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

Digamma function