Gauss hypergeometric function

Symbol: Hypergeometric2F1 —

\,{}_2F_1\!\left(a, b, c, z\right)

— Gauss hypergeometric function

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function

Source code for this entry:

Entry(ID("e03016"),
    SymbolDefinition(Hypergeometric2F1, Hypergeometric2F1(a, b, c, z), "Gauss hypergeometric function"))

c43abd

Symbol: Hypergeometric2F1Regularized —

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)

— Regularized Gauss hypergeometric function

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function

Source code for this entry:

Entry(ID("c43abd"),
    SymbolDefinition(Hypergeometric2F1Regularized, Hypergeometric2F1Regularized(a, b, c, z), "Regularized Gauss hypergeometric function"))

ad8db2

\,{}_2F_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}

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} \;\mathbin{\operatorname{and}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\}\right)

TeX:

\,{}_2F_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}

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} \;\mathbin{\operatorname{and}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Sum`	$\sum_{n} f(n)$	Sum
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("ad8db2"),
    Formula(Equal(Hypergeometric2F1(a, b, c, z), Sum(Mul(Div(Mul(RisingFactorial(a, k), RisingFactorial(b, k)), RisingFactorial(c, k)), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, CC), Or(Less(Abs(z), 1), Element(a, ZZLessEqual(0)), Element(b, ZZLessEqual(0))))))

306ef7

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\Gamma\!\left(c + k\right)} \frac{{z}^{k}}{k !}

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} \;\mathbin{\operatorname{and}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; c \in \{0, -1, \ldots\}\right)

TeX:

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\Gamma\!\left(c + k\right)} \frac{{z}^{k}}{k !}

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} \;\mathbin{\operatorname{and}}\; \left(\left|z\right| < 1 \;\mathbin{\operatorname{or}}\; a \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; b \in \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; c \in \{0, -1, \ldots\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Sum`	$\sum_{n} f(n)$	Sum
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("306ef7"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Sum(Mul(Div(Mul(RisingFactorial(a, k), RisingFactorial(b, k)), Gamma(Add(c, k))), Div(Pow(z, k), Factorial(k))), For(k, 0, Infinity)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), Or(Less(Abs(z), 1), Element(a, ZZLessEqual(0)), Element(b, ZZLessEqual(0)), Element(c, ZZLessEqual(0))))))

fe6e74

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{\,{}_2F_1\!\left(a, b, c, z\right)}{\Gamma(c)}

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}

TeX:

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{\,{}_2F_1\!\left(a, b, c, z\right)}{\Gamma(c)}

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}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("fe6e74"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Div(Hypergeometric2F1(a, b, c, z), Gamma(c)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, CC))))

65693e

\,{}_2{\textbf F}_1\!\left(a, b, -n, z\right) = \frac{\left(a\right)_{n + 1} \left(b\right)_{n + 1} {z}^{n + 1}}{\left(n + 1\right)!} \,{}_2F_1\!\left(a + n + 1, b + n + 1, n + 2, z\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}

TeX:

\,{}_2{\textbf F}_1\!\left(a, b, -n, z\right) = \frac{\left(a\right)_{n + 1} \left(b\right)_{n + 1} {z}^{n + 1}}{\left(n + 1\right)!} \,{}_2F_1\!\left(a + n + 1, b + n + 1, n + 2, z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("65693e"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, Neg(n), z), Mul(Div(Mul(Mul(RisingFactorial(a, Add(n, 1)), RisingFactorial(b, Add(n, 1))), Pow(z, Add(n, 1))), Factorial(Add(n, 1))), Hypergeometric2F1(Add(Add(a, n), 1), Add(Add(b, n), 1), Add(n, 2), z)))),
    Variables(a, b, n, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(1))))))

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))))))

18d955

\,{}_2F_1\!\left(a, b, c, 0\right) = 1

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\,{}_2F_1\!\left(a, b, c, 0\right) = 1

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`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

Source code for this entry:

Entry(ID("18d955"),
    Formula(Equal(Hypergeometric2F1(a, b, c, 0), 1)),
    Variables(a, b, c),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))))))

