We provide two completely different proofs of some q-series identities of the form (infinite product) = (infinite sum). One is an analytical method that exploits the fact that q-series are modular forms. The proof of modularity on the infinite sum side uses the theory of indefinite theta functions proposed by Zwegers et al. The other proof is an algebraic method that uses the denominator formula of affine Lie superalgebras, and is based on the idea presented in Kac-Wakimoto’s 1994 paper. This is joint work with Toshiki Matsusaka (Kyushu Univ.).