# Fungrim entry: 853a62

$\left|\frac{{f}^{(k)}(z)}{k !}\right| \le A {N + k \choose k} {\nu}^{k}\; \text{ where } f(z) = \,{}_2{\textbf F}_1\!\left(a, b, c, z\right),\;\nu = \max\!\left(\frac{1}{\left|z - 1\right|}, \frac{1}{\left|z\right|}\right),\;N = 2 \max\!\left(\sqrt{{\nu}^{-1} \left|a b\right|}, \left|a + b + 1\right| + 2 \left|c\right|\right),\;A = \max\!\left(\left|f(z)\right|, \frac{\left|f'(z)\right|}{\nu \left(N + 1\right)}\right)$
Actually valid when $f(z)$ is any branch of any solution of the hypergeometric ODE, away from the branch points $z = 0$ and $z = 1$. The variables $\nu$, $N$, and $A$ can be replaced by any upper bounds.
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \cup \left[1, \infty\right) \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0}$
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(z) = \,{}_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(z)\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
Abs$\left|z\right|$ Absolute value
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Factorial$n !$ Factorial
Binomial${n \choose k}$ Binomial coefficient
Pow${a}^{b}$ Power
Hypergeometric2F1Regularized$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$ Regularized Gauss hypergeometric function
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("853a62"),
Formula(Where(LessEqual(Abs(Div(ComplexDerivative(f(z), For(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(ComplexDerivative(f(z), For(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"))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC