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

Digamma function