Transformation formulas for the number of representations of \(n\) by linear combinations of four triangular numbers

Authors

Keywords:

theta function, triangular number, quadratic form

Abstract

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2 $ $(x,y,z,w\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $\varphi(q)$ and $\psi(q)$ we present some transformation formulas for $t(a,b,c,d;n)$, and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.

Published

2020-07-21

Issue

Section

Articles