A few theta-function identities and some of Ramanujan's modular equations
Let \(\varphi(q)=\sum_{k=-\infty}^\infty q^{k^2}\), \(\psi(q)=\sum_{k=0}^\infty q^{k(k+1)/2}\) and \[ \alpha(q)=16q\frac{\psi^4(q^2)}{\varphi^4(q)}. \] A modular equation of degree \(n\) is defined as an identity satisfied by \(\alpha(q)\) and \(\alpha(q^n)\). A modular equation of composite degrees \(r,s,rs\) is an identity satisfied by \(\alpha(q^j)\) with \(j=1,r,s,\) and \(rs\). In his notebooks, S. Ramanujan recorded many modular equations without proofs. The proofs of these modular equations are now available, thanks to \textit{B. C. Berndt} and his collaborators [see Ramanujan's notebooks. III, Springer, New York (1991; Zbl 0733.11001); IV (1994; Zbl 0785.11001); V (1998; Zbl 0886.11001)]. Unfortunately, some of the proofs provided by Berndt and others are verifications which require the knowledge of the modular equations in advance. As such, new proofs are needed in order to shed some light on the methods used by Ramanujan in his discovery of these identities. In this paper, the author succeeds in using elementary methods and theta-functions identities to derive elegant modular equations such as \[ Q^4 +\frac{1}{Q^4}-2\left(P^2+\frac{1}{P^2}\right)+3=0, \] where \[ P=\{256 \alpha\beta\gamma\delta(1-\alpha)(1-\beta)(1-\gamma)(1-\delta)\}^{1/48} \] and \[ Q=\left(\frac{\beta\delta(1-\beta)(1-\delta)}{\alpha\gamma(1-\alpha)(1-\gamma)}\right)^{1/48}, \] with \(\alpha=\alpha(q),\beta=\alpha(q^3),\gamma=\alpha(q^5),\) and \(\delta=\alpha(q^{15}).\) The reviewer wishes to highlight a paper written by \textit{M. L. Lang} and himself [Isr. J. Math. 103, 1-16 (1998; Zbl 0922.11040)] which indicates that the theory of modular forms can be used to derive (instead of just verifying) many new modular equations analogous to those considered by S. Ramanujan.
scientific article; zbMATH DE number 1563049