On triangular numbers, forms of mixed type and their representation numbers

TL;DR

This paper explores the representation of natural numbers by triangular numbers with coefficients and mixed forms, using modular forms to derive formulas and connect to Eisenstein series, extending previous work with new methods.

Contribution

It introduces a modular form approach to derive representation formulas for triangular numbers with coefficients and general mixed forms, providing new proofs and parametrizations.

Findings

Derived formulas for representation numbers using modular forms.

Established modular properties for generating functions of mixed forms.

Connected representation formulas to Eisenstein series parametrizations.

Abstract

In \cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of determining formulas for the number of representations of a natural number $n$ by a sum of $k$ triangular numbers and derived many applications, including the one connecting these numbers with the number of representations of $n$ as a sum of $k$ odd square integers. They also obtained an application to the number of lattice points in the $k$-dimensional sphere. In this paper, we consider triangular numbers with positive integer coefficients. First we show that if the sum of these coefficients is a multiple of $8$, then the associated generating function gives rise to a modular form of integral weight (when even number of triangular numbers are taken). We then use the theory of modular forms to get the representation number formulas corresponding to the triangular numbers with coefficients. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in \cite{ono}. In the second part of the paper, we consider more general mixed forms (as done in Xia-Ma-Tian \cite{xia}) and derive modular properties for the corresponding generating functions associated to these mixed forms. Using our method we deduce all the 21 formulas proved in \cite[Theorem 1.1]{xia} and show that our method of deriving the 21 formulas together with the $(p,k)$ parametrization of the generating functions of the three mixed forms imply the $(p,k)$ parametrization of the Eisenstein series $E_4(\tau)$ and its duplications. It is to be noted that the $(p,k)$ parametrization of $E_4$ and its duplications were derived by a different method by K. S. Williams and his co-authors. In the final section, we provide sample formulas for these representation numbers in the case of 4 and 6 variable forms.