Proof of a conjecture of Das on the coefficients of mock theta functionsProof of a conjecture of Das on mock theta functionsD. Tang

Ramanujan recorded seventeen mock theta functions in his last letter to Hardy. Quite recently, Das (J Math Anal Appl 543(2):128913, 2025) proved some congruences modulo small powers of 3 between the coefficients of the second order mock theta functions μ2(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _2(q)$$\end{document} and A2(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2(q)$$\end{document}, introduced by Ramanujan and McIntosh, respectively. Moreover, Das conjectured a congruence modulo 2187 and three congruences modulo 6561 between the coefficients of μ2(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _2(q)$$\end{document} and A2(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2(q)$$\end{document}. In this paper, we prove these conjectural congruences by employing some q-series manipulations. Further, we conjecture an infinite family of congruences modulo high powers of 3 between the coefficients of μ2(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _2(q)$$\end{document} and A2(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2(q)$$\end{document}.