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Fungrim entry: 2443de

Ra ⁣([b1,b2,,bn],[z1,z2,,zn])=znaN=0(a)N(c)NTN ⁣([b1,b2,,bn1],[1z1zn,1z2zn,,1zn1zn])   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) = z_{n}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n - 1}\right], \left[1 - \frac{z_{1}}{z_{n}}, 1 - \frac{z_{2}}{z_{n}}, \ldots, 1 - \frac{z_{n - 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  znC{0}  and  (zkC  and  1zkzn<1   for all k{1,2,,n1})  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}}\; z_{n} \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{z_{k}}{z_{n}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n - 1\}\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) = z_{n}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n - 1}\right], \left[1 - \frac{z_{1}}{z_{n}}, 1 - \frac{z_{2}}{z_{n}}, \ldots, 1 - \frac{z_{n - 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}}\; z_{n} \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{z_{k}}{z_{n}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n - 1\}\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
Powab{a}^{b} Power
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("2443de"),
    Formula(Where(Equal(CarlsonHypergeometricR(Neg(a), List(b_(k), For(k, 1, n)), List(z_(k), For(k, 1, n))), Mul(Pow(z_(n), Neg(a)), Sum(Mul(Div(RisingFactorial(a, N), RisingFactorial(c, N)), CarlsonHypergeometricT(N, List(b_(k), For(k, 1, Sub(n, 1))), List(Sub(1, Div(z_(k), z_(n))), For(k, 1, Sub(n, 1))))), 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))), Element(z_(n), SetMinus(CC, Set(0))), All(And(Element(z_(k), CC), Less(Abs(Sub(1, Div(z_(k), z_(n)))), 1)), ForElement(k, Range(1, Sub(n, 1)))), 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