Proofs of some conjectures of Sun on the relations between t(a, b, c; n) and N(a, b, c; n)

Liping Cao () and Bernard L. S. Lin ()
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Liping Cao: Jimei University
Bernard L. S. Lin: Jimei University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 4, 1081-1098

Abstract: Abstract Let $${\mathbb {Z}}$$ Z and $${\mathbb {Z}}^{+}$$ Z + be the set of integers and the set of positive integers, respectively. For $$a, b, c, n\in {\mathbb {Z}}^{+}$$ a , b , c , n ∈ Z + and $$x, y, z\in {\mathbb {Z}}$$ x , y , z ∈ Z , let N(a, b, c; n) be the number of representations of n as $$ax^2+by^2+cz^2$$ a x 2 + b y 2 + c z 2 and t(a, b, c; n) be the number of representations of n as $$a\frac{x(x+1)}{2}+b\frac{y(y+1)}{2}+c\frac{z(z+1)}{2}$$ a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 . Recently, Sun established many relations between t(a, b, c; n) and $$N(a,b,c;8n+a+b+c)$$ N ( a , b , c ; 8 n + a + b + c ) and listed 43 relations need to be confirmed. More recently, Xia and Zhang proved 19 relations conjectured by Sun. In this paper, by employing Ramanujan’s theta function identities, we prove the remaining 24 relations.

Keywords: Ramanujan’s theta function identities; Sum of squares; Sum of triangular numbers; 11D85; 11E25 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00324-8

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