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Fungrim entry: f42042

n=01(n+a)r=(1)r(r1)!ψ(r1) ⁣(a)\sum_{n=0}^{\infty} \frac{1}{{\left(n + a\right)}^{r}} = \frac{{\left(-1\right)}^{r}}{\left(r - 1\right)!} \psi^{(r - 1)}\!\left(a\right)
Assumptions:rZ2  and  aC  and  a{0,1,}r \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
\sum_{n=0}^{\infty} \frac{1}{{\left(n + a\right)}^{r}} = \frac{{\left(-1\right)}^{r}}{\left(r - 1\right)!} \psi^{(r - 1)}\!\left(a\right)

r \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Infinity\infty Positive infinity
Factorialn!n ! Factorial
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(Sum(Div(1, Pow(Add(n, a), r)), For(n, 0, Infinity)), Mul(Div(Pow(-1, r), Factorial(Sub(r, 1))), DigammaFunction(a, Sub(r, 1))))),
    Variables(r, a),
    Assumptions(And(Element(r, ZZGreaterEqual(2)), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC