Fungrim entry: c687d6

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

Assumptions:

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

TeX:

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

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; 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
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Sum`	$\sum_{n} f(n)$	Sum
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("c687d6"),
    Formula(Equal(DigammaFunction(Sub(z, n), m), Sub(DigammaFunction(z, m), Mul(Mul(Pow(-1, m), Factorial(m)), Sum(Div(1, Pow(Sub(z, k), Add(m, 1))), For(k, 1, n)))))),
    Variables(m, z, n),
    Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), Element(n, ZZGreaterEqual(0)), NotElement(Sub(z, n), ZZLessEqual(0)))))

Topics using this entry

Digamma function