Revisiting a Classic Identity That Implies the Rogers–Ramanujan Identities II

Hei-Chi Chan

doi:10.3390/axioms10040239

Abstract

We give a new proof of an identity due to Ramanujan. From this identity, he deduced the famous Rogers–Ramanujan identities. We prove this identity by establishing a simple recursion Jk=qkJk−1, where |q|<1. This is a sequel to our recent work.

Highlights

IntroductionClassic Identity That Implies the Rogers-Ramanujan Identities II. n =1 are perhaps one of the most important and celebrated results in the theory of partition
The Rogers–Ramanujan identities, given by ∞Citation: Chan, H.-C
This, in turn, gives a new proof of (3), from which the Rogers–Ramanujan identities are derived

Summary

Introduction

Classic Identity That Implies the Rogers-Ramanujan Identities II. n =1 are perhaps one of the most important and celebrated results in the theory of partition. Note that it was Hardy who arranged the publication of [5], in which Rogers and Ramanujan independently contributed proofs of (1) and (2). Many terms, when we multiply out the right-hand side, should cancel each other. For each of these functions, if one or more of its arguments lies outside the range specified on the right-hand side, the function is defined to be zero It turns out α± (k)r are canceled by their counterparts (see Theorem 1.2 and Lemma 3.2 in [3]). The proof in [3] focuses on cancellations: it is based on how the terms on the right-hand side of (5), when multiplied out, cancel each other. This is avoided and the main calculations involved are those of proving (13) and (14), which are relatively simpler when compared with the proof in [3]

Proof of the Main Result

Conclusions