659ce8

\,{}_2F_1\!\left(a, b, c, 1\right) = \frac{\Gamma(c) \Gamma\!\left(c - a - b\right)}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\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}}\; \operatorname{Re}\!\left(c - a - b\right) > 0

TeX:

\,{}_2F_1\!\left(a, b, c, 1\right) = \frac{\Gamma(c) \Gamma\!\left(c - a - b\right)}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\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}}\; \operatorname{Re}\!\left(c - a - b\right) > 0

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("659ce8"),
    Formula(Equal(Hypergeometric2F1(a, b, c, 1), Div(Mul(Gamma(c), Gamma(Sub(Sub(c, a), b))), Mul(Gamma(Sub(c, a)), Gamma(Sub(c, b)))))),
    Variables(a, b, c),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Greater(Re(Sub(Sub(c, a), b)), 0))))

a85994

\,{}_2F_1\!\left(1, 1, 2, z\right) = -\frac{\log\!\left(1 - z\right)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0, 1\right\}

TeX:

\,{}_2F_1\!\left(1, 1, 2, z\right) = -\frac{\log\!\left(1 - z\right)}{z}

z \in \mathbb{C} \setminus \left\{0, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a85994"),
    Formula(Equal(Hypergeometric2F1(1, 1, 2, z), Neg(Div(Log(Sub(1, z)), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0, 1)))))

20bf69

\,{}_2F_1\!\left(a, b, b, z\right) = {\left(1 - z\right)}^{-a}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0, 1\right\}

TeX:

\,{}_2F_1\!\left(a, b, b, z\right) = {\left(1 - z\right)}^{-a}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("20bf69"),
    Formula(Equal(Hypergeometric2F1(a, b, b, z), Pow(Sub(1, z), Neg(a)))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, SetMinus(CC, ZZLessEqual(0))), Element(z, SetMinus(CC, Set(0, 1))))))

0e0393

\,{}_2F_1\!\left(a, b, c, z\right) = \,{}_2F_1\!\left(b, a, 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}

TeX:

\,{}_2F_1\!\left(a, b, c, z\right) = \,{}_2F_1\!\left(b, a, 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}

Definitions:

Fungrim symbol	Notation	Short description
`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

Source code for this entry:

Entry(ID("0e0393"),
    Formula(Equal(Hypergeometric2F1(a, b, c, z), Hypergeometric2F1(b, a, c, z))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, CC))))

3d6d7e

\,{}_2F_1\!\left(a, b, c, z\right) = \overline{\,{}_2F_1\!\left(\overline{a}, \overline{b}, \overline{c}, \overline{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:

\,{}_2F_1\!\left(a, b, c, z\right) = \overline{\,{}_2F_1\!\left(\overline{a}, \overline{b}, \overline{c}, \overline{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
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Conjugate`	$\overline{z}$	Complex conjugate
`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("3d6d7e"),
    Formula(Equal(Hypergeometric2F1(a, b, c, z), Conjugate(Hypergeometric2F1(Conjugate(a), Conjugate(b), Conjugate(c), Conjugate(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))))))

651a4a

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{c - a - b} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c, z\right)

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} \;\mathbin{\operatorname{and}}\; z \ne 1

TeX:

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{c - a - b} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c, z\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} \;\mathbin{\operatorname{and}}\; z \ne 1

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("651a4a"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Mul(Pow(Sub(1, z), Sub(Sub(c, a), b)), Hypergeometric2F1Regularized(Sub(c, a), Sub(c, b), c, z)))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotEqual(z, 1))))

b25089

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-a} \,{}_2{\textbf F}_1\!\left(a, c - b, c, \frac{z}{z - 1}\right)

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} \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

TeX:

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-a} \,{}_2{\textbf F}_1\!\left(a, c - b, c, \frac{z}{z - 1}\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} \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("b25089"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Mul(Pow(Sub(1, z), Neg(a)), Hypergeometric2F1Regularized(a, Sub(c, b), c, Div(z, Sub(z, 1)))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, ClosedOpenInterval(1, Infinity)))))

504717

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-b} \,{}_2{\textbf F}_1\!\left(c - a, b, c, \frac{z}{z - 1}\right)

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} \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

TeX:

