On the q-binomial identities involving the Legendre symbol modulo 3

TL;DR

This paper proves new q-binomial identities involving the Legendre symbol modulo 3 using polynomial analogues of classical identities and extends them into infinite hierarchies of q-series identities, connecting to known identities by Ramanujan and others.

Contribution

It introduces novel q-binomial identities involving the Legendre symbol modulo 3 and extends them into infinite hierarchies using Bailey's lemma, linking to classical identities.

Findings

Proved six new q-binomial identities involving the Legendre symbol modulo 3

Extended identities to infinite hierarchies of q-series identities

Connected new identities to classical results by Ramanujan, Slater, McLaughlin, and Sills

Abstract

I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.