# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-12-30 15:00:46.909060 UTC