\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-b} \,{}_2{\textbf F}_1\!\left(c - a, b, c, \frac{z}{z - 1}\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} \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("504717"),
    Formula(Equal(Hypergeometric2F1Regularized(a, b, c, z), Mul(Pow(Sub(1, z), Neg(b)), Hypergeometric2F1Regularized(Sub(c, a), b, c, Div(z, Sub(z, 1)))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, ClosedOpenInterval(1, Infinity)))))

90ac58

\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma(b) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a - b + 1, \frac{1}{z}\right) - \frac{{\left(-z\right)}^{-b}}{\Gamma(a) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, b - c + 1, b - a + 1, \frac{1}{z}\right)

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} \;\mathbin{\operatorname{and}}\; z \notin \left[0, \infty\right)

TeX:

\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma(b) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a - b + 1, \frac{1}{z}\right) - \frac{{\left(-z\right)}^{-b}}{\Gamma(a) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, b - c + 1, b - a + 1, \frac{1}{z}\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} \;\mathbin{\operatorname{and}}\; z \notin \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("90ac58"),
    Formula(Equal(Mul(Div(Sin(Mul(Pi, Sub(b, a))), Pi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(Pow(Neg(z), Neg(a)), Mul(Gamma(b), Gamma(Sub(c, a)))), Hypergeometric2F1Regularized(a, Add(Sub(a, c), 1), Add(Sub(a, b), 1), Div(1, z))), Mul(Div(Pow(Neg(z), Neg(b)), Mul(Gamma(a), Gamma(Sub(c, b)))), Hypergeometric2F1Regularized(b, Add(Sub(b, c), 1), Add(Sub(b, a), 1), Div(1, z)))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, ClosedOpenInterval(0, Infinity)))))

27bc34

\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(1 - z\right)}^{-a}}{\Gamma(b) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, c - b, a - b + 1, \frac{1}{1 - z}\right) - \frac{{\left(1 - z\right)}^{-b}}{\Gamma(a) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, c - a, b - a + 1, \frac{1}{1 - z}\right)

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} \;\mathbin{\operatorname{and}}\; z \notin \left[0, \infty\right)

TeX:

\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(1 - z\right)}^{-a}}{\Gamma(b) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, c - b, a - b + 1, \frac{1}{1 - z}\right) - \frac{{\left(1 - z\right)}^{-b}}{\Gamma(a) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, c - a, b - a + 1, \frac{1}{1 - z}\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} \;\mathbin{\operatorname{and}}\; z \notin \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("27bc34"),
    Formula(Equal(Mul(Div(Sin(Mul(Pi, Sub(b, a))), Pi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(Pow(Sub(1, z), Neg(a)), Mul(Gamma(b), Gamma(Sub(c, a)))), Hypergeometric2F1Regularized(a, Sub(c, b), Add(Sub(a, b), 1), Div(1, Sub(1, z)))), Mul(Div(Pow(Sub(1, z), Neg(b)), Mul(Gamma(a), Gamma(Sub(c, b)))), Hypergeometric2F1Regularized(b, Sub(c, a), Add(Sub(b, a), 1), Div(1, Sub(1, z))))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, ClosedOpenInterval(0, Infinity)))))

db3eb9

\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{1}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, b, a + b - c + 1, 1 - z\right) - \frac{{\left(1 - z\right)}^{c - a - b}}{\Gamma(a) \Gamma(b)} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c - a - b + 1, 1 - z\right)

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} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

TeX:

\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{1}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, b, a + b - c + 1, 1 - z\right) - \frac{{\left(1 - z\right)}^{c - a - b}}{\Gamma(a) \Gamma(b)} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c - a - b + 1, 1 - z\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} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("db3eb9"),
    Formula(Equal(Mul(Div(Sin(Mul(Pi, Sub(Sub(c, a), b))), Pi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(1, Mul(Gamma(Sub(c, a)), Gamma(Sub(c, b)))), Hypergeometric2F1Regularized(a, b, Add(Sub(Add(a, b), c), 1), Sub(1, z))), Mul(Div(Pow(Sub(1, z), Sub(Sub(c, a), b)), Mul(Gamma(a), Gamma(b))), Hypergeometric2F1Regularized(Sub(c, a), Sub(c, b), Add(Sub(Sub(c, a), b), 1), Sub(1, z)))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), NotElement(z, ClosedOpenInterval(1, Infinity)))))

ca9123

\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{z}^{-a}}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a + b - c + 1, 1 - \frac{1}{z}\right) - \frac{{z}^{a - c} {\left(1 - z\right)}^{c - a - b}}{\Gamma(a) \Gamma(b)} \,{}_2{\textbf F}_1\!\left(c - a, 1 - a, c - a - b + 1, 1 - \frac{1}{z}\right)

