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# Fungrim entry: ddc7e1

$\psi^{(m)}\!\left(-n + z\right) = \frac{{\left(-1\right)}^{m + 1} m !}{{z}^{m + 1}} + \sum_{k=0}^{\infty} \left(k + 1\right)_{m} \left({\left(-1\right)}^{m + k + 1} \zeta\!\left(m + k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + m + 1}}\right) {z}^{k}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1$
TeX:
\psi^{(m)}\!\left(-n + z\right) = \frac{{\left(-1\right)}^{m + 1} m !}{{z}^{m + 1}} + \sum_{k=0}^{\infty} \left(k + 1\right)_{m} \left({\left(-1\right)}^{m + k + 1} \zeta\!\left(m + k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + m + 1}}\right) {z}^{k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1
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
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("ddc7e1"),
Formula(Equal(DigammaFunction(Add(Neg(n), z), m), Add(Div(Mul(Pow(-1, Add(m, 1)), Factorial(m)), Pow(z, Add(m, 1))), Sum(Mul(Mul(RisingFactorial(Add(k, 1), m), Add(Mul(Pow(-1, Add(Add(m, k), 1)), RiemannZeta(Add(Add(m, k), 1))), Sum(Div(1, Pow(j, Add(Add(k, m), 1))), For(j, 1, n)))), Pow(z, k)), For(k, 0, Infinity))))),
Variables(n, m, z),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(1)), Element(z, CC), Less(Abs(z), 1))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC