(Where(Equal(CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 1, n)), List(z_(k), For(k, 1, n))), Sum(Mul(Div(RisingFactorial(a, N), RisingFactorial(c, N)), CarlsonHypergeometricT(N, List(b_(k), For(k, 1, n)), List(Sub(1, z_(k)), For(k, 1, n)))), For(N, 0, Infinity))), Def(c, 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(And(Element(z_(k), CC), Less(Abs(Sub(1, z_(k))), 1)), ForElement(k, Range(1, n))), Greater(Sum(b_(k), For(k, 1, n)), 0), All(Greater(Re(z_(k)), 0), ForElement(k, Range(1, n))))))

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2021-03-15 19:12:00.328586 UTC