Some relations between t(a,b,c,d;n) and N(a,b,c,d;n)

TL;DR

This paper explores the mathematical relationships between two functions counting representations of integers by quadratic forms and their related triangular form representations, revealing new connections between these counting functions.

Contribution

It establishes novel links between the functions t(a,b,c,d;n) and N(a,b,c,d;n), enhancing understanding of their interrelations in number theory.

Findings

Identifies explicit formulas connecting t and N functions.

Provides new insights into representations of integers by quadratic and triangular forms.

Enhances theoretical understanding of form representations in number theory.

Abstract

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and 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 we reveal some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$.