Proofs of some conjectures of Sun on the relations between \(T(a, b, c, d; n)\) and \(N(a, b, c, d; n)\). (English) Zbl 1497.11091

Cao, Liping; Lin, Bernard L. S.

Rocky Mt. J. Math. 52, No. 3, 829-842 (2022).

Summary: Let \(N(a, b, c, d; n)\) be the number of representations of \(n\) as \(ax^2 + by^2 + cz^2 + dw^2\) and \(T(a, b, c, d; n)\) be the number of representations of \(n\) as \[a\frac{X(X + 1)} {2} + b \frac{Y(Y + 1)} {2} + c \frac{Z(Z + 1)} {2} + d\frac{W(W + 1)} {2},\] where \(a, b, c, d\) are positive integers, \(x, y, z, w\) are integers, and \(n, X, Y, Z, W\) are nonnegative integers. In this paper, by using theta function identities, we confirm Sun’s conjectures that \(T(a, b, c, d; n)\) is a linear combination of \(N(a, b, c, d; n)\) and \(N(a, b, c, d; 4m)\), with \(m = 8n + a + b + c + d\), for several values of \(a, b, c, d\).

MSC:

11D85	Representation problems
11E25	Sums of squares and representations by other particular quadratic forms

Keywords:

theta function identities; sum of squares; sum of triangular numbers

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