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Fungrim entry: d69b41

a=1qχ ⁣(a)zeazeqz1=n=0Bn,χznn!\sum_{a=1}^{q} \chi\!\left(a\right) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}
Assumptions:qZ1andχGqandzCandz0andz<2πqq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, \left|z\right| < \frac{2 \pi}{q}
TeX:
\sum_{a=1}^{q} \chi\!\left(a\right) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, \left|z\right| < \frac{2 \pi}{q}
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Expez{e}^{z} Exponential function
GeneralizedBernoulliBBn,χB_{n,\chi} Generalized Bernoulli number
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
ConstPiπ\pi The constant pi (3.14...)
Source code for this entry:
Entry(ID("d69b41"),
    Formula(Equal(Sum(Mul(chi(a), Div(Mul(z, Exp(Mul(a, z))), Sub(Exp(Mul(q, z)), 1))), Tuple(a, 1, q)), Sum(Mul(GeneralizedBernoulliB(n, chi), Div(Pow(z, n), Factorial(n))), Tuple(n, 0, Infinity)))),
    Variables(q, chi, z),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(z, CC), Unequal(z, 0), Less(Abs(z), Div(Mul(2, ConstPi), q)))))

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

2019-08-17 11:32:46.829430 UTC