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Fungrim entry: 5540a1

G2k ⁣(τ)=3(2k+1)(k3)(2k1)r=2k2(2r1)(2k2r1)G2r ⁣(τ)G2k2r ⁣(τ)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:kZ4andτHk \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
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
HHH\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)), Tuple(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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2019-08-19 14:38:23.809000 UTC