Figurate numbers, forms of mixed type and their representation numbers

TL;DR

This paper develops formulas for counting representations of natural numbers by sums of figurate numbers with coefficients, establishing their modular properties and deriving explicit formulas using modular form bases.

Contribution

It introduces a novel method to derive formulas for representation numbers of mixed figurate forms via modular forms, extending previous results and providing explicit parametrizations.

Findings

Proves generating functions are modular forms under certain conditions.

Derives explicit formulas for representation counts of mixed figurate forms.

Provides $(p,k)$ parametrization of Eisenstein series $E_4( au)$ and its duplications.

Abstract

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 condition on the coefficients when even number of higher figurate numbers are considered. In particular, we obtain modular property of the generating function corresponding to a sum of even number of triangular numbers with coefficients. We also obtain 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 $m^2+mn+n^2$ 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 \cite{ono}. In \cite{xia}, Xia-Ma-Tian considered some special cases of mixed forms and obtained the number of representations of these 21 mixed forms using the $(p,k)$ parametrisation method. We also derive these 21 formulas using our method and further obtain as a consequence, the $(p,k)$ parametrisation of the Eisenstein series $E_4(\tau)$ and its duplications. It is to be noted that the $(p,k)$ parametrisation of $E_4$ and its duplications were derived by a different method in \cite{{aw},{aaw}}. We illustrate our method with several examples.