Fungrim entry: f1bd89

z \left(1 - z\right) y''(z) + \left(c - \left(a + b + 1\right) z\right) y'(z) - a b y(z) = 0\; \text{ where } y(z) = \,{}_2F_1\!\left(a, b, c, z\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left[1, \infty\right)

TeX:

z \left(1 - z\right) y''(z) + \left(c - \left(a + b + 1\right) z\right) y'(z) - a b y(z) = 0\; \text{ where } y(z) = \,{}_2F_1\!\left(a, b, c, z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("f1bd89"),
    Formula(Where(Equal(Sub(Add(Mul(Mul(z, Sub(1, z)), ComplexDerivative(y(z), For(z, z, 2))), Mul(Sub(c, Mul(Add(Add(a, b), 1), z)), ComplexDerivative(y(z), For(z, z, 1)))), Mul(Mul(a, b), y(z))), 0), Equal(y(z), Hypergeometric2F1(a, b, c, z)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, SetMinus(CC, ClosedOpenInterval(1, Infinity))))))

Topics using this entry

Gauss hypergeometric function