Engel expansions and the Rogers-Ramanujan identities
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 \(\prod^\infty_{n=1} {1\over 1-q^n}= 1+\sum^\infty_{n=1} {q^{n^2}\over (1-q)^2(1-q^2)^2 \cdots (1-q^n)^2}\).
scientific article; zbMATH DE number 1423160
scientific article