Let N(a, b, c, d; n) be the number of representations of n as ax²+by²+cz²+dw² 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, n, X, Y, Z, W are nonnegative integers, and x, y, z, w are integers. Recently, Z.-H. Sun found many relations between N(a, b, c, d, n) and T(a, b, c, d, n) and conjectured 23 more relations. Yao proved five of Sun’s conjectures by using (p, k)-parametrization of theta functions and stated that six more could be proved by using the same method. More recently, Sun himself confirmed two more conjectures by proving a general result whereas Xia and Zhong proved three more conjectures of Sun by employing theta function identities. In this paper, we prove the remaining seven conjectures. Six are proved by employing Ramanujan’s theta function identities and one is proved by elementary techniques.

中文翻译：

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Z. -H。的一些猜想的证明。关于平方和与三角数和之间的关系的太阳

令N（a，b，c，d ; n）为n的表示数，表示为ax ² + by ² + cz ² + dw ²，T（a，b，c，d，n）为表示数。的ñ作为\（一个\压裂{{X（X + 1）}} {2} + b \压裂{{Y（Y + 1）}} {2} + C \压裂{{Z（Z + 1） }} {2} + d \ frac {{W（W + 1）}} {2} \），其中a，b，c，d为正整数，n，X，Y，Z，W为非负整数，x，y，z，w为整数。最近，Z.-H。Sun发现N（a，b，c，d，n）与T（a，b，c，d，n）之间有许多关系），并推测还有23个关系。姚通过使用theta函数的（p，k）参数化证明了Sun的五个猜想，并说使用相同的方法可以证明另外六个。最近，Sun自己通过证明一个一般结果又证实了两个猜想，而Xia和Zhong通过使用theta函数恒等式证明了Sun的另外三个猜想。在本文中，我们证明了剩余的七个猜想。通过使用Ramanujan的theta函数恒等式证明了六种，通过基础技术证明了一种。