# Fungrim entry: 65693e

$\,{}_2{\textbf F}_1\!\left(a, b, -n, z\right) = \frac{\left(a\right)_{n + 1} \left(b\right)_{n + 1} {z}^{n + 1}}{\left(n + 1\right)!} \,{}_2F_1\!\left(a + n + 1, b + n + 1, n + 2, z\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}$
TeX:
\,{}_2{\textbf F}_1\!\left(a, b, -n, z\right) = \frac{\left(a\right)_{n + 1} \left(b\right)_{n + 1} {z}^{n + 1}}{\left(n + 1\right)!} \,{}_2F_1\!\left(a + n + 1, b + n + 1, n + 2, z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}
Definitions:
Fungrim symbol Notation Short description
Hypergeometric2F1Regularized$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$ Regularized Gauss hypergeometric function
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("65693e"),
Assumptions(And(Element(a, CC), Element(b, CC), Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(1))))))