# 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
Sum$\sum_{n} f(n)$ Sum
Gamma$\Gamma(z)$ 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, Gamma(Add(a, k))), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
Variables(a, z),
Assumptions(And(Element(a, CC), Element(z, CC))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC