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."))

Topics using this entry

Eisenstein series