# Fungrim entry: a61f01

$\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
TeX:
\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Hypergeometric1F1$\,{}_1F_1\!\left(a, b, z\right)$ Kummer confluent hypergeometric function
Sum$\sum_{n} f(n)$ Sum
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("a61f01"),
Formula(Equal(Hypergeometric1F1(a, b, z), Sum(Mul(Div(RisingFactorial(a, k), RisingFactorial(b, k)), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
Variables(a, b, z),
Assumptions(And(Element(a, CC), Element(b, SetMinus(CC, ZZLessEqual(0))), 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