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"),
    Formula(Equal(DigammaFunction(z, m), Add(Mul(Div(Pow(-1, Add(m, 1)), Factorial(m)), Add(Add(Div(1, Mul(m, Pow(z, m))), Div(1, Mul(2, Pow(z, Add(m, 1))))), Sum(Mul(Div(RisingFactorial(Add(m, 1), Sub(Mul(2, n), 1)), Factorial(Mul(2, n))), Div(BernoulliB(Mul(2, n)), Pow(z, Add(m, Mul(2, n))))), For(n, 1, Sub(N, 1))))), Derivative(StirlingSeriesRemainder(N, z), For(z, z, Add(m, 1)))))),
    Variables(m, z, N),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(N, ZZGreaterEqual(0)))))

Topics using this entry

Digamma function