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`	$\sum_{n} f(n)$	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))), For(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

Confluent hypergeometric functions