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Fungrim entry: 853a62

f(k)(z)k!A(N+kk)νk   where f ⁣(z)=2F1 ⁣(a,b,c,z),ν=max ⁣(1z1,1z),N=2max ⁣(ν1ab,a+b+1+2c),A=max ⁣(f ⁣(z),f(z)ν(N+1))\left|\frac{{f}^{(k)}(z)}{k !}\right| \le A {N + k \choose k} {\nu}^{k}\; \text{ where } f\!\left(z\right) = \,{}_2{\textbf F}_1\!\left(a, b, c, z\right),\,\nu = \max\!\left(\frac{1}{\left|z - 1\right|}, \frac{1}{\left|z\right|}\right),\,N = 2 \max\!\left(\sqrt{{\nu}^{-1} \left|a b\right|}, \left|a + b + 1\right| + 2 \left|c\right|\right),\,A = \max\!\left(\left|f\!\left(z\right)\right|, \frac{\left|f'(z)\right|}{\nu \left(N + 1\right)}\right)
Actually valid when f ⁣(z)f\!\left(z\right) is any branch of any solution of the hypergeometric ODE, away from the branch points z=0z = 0 and z=1z = 1. The variables ν\nu, NN, and AA can be replaced by any upper bounds.
Assumptions:aCandbCandcCandzC{0}[1,)andkZ0a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \cup \left[1, \infty\right) \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
References:
  • F. Johansson, Computing hypergeometric functions rigorously, https://arxiv.org/abs/1606.06977
TeX:
\left|\frac{{f}^{(k)}(z)}{k !}\right| \le A {N + k \choose k} {\nu}^{k}\; \text{ where } f\!\left(z\right) = \,{}_2{\textbf F}_1\!\left(a, b, c, z\right),\,\nu = \max\!\left(\frac{1}{\left|z - 1\right|}, \frac{1}{\left|z\right|}\right),\,N = 2 \max\!\left(\sqrt{{\nu}^{-1} \left|a b\right|}, \left|a + b + 1\right| + 2 \left|c\right|\right),\,A = \max\!\left(\left|f\!\left(z\right)\right|, \frac{\left|f'(z)\right|}{\nu \left(N + 1\right)}\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \cup \left[1, \infty\right) \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
Factorialn!n ! Factorial
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
Hypergeometric2F1Regularized2F1 ⁣(a,b,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) Regularized Gauss hypergeometric function
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("853a62"),
    Formula(Where(LessEqual(Abs(Div(Derivative(f(z), Tuple(z, z, k)), Factorial(k))), Mul(Mul(A, Binomial(Add(N, k), k)), Pow(nu, k))), Equal(f(z), Hypergeometric2F1Regularized(a, b, c, z)), Equal(nu, Max(Div(1, Abs(Sub(z, 1))), Div(1, Abs(z)))), Equal(N, Mul(2, Max(Sqrt(Mul(Pow(nu, -1), Abs(Mul(a, b)))), Add(Abs(Add(Add(a, b), 1)), Mul(2, Abs(c)))))), Equal(A, Max(Abs(f(z)), Div(Abs(Derivative(f(z), Tuple(z, z, 1))), Mul(nu, Add(N, 1))))))),
    Description("Actually valid when", f(z), "is any branch of any solution of the hypergeometric ODE, away from the branch points", Equal(z, 0), "and", Equal(z, 1), ".", "The variables", nu, ",", N, ", and", A, "can be replaced by any upper bounds."),
    Variables(a, b, c, z, k),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, SetMinus(CC, Union(Set(0), ClosedOpenInterval(1, Infinity)))), Element(k, ZZGreaterEqual(0)))),
    References("F. Johansson, Computing hypergeometric functions rigorously, https://arxiv.org/abs/1606.06977"))

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

2019-08-19 14:38:23.809000 UTC