Signed Enumeration of Partitions with Exactly k Distinct Magnitudes of Odd Multiplicity

Abstract

Abstract We introduce and study a family of q -series $$\{A_k(q)\}_{k\ge 0}$$ { A k ( q ) } k ≥ 0 arising from partitions whose distinct parts occur with odd multiplicities. Starting from their finite truncations $$A_{k,m}(q)$$ A k , m ( q ) , we derive several exact formulas expressing $$A_k(q)$$ A k ( q ) in terms of Gaussian coefficients, partial theta functions, and classical q -products. The main results establish new identities connecting these functions to the generating function for partitions in which all odd parts are distinct. Analytically, $$A_k(q)$$ A k ( q ) admits a representation involving Chebyshev polynomials and the Jacobi Triple Product, revealing a theta-like structure reminiscent of the Rogers–Ramanujan identities. We further conjecture that the coefficients of $$A_k(q)$$ A k ( q ) are non-negative, suggesting the existence of a direct combinatorial model. These results highlight a close interplay between partition theory, q -hypergeometric transformations, and analytic q -series of Rogers–Ramanujan type.