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"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, Neg(n), z), Mul(Div(Mul(Mul(RisingFactorial(a, Add(n, 1)), RisingFactorial(b, Add(n, 1))), Pow(z, Add(n, 1))), Factorial(Add(n, 1))), Hypergeometric2F1(Add(Add(a, n), 1), Add(Add(b, n), 1), Add(n, 2), z)))),
    Variables(a, b, n, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(1))))))

Topics using this entry

Gauss hypergeometric function