On conjectural fermionic formulas for the Macdonald index in Argyres-Douglas theories

Abstract: We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type $(A_1, D_{2k+1})$, thereby yielding a conjectural fermionic formula due to Andrews et al. Our duality is built upon a new conjugate Bailey pair to be established using techniques from orthogonal polynomials and basic hypergeometric series. In addition, this fermionic formula implies another sum-like expression for the Macdonald index conjectured by Kim, Kim, and Song.