Fungrim entry: be533c

\,{}_1F_1\!\left(a, b, z\right) = {e}^{z} \,{}_1F_1\!\left(b - a, b, -z\right)

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) = {e}^{z} \,{}_1F_1\!\left(b - a, b, -z\right)

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
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("be533c"),
    Formula(Equal(Hypergeometric1F1(a, b, z), Mul(Exp(z), Hypergeometric1F1(Sub(b, a), b, Neg(z))))),
    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