# Fungrim entry: 5540a1

$G_{2 k}\!\left(\tau\right) = \frac{3}{\left(2 k + 1\right) \left(k - 3\right) \left(2 k - 1\right)} \sum_{r=2}^{k - 2} \left(2 r - 1\right) \left(2 k - 2 r - 1\right) G_{2 r}\!\left(\tau\right) G_{2 k - 2 r}\!\left(\tau\right)$
Assumptions:$k \in \mathbb{Z}_{\ge 4} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
References:
• T. Apostol (1976), Modular Functions and Dirichlet Series in Number Theory, Springer. Theorem 1.13.
TeX:
G_{2 k}\!\left(\tau\right) = \frac{3}{\left(2 k + 1\right) \left(k - 3\right) \left(2 k - 1\right)} \sum_{r=2}^{k - 2} \left(2 r - 1\right) \left(2 k - 2 r - 1\right) G_{2 r}\!\left(\tau\right) G_{2 k - 2 r}\!\left(\tau\right)

k \in \mathbb{Z}_{\ge 4} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
EisensteinG$G_{k}\!\left(\tau\right)$ Eisenstein series
Sum$\sum_{n} f(n)$ Sum
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("5540a1"),
Formula(Equal(EisensteinG(Mul(2, k), tau), Mul(Div(3, Mul(Mul(Add(Mul(2, k), 1), Sub(k, 3)), Sub(Mul(2, k), 1))), Sum(Mul(Mul(Mul(Sub(Mul(2, r), 1), Sub(Sub(Mul(2, k), Mul(2, r)), 1)), EisensteinG(Mul(2, r), tau)), EisensteinG(Sub(Mul(2, k), Mul(2, r)), tau)), For(r, 2, Sub(k, 2)))))),
Variables(k, tau),
Assumptions(And(Element(k, ZZGreaterEqual(4)), Element(tau, HH))),
References("T. Apostol (1976), Modular Functions and Dirichlet Series in Number Theory, Springer. Theorem 1.13."))

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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