Engel expansions and the Rogers-Ramanujan identities
type: Item
instance of: scholarly article
title: Engel expansions and the Rogers-Ramanujan identities
zbMATH Open document ID: 0954.11028
author: George E. Andrews
author: Arnold Knopfmacher
author: John Knopfmacher
published in: Journal of Number Theory
publication date: 2000-06-07T00:00:00Z
review text: The authors introduce a generalization of the notion of Engel series and use it to give new proofs of identities first proven by Euler, Jacobi and Rogers and Ramanujan. The Engel series is an algorithm for representing real numbers and is of the form \(t=\sum^\infty_{n=1} {1\over a_1a_2\cdots a_n}\), where \(a_1,a_2, \dots\) is a non-decreasing sequence of natural numbers with \(a_1\geq 2\). The generalization of this algorithm is to the formal Laurent series over the field of complex numbers \(\mathbb{C}\). If \(A= \sum^\infty_{n= \nu}c_nq^n\), \(\nu=\nu(A)\) is called the order of \(A\) and \(\|A\|= 2^{-\nu(A)}\) is called the norm of \(A\); the integral part of \(A\) is defined as \(\sum_{\nu\leq n\leq 0}c_nq^n\). Among the applications, first a conjecture of Ramanujan is mentioned. It is an identity of an infinite product and an infinite series, where the sequence of primes occurs. Ramanujan crossed out this conjecture in his notebook, and the first author of the present paper disproved it, using a computer program. The remainder of the paper gives new proofs of identities known in the theory of partitions. Let us quote a few of them. Theorem 1. We have \(\prod^\infty_{n=1}(1+zq^n)= 1+\sum^\infty_{n=1} {z^nq^{n(n+1)/2} \over(1-q)(1-q^2)\cdots (1-q^n)}\). Theorem 4. We have \(\prod^\infty_{n=1} {1\over 1-q^n}= 1+\sum^\infty_{n=1} {q^{n^2}\over (1-q)^2(1-q^2)^2 \cdots (1-q^n)^2}\).
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 11P82
zbMATH DE Number: 1423160
zbMATH Keywords: Engel series
zbMATH Keywords: formal Laurent series
zbMATH Keywords: identities
zbMATH Keywords: partitions
MaRDI profile type: Publication
full work available at URL: jnth.1999.2453
OpenAlex ID: W2088826234
cites work: Asymptotic analysis of Ramanujan pairs
cites work: An Analytic Generalization of the Rogers-Ramanujan Identities for Odd Moduli
cites work: Q4146776
cites work: Q3861471
cites work: Q3218192
cites work: A Computer-Assisted Investigation of Ramanujan Pairs
cites work: Q4040189
cites work: Q3887469
cites work: Inverse polynomial expansions of Laurent series
cites work: Inverse polynomial expansions of Laurent series. II
cites work: Q4841803
cites work: Further Identities of the Rogers-Ramanujan Type
DOI: 10.1006/JNTH.1999.2453
description: scientific article; zbMATH DE number 1423160