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Fungrim entry: 411f3b

γn ⁣(a)=limN[(k=0Nlogn ⁣(k+a)k+a)logn+1 ⁣(N+a)n+1]\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]
Assumptions:nZ0  and  aC  and  a{0,1,}n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
Fungrim symbol Notation Short description
StieltjesGammaγn ⁣(a)\gamma_{n}\!\left(a\right) Stieltjes constant
SequenceLimitlimnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Loglog(z)\log(z) Natural logarithm
Infinity\infty Positive infinity
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(StieltjesGamma(n, a), SequenceLimit(Brackets(Sub(Parentheses(Sum(Div(Pow(Log(Add(k, a)), n), Add(k, a)), For(k, 0, N))), Div(Pow(Log(Add(N, a)), Add(n, 1)), Add(n, 1)))), For(N, Infinity)))),
    Variables(n, a),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), 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