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Fungrim entry: 21e21a

n=01(n+a)(n+b)={1ab(ψ ⁣(a)ψ ⁣(b)),abψ ⁣(a),a=b\sum_{n=0}^{\infty} \frac{1}{\left(n + a\right) \left(n + b\right)} = \begin{cases} \frac{1}{a - b} \left(\psi\!\left(a\right) - \psi\!\left(b\right)\right), & a \ne b\\\psi'\!\left(a\right), & a = b\\ \end{cases}
Assumptions:aCandbCanda{0,1,}andb{0,1,}a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, a \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b \notin \{0, -1, \ldots\}
TeX:
\sum_{n=0}^{\infty} \frac{1}{\left(n + a\right) \left(n + b\right)} = \begin{cases} \frac{1}{a - b} \left(\psi\!\left(a\right) - \psi\!\left(b\right)\right), & a \ne b\\\psi'\!\left(a\right), & a = b\\ \end{cases}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, a \notin \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Infinity\infty Positive infinity
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("21e21a"),
    Formula(Equal(Sum(Div(1, Mul(Add(n, a), Add(n, b))), For(n, 0, Infinity)), Cases(Tuple(Mul(Div(1, Sub(a, b)), Sub(DigammaFunction(a), DigammaFunction(b))), NotEqual(a, b)), Tuple(DigammaFunction(a, 1), Equal(a, b))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotElement(a, ZZLessEqual(0)), NotElement(b, 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.

2020-01-31 18:09:28.494564 UTC