Fungrim entry: 2df3e3

\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 a \notin \{1, 0, \ldots\}

TeX:

\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 a \notin \{1, 0, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`Exp`	${e}^{z}$	Exponential function
`Sqrt`	$\sqrt{z}$	Principal square root
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric 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("2df3e3"),
    Formula(Equal(Hypergeometric0F1(a, z), Mul(Exp(Neg(Mul(2, Sqrt(z)))), Hypergeometric1F1(Sub(a, Div(1, 2)), Sub(Mul(2, a), 1), Mul(4, Sqrt(z)))))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC), NotElement(Mul(2, a), ZZLessEqual(1)))))

Topics using this entry

Confluent hypergeometric functions