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Fungrim entry: 24c9e9

ψ(m) ⁣(z)=(1)m+1m!(1mzm+12zm+1+n=1N1(m+1)2n1(2n)!B2nzm+2n)+RN(m+1)(z)\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:mZ1  and  zC(,0]  and  NZ0m \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}
\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}
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Powab{a}^{b} Power
Factorialn!n ! Factorial
Sumnf(n)\sum_{n} f(n) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
BernoulliBBnB_{n} Bernoulli number
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
StirlingSeriesRemainderRn ⁣(z)R_{n}\!\left(z\right) Remainder term in the Stirling series for the logarithmic gamma function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
Source code for this entry:
    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)))))

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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