Fungrim entry: 6547da

\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:

r \in \mathbb{Z}_{\ge 1}

References:

https://doi.org/10.1080%2F10652469.2017.1376193

TeX:

\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}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`DigammaFunctionZero`	$x_{n}$	Zero of the digamma function
`Infinity`	$\infty$	Positive infinity
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Factorial`	$n !$	Factorial
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("6547da"),
    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)))))),
    Variables(r),
    Assumptions(Element(r, ZZGreaterEqual(1))),
    References("https://doi.org/10.1080%2F10652469.2017.1376193"))

Topics using this entry

Specific values of the digamma function