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Journal of the Ramanujan Mathematical Society

Volume 35, Issue 4, December 2020  pp. 373–389.

Representations by linear combinations of triangular numbers

Authors:  Zhi-Hong Sun
Author institution:School of Mathematics and Statistics, Huaiyin Normal University, Huaian, Jiangsu 223300, P.R. China

Summary:  Let Z and Z{+} be the set of integers and the set of positive integers, respectively. For a{1},…,a{k},n ∈ Z{+} let t(a{1},…,a{k};n) be the number of representations of n by a{1} x{1} (x{1} + 1)/2 + a{2} x{2} (x{2} + 1)/2+… +a{k}x{k} (x{k}+1)/2, and let N(a{1},…,a{k};n) be the number of representations of n by a{1} x{1}{2} + a{2} x{2}{2} + … + a{k} x{k}{2}, where x{1}, x{2}, …, x{k} ∈ Z. In this paper, using theta function identities we establish seventeen transformation formulas for t(a{1}, …,a{k}; n), and reveal many relations between t(a, b, c, d; n) and N (a, b, c, d; n), where a,b,c,d∈ Z{+}.

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