Fungrim entry: 554ac2

\psi\!\left(z - n\right) = \psi\!\left(z\right) - \sum_{k=1}^{n} \frac{1}{z - k}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - n \notin \{0, -1, \ldots\}

TeX:

\psi\!\left(z - n\right) = \psi\!\left(z\right) - \sum_{k=1}^{n} \frac{1}{z - k}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - n \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Sum`	$\sum_{n} f(n)$	Sum
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("554ac2"),
    Formula(Equal(DigammaFunction(Sub(z, n)), Sub(DigammaFunction(z), Sum(Div(1, Sub(z, k)), For(k, 1, n))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(0)), NotElement(Sub(z, n), ZZLessEqual(0)))))

Topics using this entry

Digamma function