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Fungrim entry: f1bd89

z(1z)y(z)+(c(a+b+1)z)y(z)aby ⁣(z)=0   where y ⁣(z)=2F1 ⁣(a,b,c,z)z \left(1 - z\right) y''(z) + \left(c - \left(a + b + 1\right) z\right) y'(z) - a b y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = \,{}_2F_1\!\left(a, b, c, z\right)
Assumptions:aCandbCandcC{0,1,}andzC[1,)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\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = \,{}_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
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ClosedOpenInterval[a,b)\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)), Derivative(y(z), Tuple(z, z, 2))), Mul(Sub(c, Mul(Add(Add(a, b), 1), z)), Derivative(y(z), Tuple(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))))))

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

2019-09-15 11:00:55.020619 UTC