Fungrim entry: 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))))

Topics using this entry

Gauss hypergeometric function