Liu, Zhi-Guo A theta function identity and the Eisenstein series on \(\Gamma_0(5)\). (English) Zbl 1143.11017 J. Ramanujan Math. Soc. 22, No. 3, 283-298 (2007). 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. Reviewer: Tom M. Apostol (Pasadena) Cited in 10 Documents 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 Keywords:Rogers-Ramanujan continued fraction PDFBibTeX XMLCite \textit{Z.-G. Liu}, J. Ramanujan Math. Soc. 22, No. 3, 283--298 (2007; Zbl 1143.11017)