Fungrim entry: dfb55b

\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \sum_{n=0}^{\infty} \frac{1}{{\left(n + z\right)}^{m + 1}}

Assumptions:

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

TeX:

\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \sum_{n=0}^{\infty} \frac{1}{{\left(n + z\right)}^{m + 1}}

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \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
`Infinity`	$\infty$	Positive infinity
`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("dfb55b"),
    Formula(Equal(DigammaFunction(z, m), Mul(Mul(Pow(-1, Add(m, 1)), Factorial(m)), Sum(Div(1, Pow(Add(n, z), Add(m, 1))), For(n, 0, Infinity))))),
    Variables(m, z),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

Topics using this entry

Digamma function