Ramanujan's theta functions and sums of triangular numbers

2016/01/24 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.1601.06378

Abstract

Let \Bbb Z and \Bbb N be the set of integers and the set of positive integers, respectively. For a₁,a₂,…,a_k,n∈\Bbb N let N(a₁,a₂,…,a_k;n) be the number of representations of n by a₁x₁²+a₂x₂²+⋯+a_kx_k², and let t(a₁,a₂,…,a_k;n) be the number of representations of n by a₁\fracx₁(x₁-1)2+a₂\fracx₂(x₂-1)2+⋯+a_k\fracx_k(x_k-1)2 (x₁,…,x_k∈\Bbb Z). In this paper, by using Ramanujan's theta functions φ(q) and ψ(q) we reveal many relations between t(a₁,a₂,…,a_k;n) and N(a₁,a₂,…,a_k;8n+a₁+⋯+a_k) for k=3,4.

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