Fungrim entry: 0a7f30

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}

Assumptions:

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
`CarlsonHypergeometricT`	$T_{N}\!\left(b, z\right)$	Term in expansion of Carlson multivariate hypergeometric function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Product`	$\prod_{n} f(n)$	Product
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Range`	$\{a, a + 1, \ldots, b\}$	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))))))

Topics using this entry

Series representations of Carlson symmetric elliptic integrals