Fungrim entry: 90b26f

\psi^{(m)}\!\left(n\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, n\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}

TeX:

\psi^{(m)}\!\left(n\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, n\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("90b26f"),
    Formula(Equal(DigammaFunction(n, m), Mul(Mul(Pow(-1, Add(m, 1)), Factorial(m)), HurwitzZeta(Add(m, 1), n)))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(m, ZZGreaterEqual(1)))))

Topics using this entry

Digamma function
Specific values of the digamma function