Fungrim entry: b16177

\begin{cases} \psi^{(m)}\!\left(x + y\right) > \psi\!\left(x\right), & m \text{ even}\\\psi^{(m)}\!\left(x + y\right) < \psi\!\left(x\right), & m \text{ odd}\\ \end{cases}

Assumptions:

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)

TeX:

\begin{cases} \psi^{(m)}\!\left(x + y\right) > \psi\!\left(x\right), & m \text{ even}\\\psi^{(m)}\!\left(x + y\right) < \psi\!\left(x\right), & m \text{ odd}\\ \end{cases}

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("b16177"),
    Formula(Cases(Tuple(Greater(DigammaFunction(Add(x, y), m), DigammaFunction(x)), Even(m)), Tuple(Less(DigammaFunction(Add(x, y), m), DigammaFunction(x)), Odd(m)))),
    Variables(x, y, m),
    Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(x, OpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity)))))

Topics using this entry

Digamma function