一个Ramanujan Tau函数的新表达式

A new representation of the Ramanujan′s Tau function

通过对Ramanujan Tau函数的研究,并借助Ewell的一个关于q-级数的恒等式结果,发现了关于Ramanujan Tau函数的生成函数的恒等式,该恒等式中含有两类重要的算术函数,即表正整数为若干三角数的和的表法数及表正整数为若干平方数的和的表法数.从而得到了一个关于Ramanujan Tau函数新的显式表达式.最后作为结果的应用,该文还给出了一个关于Ramanujan Tau函数新的简洁同余恒等式.

Utilizing one result of Ewell, we investigate the Ramanujan＇s tau function. We then find a new identity of the generating function for the Ramanujan＇s function involving sums of square numbers and sums of triangle numbers, which deduces a new representation of the Ramanujan＇s tau function. As one application of our result, a concise congruence of the Ramanujan＇s tau function follows lastly.