On some new explicit evaluations of class invariants
type: Item
instance of: scholarly article
title: On some new explicit evaluations of class invariants
zbMATH Open document ID: 1177.11036
author: K. Sushan Bairy
author: M. S. Mahadeva Naika
published in: Vietnam Journal of Mathematics
publication date: 2008-08-12T00:00:00Z
review text: The Ramanujan-Weber class invariants \(G_n\) and \(g_n\) are \[ G_n := 2^{-1/4}q^{-1/24}\chi(q) \qquad\text{ and }\qquad g_n := 2^{-1/4}q^{-1/24}\chi(-q), \] where \(\chi(q) = (1+q)(1+q^3)(1+q^5)\cdots \) and \( q = e^{-\pi \sqrt{n}}. \) A number of values of \(G_n\) and \(g_n\) are given in this paper. Specifically, \(G_n\) is evaluated for \[ n = 30,10/3,42,14/3,78,26/3,102,34/3, \] while \(g_n\) is evaluated for \[ \begin{gathered} n = 15,5/3,21,7/3,33,11/3,39,13/3,57,19/3,93,31/3,177,59/3, \\ 60,20/3,84,28/3,120,40/3,132,44/3,156,52/3,168,56/3, \\ 228,76/3,312,104/3, 372,124/3,408,136/3. \end{gathered} \] The evaluations of \(G_n\), and \(g_n\) with \(n\) odd, are obtained using modular equations, other known class invariant evaluations, and relations involving \(G_n\), \(G_{9n}\), \(g_n\), \(g_{4n}\), \(g_{9n}\), and \(g_{36n}\). Details are given for \(G_{30}\), \(G_{10/3}\), \(g_{15}\) and \(g_{5/3}\), and the remaining cases are left to the reader. The evaluations of \(g_n\) with \(n\) even follow from the fact that \(g_{4n} = 2^{1/4}G_ng_n\). Values of the singular modulus \(\alpha_n\) (whose definition we do not give here, but which is equal to \((G_ng_{4n})^{-8}\)) are also recorded, for \[ \begin{gathered} n = 15,5/3,21,7/3,30,10/3,33,11/3,39,13/3,42,14/3, \\ 57,19/3,78,26/3,93,31/3,102,34/3.\end{gathered} \] The reader should be alerted that it will be necessary to consult Section 34 of [\textit{B. C. Berndt}, Ramanujan's notebooks. Part V. New York, NY: Springer (1998; Zbl 0886.11001)] for the known class invariant evaluations required for those proofs which are omitted. It might also be pointed out that in the authors' remark on p. \(123\), the identities numbered \((6.x)\) should be \((7.x)\).
reviewed by: Jeremy Lovejoy
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 5310220
zbMATH Keywords: Ramanujan-Weber class invariant
zbMATH Keywords: modular equations
zbMATH Keywords: hypergeometric series
zbMATH Keywords: singular moduli
zbMATH Keywords: elliptic integrals
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5310220