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Fungrim entry: 0a7f30

TN ⁣([β,,βn times],[z1,z2,,zn])=(m1,m2,,mn)(Z0)nk=1nkmk=N(1)M+N(β)Mk=1nekmk ⁣([z1,z2,,zn])(mk)!   where M=k=1nmkT_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{\textstyle{\left(m_{1}, m_{2}, \ldots, m_{n}\right) \in {\left(\mathbb{Z}_{\ge 0}\right)}^{n} \atop \sum_{k=1}^{n} k m_{k} = N}} {\left(-1\right)}^{M + N} \left(\beta\right)_{M} \prod_{k=1}^{n} \frac{e_{k}^{m_{k}}\!\left(\left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)}{\left(m_{k}\right)!}\; \text{ where } M = \sum_{k=1}^{n} m_{k}
Assumptions:nZ1  and  β(0,)  and  NZ0  and  (zkC   for all k{1,2,,n})n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)
TeX:
T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{\textstyle{\left(m_{1}, m_{2}, \ldots, m_{n}\right) \in {\left(\mathbb{Z}_{\ge 0}\right)}^{n} \atop \sum_{k=1}^{n} k m_{k} = N}} {\left(-1\right)}^{M + N} \left(\beta\right)_{M} \prod_{k=1}^{n} \frac{e_{k}^{m_{k}}\!\left(\left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)}{\left(m_{k}\right)!}\; \text{ where } M = \sum_{k=1}^{n} m_{k}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonHypergeometricTTN ⁣(b,z)T_{N}\!\left(b, z\right) Term in expansion of Carlson multivariate hypergeometric function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Productnf(n)\prod_{n} f(n) Product
Factorialn!n ! Factorial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Source code for this entry:
Entry(ID("0a7f30"),
    Formula(Equal(CarlsonHypergeometricT(N, List(Repeat(beta, n)), List(z_(k), For(k, 1, n))), Sum(Where(Mul(Mul(Pow(-1, Add(M, N)), RisingFactorial(beta, M)), Product(Div(Pow(SymmetricPolynomial(k, List(z_(k), For(k, 1, n))), m_(k)), Factorial(m_(k))), For(k, 1, n))), Def(M, Sum(m_(k), For(k, 1, n)))), ForElement(Tuple(m_(k), For(k, 1, n)), CartesianPower(Parentheses(ZZGreaterEqual(0)), n)), Equal(Sum(Mul(k, m_(k)), For(k, 1, n)), N)))),
    Variables(n, beta, N, z_),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(beta, OpenInterval(0, Infinity)), Element(N, ZZGreaterEqual(0)), All(Element(z_(k), CC), ForElement(k, Range(1, n))))))

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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