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

Abstract

We introduce and study a family of q-series {Ak(q)}k≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{A_k(q)\}_{k\ge 0}$$\end{document} arising from partitions whose distinct parts occur with odd multiplicities. Starting from their finite truncations Ak,m(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{k,m}(q)$$\end{document}, we derive several exact formulas expressing Ak(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_k(q)$$\end{document} 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, Ak(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_k(q)$$\end{document} 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 Ak(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_k(q)$$\end{document} 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.