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Fungrim entry: 19e67f

n=AB1n+a=ψ ⁣(a+B+1)ψ ⁣(a+A)\sum_{n=A}^{B} \frac{1}{n + a} = \psi\!\left(a + B + 1\right) - \psi\!\left(a + A\right)
Assumptions:aC  and  AZ  and  BZ  and  AB  and  a{A,A+1,,B}a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; A \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; B \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; A \le B \;\mathbin{\operatorname{and}}\; -a \notin \{A, A + 1, \ldots, B\}
TeX:
\sum_{n=A}^{B} \frac{1}{n + a} = \psi\!\left(a + B + 1\right) - \psi\!\left(a + A\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; A \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; B \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; A \le B \;\mathbin{\operatorname{and}}\; -a \notin \{A, A + 1, \ldots, B\}
Definitions:
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Source code for this entry:
Entry(ID("19e67f"),
    Formula(Equal(Sum(Div(1, Add(n, a)), For(n, A, B)), Sub(DigammaFunction(Add(Add(a, B), 1)), DigammaFunction(Add(a, A))))),
    Variables(a, A, B),
    Assumptions(And(Element(a, CC), Element(A, ZZ), Element(B, ZZ), LessEqual(A, B), NotElement(Neg(a), Range(A, B)))))

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

2020-08-27 09:56:25.682319 UTC