New modular equations in the spirit of Ramanujan
From the introduction: In his notebook ([Notebooks of Srinivasa Ramanujan. Vols. 1, 2. Bombay: Tata Institute of Fundamental Research (1957; Zbl 0138.24201)], [\textit{B. C. Berndt}, Ramanujan's Notebooks. III. New York etc.: Springer-Verlag (1991; Zbl 0733.11001), pp. 204--237]), \textit{S. Ramanujan} stated 23 beautiful modular equations of a type that we now describe. Following Ramanujan, set \[ f(-q):=(q;q)_\infty:=\prod_{n=1}^\infty (1-q^^n),\quad|q|<1. \] Ramanujan's modular equations involve quotients of the function \(f(-q)\) at certain arguments. For example [Berndt (op. cit), p. 206], let \[ P=\frac{f(-q)}{q^{1/6}f(-q^5)} \quad\text{and}\quad Q=\frac{f(-q^2)}{q^{1/3}f(-q^{10})}. \] Then \[ PQ+\frac{5}{PQ}=\left(\frac QP\right)^3+\left(\frac PQ\right)^3.\tag{1.1} \] Note that these modular equations are also called Schläfli-type. Since the publication of [Berndt (op. cit)], several authors, including N. D. Baruah, M.S. Mahadeva Naika, and K. R. Vasuki, have found additional modular equations of the type (1.1). Michael Somos [personal communication] recently used a computer to discover 21 new elegant modular equations in the spirit of the 23 original modular equations found by Ramanujan. He has a large list of eta-product identities, and he runs PARI/GP scripts to look at each identity to see if it is equivalent to an identity in \(P\)-\(Q\) forms. The purpose of this paper is to prove these modular equations found by Somos. Some of Somos's equations are simple consequences of others, and we do record only ten of them in Sect. 3. Moreover, some of our proofs are similar to each other, and so, in the interest of brevity and avoiding the repetition of ideas, we provide proofs for some of these equations and simply indicate the basic ideas for remaining proofs. In Sect. 4, we establish combinatorial interpretations of some of our theorems.
scientific article; zbMATH DE number 6120466
scientific article