# Fungrim entry: 24c9e9

$\psi^{(m)}\!\left(z\right) = \frac{{\left(-1\right)}^{m + 1}}{m !} \left(\frac{1}{m {z}^{m}} + \frac{1}{2 {z}^{m + 1}} + \sum_{n=1}^{N - 1} \frac{\left(m + 1\right)_{2 n - 1}}{\left(2 n\right)!} \frac{B_{2 n}}{{z}^{m + 2 n}}\right) + {R}^{(m + 1)}_{N}(z)$
Assumptions:$m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0}$
TeX:
\psi^{(m)}\!\left(z\right) = \frac{{\left(-1\right)}^{m + 1}}{m !} \left(\frac{1}{m {z}^{m}} + \frac{1}{2 {z}^{m + 1}} + \sum_{n=1}^{N - 1} \frac{\left(m + 1\right)_{2 n - 1}}{\left(2 n\right)!} \frac{B_{2 n}}{{z}^{m + 2 n}}\right) + {R}^{(m + 1)}_{N}(z)

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0}
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
RisingFactorial$\left(z\right)_{k}$ Rising factorial
BernoulliB$B_{n}$ Bernoulli number
Derivative$\frac{d}{d z}\, f\!\left(z\right)$ Derivative
StirlingSeriesRemainder$R_{n}\!\left(z\right)$ Remainder term in the Stirling series for the logarithmic gamma function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("24c9e9"),
Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(N, ZZGreaterEqual(0)))))