Fungrim entry: 306ef7

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\Gamma\!\left(c + k\right)} \frac{{z}^{k}}{k !}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; c \in \{0, -1, \ldots\}\right)

TeX:

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\Gamma\!\left(c + k\right)} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; c \in \{0, -1, \ldots\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Sum`	$\sum_{n} f(n)$	Sum
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("306ef7"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Sum(Mul(Div(Mul(RisingFactorial(a, k), RisingFactorial(b, k)), Gamma(Add(c, k))), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), Or(Less(Abs(z), 1), Element(a, ZZLessEqual(0)), Element(b, ZZLessEqual(0)), Element(c, ZZLessEqual(0))))))

Topics using this entry

Gauss hypergeometric function