Fungrim entry: 0a0aec

\,{}_0{\textbf F}_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\Gamma\!\left(a + k\right)} \frac{{z}^{k}}{k !}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_0{\textbf F}_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\Gamma\!\left(a + k\right)} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0a0aec"),
    Formula(Equal(Hypergeometric0F1Regularized(a, z), Sum(Mul(Div(1, GammaFunction(Add(a, k))), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, Infinity)))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC))))

Topics using this entry

Confluent hypergeometric functions