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

Confluent hypergeometric functions