Polynomial identities of the Rogers--Ramanujan type

TL;DR

This paper introduces polynomial identities that generalize classical Euler and Rogers--Ramanujan identities, linking restricted partitions with generating functions and proving these identities through a graphical bijection.

Contribution

It presents new polynomial identities of Rogers--Ramanujan type and establishes a graphical bijection to prove these identities, extending classical partition identities.

Findings

Polynomial identities generalizing Euler and Rogers--Ramanujan identities.

A graphical bijection between two types of restricted partitions.

Proof of identities via combinatorial correspondence.

Abstract

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by establishing a graphical one-to-one correspondence between those two kinds of restricted partitions.