Proofs of four Takigiku--Tsuchioka conjectures on level $2$ principal characters of $A_{2\ell-1}^{(2)}$

Abstract

We prove four conjectures of Takigiku and Tsuchioka consisting of eight Rogers--Ramanujan type identities. Two conjectures involve triple sums and a third involves quadruple sums, and their product sides are certain level $2$ principal characters of $A_{13}^{(2)}$. The fourth conjecture gives multisum representations of certain level $2$ principal characters of $A_{2\ell-1}^{(2)}$. The proofs rely on the Bailey pair method, finite sum identities involving Gaussian polynomials, and the quintuple product identity.