Umbral Calculus, Bailey Chains, and Pentagonal Number Theorems

We revisit the umbral methods used by L. J. Rogers in his second proof of the Rogers–Ramanujan identities. We shall study how subsequent methods such as the Bailey chains and their variants arise naturally from Rogersʹ insights. We conclude with the introduction of multi-dimensional Bailey chains and apply them to prove some new Pentagonal Number Theorems.