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A theta function identity and the Eisenstein series on \(\Gamma_0(5)\). (English) Zbl 1143.11017

Let \(q= \exp(2\pi i\tau)\), where \(\text{Im}(\tau)> 0\). For integer \(k\geq 1\) the author defines
\[ \begin{aligned} L_k(\tau)&= \sum^\infty_{n=1} {n^{k-1}(q^n- q^{2n}- q^{3n}+ q^{4n})\over 1- q^{5n}},\\ V(z|\tau)&= 4 \sum^\infty_{k=1} (-1)^{k-1} {(2z)^{2k-1}\over (2k-1)!} L_{2k}(\tau).\end{aligned} \]
Two differential equations for \(V(z|\tau)\) lead to two recurrence relations for \(L_{2k}(\tau)\). These, in turn, express \(L_{2k}(\tau)\) in terms of \(h(\tau)= \eta^6(5\tau)/\eta^6(\tau)\) and \(k(\tau)= \eta^5(\tau)/\eta(5\tau)\), where \(\eta(\tau)\) is Dedekind’s eta function.
Connections with the Rogers-Ramanujan continued fraction are also given.

MSC:

11F27 Theta series; Weil representation; theta correspondences
11F20 Dedekind eta function, Dedekind sums
33E05 Elliptic functions and integrals
11F11 Holomorphic modular forms of integral weight
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