Fungrim entry: d69b41

$\sum_{a=1}^{q} \chi(a) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; \left|z\right| < \frac{2 \pi}{q}$
TeX:
\sum_{a=1}^{q} \chi(a) \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
Sum$\sum_{n} f(n)$ Sum
Exp${e}^{z}$ Exponential function
GeneralizedBernoulliB$B_{n,\chi}$ Generalized Bernoulli number
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Pi$\pi$ 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))), For(a, 1, q)), Sum(Mul(GeneralizedBernoulliB(n, chi), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)))),
Variables(q, chi, z),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(z, CC), NotEqual(z, 0), Less(Abs(z), Div(Mul(2, Pi), q)))))

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