Fungrim entry: 039051

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

Assumptions:

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

TeX:

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

m \in \mathbb{Z}_{\ge 0} \;\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
`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("039051"),
    Formula(Equal(DigammaFunction(Add(z, 1), m), Add(DigammaFunction(z, m), Div(Mul(Pow(-1, m), Factorial(m)), Pow(z, Add(m, 1)))))),
    Variables(m, z),
    Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

Topics using this entry

Digamma function