Analogues of Ramanujan’s partition identities and congruences arising from his theta functions and modular equations

In this paper, we study the partition function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{[c^{l}d^{m}]}(n)$\end{document} defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum_{n=0}^{\infty}p_{[c^{l}d^{m}]}(n)q^{n}=(q^{c};q^{c})_{\infty}^{-l}(q^{d};q^{d})_{\infty}^{-m}$\end{document} and prove some analogues of Ramanujan’s partition identities. We also deduce some interesting partition congruences.