Figurate numbers, forms of mixed type, and their representation numbers

In this article, we consider the problem of determining formulas for the number of representations of a natural number n by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the associated generating function gives rise to a modular form of integral weight under certain conditions on the coefficients when even number of higher figurate numbers are considered. In particular, we obtain the modular property of the generating function corresponding to a sum of even number of triangular numbers with coefficients. We also obtain the modularity property of the generating function of mixed forms involving figurate numbers (including the squares and triangular numbers) with coefficients and forms of the type m2+mn+n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m<^>2+mn+n<^>2$$\end{document} with coefficients. In particular, we show the modularity of the generating function of odd number of squares and odd number of triangular numbers (with coefficients). As a consequence, explicit formulas for the number of representations of these mixed forms are obtained using a basis of the corresponding space of modular forms of integral weight. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in Ono et al. (Aequat Math 50:73-94, 1995). In 2016, Xia et al. (Int J Number Theory 12:945-954) considered some special cases of mixed forms and obtained the number of representations of these 21 mixed forms using the (p, k) parametrization method. We also derive these 21 formulas using our method and further obtain as a consequence, the (p, k) parametrization of the Eisenstein series E4(tau)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4(\tau )$$\end{document} and its duplications. It is to be noted that the (p, k) parametrization of E4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4$$\end{document} and its duplications were derived by a different method in [3, 8]. We illustrate our method with several examples.