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} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

TeX:

\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{z}^{-a}}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a + b - c + 1, 1 - \frac{1}{z}\right) - \frac{{z}^{a - c} {\left(1 - z\right)}^{c - a - b}}{\Gamma(a) \Gamma(b)} \,{}_2{\textbf F}_1\!\left(c - a, 1 - a, c - a - b + 1, 1 - \frac{1}{z}\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} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \notin \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("ca9123"),
    Formula(Equal(Mul(Div(Sin(Mul(Pi, Sub(Sub(c, a), b))), Pi), Hypergeometric2F1Regularized(a, b, c, z)), Sub(Mul(Div(Pow(z, Neg(a)), Mul(Gamma(Sub(c, a)), Gamma(Sub(c, b)))), Hypergeometric2F1Regularized(a, Add(Sub(a, c), 1), Add(Sub(Add(a, b), c), 1), Sub(1, Div(1, z)))), Mul(Div(Mul(Pow(z, Sub(a, c)), Pow(Sub(1, z), Sub(Sub(c, a), b))), Mul(Gamma(a), Gamma(b))), Hypergeometric2F1Regularized(Sub(c, a), Sub(1, a), Add(Sub(Sub(c, a), b), 1), Sub(1, Div(1, z))))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), NotElement(z, ClosedOpenInterval(1, Infinity)))))

c60679

\left|\,{}_2F_1\!\left(a, b, c, z\right) - \sum_{k=0}^{N - 1} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}\right| \le \left|\frac{\left(a\right)_{N} \left(b\right)_{N}}{\left(c\right)_{N}} \frac{{z}^{N}}{N !}\right| \begin{cases} \frac{1}{1 - D}, & D < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } D = \left|z\right| \left(1 + \frac{\left|a - c\right|}{\left|c + N\right|}\right) \left(1 + \frac{\left|b - 1\right|}{\left|1 + N\right|}\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} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(c) + N > 0

TeX:

\left|\,{}_2F_1\!\left(a, b, c, z\right) - \sum_{k=0}^{N - 1} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}\right| \le \left|\frac{\left(a\right)_{N} \left(b\right)_{N}}{\left(c\right)_{N}} \frac{{z}^{N}}{N !}\right| \begin{cases} \frac{1}{1 - D}, & D < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } D = \left|z\right| \left(1 + \frac{\left|a - c\right|}{\left|c + N\right|}\right) \left(1 + \frac{\left|b - 1\right|}{\left|1 + N\right|}\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} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(c) + N > 0

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Sum`	$\sum_{n} f(n)$	Sum
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("c60679"),
    Formula(Where(LessEqual(Abs(Sub(Hypergeometric2F1(a, b, c, z), Sum(Mul(Div(Mul(RisingFactorial(a, k), RisingFactorial(b, k)), RisingFactorial(c, k)), Div(Pow(z, k), Factorial(k))), For(k, 0, Sub(N, 1))))), Mul(Abs(Mul(Div(Mul(RisingFactorial(a, N), RisingFactorial(b, N)), RisingFactorial(c, N)), Div(Pow(z, N), Factorial(N)))), Cases(Tuple(Div(1, Sub(1, D)), Less(D, 1)), Tuple(Infinity, Otherwise)))), Equal(D, Mul(Mul(Abs(z), Add(1, Div(Abs(Sub(a, c)), Abs(Add(c, N))))), Add(1, Div(Abs(Sub(b, 1)), Abs(Add(1, N)))))))),
    Variables(a, b, c, z, N),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, CC), Less(Abs(z), 1), Element(N, ZZGreaterEqual(0)), Greater(Add(Re(c), N), 0))))

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"))

Hypergeometric series

Differential equations

Specific values

Symmetries

Linear fractional transformations

Bounds and inequalities