# Fungrim entry: b7f13b

$\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{\left(n + a\right) \left(n + b\right)} = \begin{cases} \frac{1}{2 \left(a - b\right)} \left(\psi\!\left(\frac{a}{2}\right) + \psi\!\left(\frac{b + 1}{2}\right) - \psi\!\left(\frac{b}{2}\right) - \psi\!\left(\frac{a + 1}{2}\right)\right), & a \ne b\\\frac{1}{4} \left(\psi'\!\left(\frac{a}{2}\right) - \psi'\!\left(\frac{a + 1}{2}\right)\right), & a = b\\ \end{cases}$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \notin \{0, -1, \ldots\}$
TeX:
\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{\left(n + a\right) \left(n + b\right)} = \begin{cases} \frac{1}{2 \left(a - b\right)} \left(\psi\!\left(\frac{a}{2}\right) + \psi\!\left(\frac{b + 1}{2}\right) - \psi\!\left(\frac{b}{2}\right) - \psi\!\left(\frac{a + 1}{2}\right)\right), & a \ne b\\\frac{1}{4} \left(\psi'\!\left(\frac{a}{2}\right) - \psi'\!\left(\frac{a + 1}{2}\right)\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
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
DigammaFunction$\psi\!\left(z\right)$ Digamma function
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("b7f13b"),
Formula(Equal(Sum(Div(Pow(-1, n), Mul(Add(n, a), Add(n, b))), For(n, 0, Infinity)), Cases(Tuple(Mul(Div(1, Mul(2, Sub(a, b))), Sub(Sub(Add(DigammaFunction(Div(a, 2)), DigammaFunction(Div(Add(b, 1), 2))), DigammaFunction(Div(b, 2))), DigammaFunction(Div(Add(a, 1), 2)))), NotEqual(a, b)), Tuple(Mul(Div(1, 4), Sub(DigammaFunction(Div(a, 2), 1), DigammaFunction(Div(Add(a, 1), 2), 1))), Equal(a, b))))),
Variables(a, b),
Assumptions(And(Element(a, CC), Element(b, CC), NotElement(a, ZZLessEqual(0)), NotElement(b, ZZLessEqual(0)))))

## Topics using this entry

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