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Fungrim entry: 6547da

n=01xnr+1=f(r)(0)r!   where f(z)=limtz(ψ ⁣(t)ψ ⁣(t)ψ ⁣(t))\sum_{n=0}^{\infty} \frac{1}{x_{n}^{r + 1}} = \frac{{f}^{(r)}(0)}{r !}\; \text{ where } f(z) = \lim_{t \to z} \left(\psi\!\left(t\right) - \frac{\psi'\!\left(t\right)}{\psi\!\left(t\right)}\right)
Assumptions:rZ1r \in \mathbb{Z}_{\ge 1}
\sum_{n=0}^{\infty} \frac{1}{x_{n}^{r + 1}} = \frac{{f}^{(r)}(0)}{r !}\; \text{ where } f(z) = \lim_{t \to z} \left(\psi\!\left(t\right) - \frac{\psi'\!\left(t\right)}{\psi\!\left(t\right)}\right)

r \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
DigammaFunctionZeroxnx_{n} Zero of the digamma function
Infinity\infty Positive infinity
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Factorialn!n ! Factorial
ComplexLimitlimzaf(z)\lim_{z \to a} f(z) Limiting value, complex variable
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(Sum(Div(1, Pow(DigammaFunctionZero(n), Add(r, 1))), For(n, 0, Infinity)), Where(Div(ComplexDerivative(f(z), For(z, 0, r)), Factorial(r)), Equal(f(z), ComplexLimit(Parentheses(Sub(DigammaFunction(t), Div(DigammaFunction(t, 1), DigammaFunction(t)))), For(t, z)))))),
    Assumptions(Element(r, ZZGreaterEqual(1))),

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2021-03-15 19:12:00.328586 UTC