Fungrim entry: da47f6

T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\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} m_{k} = N}} \prod_{k=1}^{n} \frac{\left(b_{k}\right)_{m_{k}}}{\left(m_{k}\right)!} z_{k}^{m_{k}}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \left(b_{k} \in \mathbb{C} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)

TeX:

T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\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} m_{k} = N}} \prod_{k=1}^{n} \frac{\left(b_{k}\right)_{m_{k}}}{\left(m_{k}\right)!} z_{k}^{m_{k}}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \left(b_{k} \in \mathbb{C} \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\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
`Product`	$\prod_{n} f(n)$	Product
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("da47f6"),
    Formula(Equal(CarlsonHypergeometricT(N, List(b_(k), For(k, 1, n)), List(z_(k), For(k, 1, n))), Sum(Product(Mul(Div(RisingFactorial(b_(k), m_(k)), Factorial(m_(k))), Pow(z_(k), m_(k))), For(k, 1, n)), ForElement(Tuple(m_(k), For(k, 1, n)), CartesianPower(Parentheses(ZZGreaterEqual(0)), n)), Equal(Sum(m_(k), For(k, 1, n)), N)))),
    Variables(n, N, b_, z_),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(N, ZZGreaterEqual(0)), All(Element(b_(k), CC), ForElement(k, Range(1, n))), All(Element(z_(k), CC), ForElement(k, Range(1, n))))))

Topics using this entry

Series representations of Carlson symmetric elliptic integrals