# Fungrim entry: dec042

$\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{not} \left(b \in \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b > -n\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}$
TeX:
\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\,  \operatorname{not} \left(b \in \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b > -n\right) \,\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$\sum_{n} f\!\left(n\right)$ Sum
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("dec042"),
Formula(Equal(Hypergeometric1F1(Neg(n), b, z), Sum(Mul(Div(RisingFactorial(Neg(n), k), RisingFactorial(b, k)), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, n)))),
Variables(n, b, z),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(b, CC), Not(And(Element(b, ZZLessEqual(0)), Greater(b, Neg(n)))), 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.

2019-08-25 15:30:03.056001 UTC