论文标题
卷积产物与连接到D'Arcais数字的组成式总和与Ramanujan Tau函数之间的同构
An isomorphism between the convolution product and the componentwise sum connected to the D'Arcais numbers and the Ramanujan tau function
论文作者
论文摘要
给定具有身份的通勤环$ r $，让$ h_r $是$ r $中的元素序列。我们研究了$（h_r， +）$和$（\ tilde h_r，*）$之间的一种新颖的同构，其中$ +$是componentwise和$*$是卷积产品（或cauchy产品）和$ \ tilde h_r $ sequiness semences of $ 1_r $。我们还对$ H_R $定义了递归转换，该转换与同构合并在一起，可以突出一些研究良好的整数序列之间的新关系。此外，这些连接允许引入连接到D'Arcais数字和Ramanujan Tau功能的多项式系列。通过这种方式，我们还推断了涉及贝尔多项式，除数函数和Ramanujan tau功能的关系。最后，我们重点介绍了Cauchy和Dirichlet产品之间的联系。
Given a commutative ring $R$ with identity, let $H_R$ be the set of sequences of elements in $R$. We investigate a novel isomorphism between $(H_R, +)$ and $(\tilde H_R,*)$, where $+$ is the componentwise sum, $*$ is the convolution product (or Cauchy product) and $\tilde H_R$ the set of sequences starting with $1_R$. We also define a recursive transform over $H_R$ that, together to the isomorphism, allows to highlight new relations among some well studied integer sequences. Moreover, these connections allow to introduce a family of polynomials connected to the D'Arcais numbers and the Ramanujan tau function. In this way, we also deduce relations involving the Bell polynomials, the divisor function and the Ramanujan tau function. Finally, we highlight a connection between Cauchy and Dirichlet products.
