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

Gauss hypergeometric function