D'Arcais数与Ramanujan tau函数的卷积积与分量和的同构
An isomorphism between the convolution product and the componentwise sum connected to the D'Arcais numbers and the Ramanujan tau function
摘要：
给定一个有单位元的交换环$R$，设$H_R$是$R$中元素序列的集合。我们研究了$(H_R, +)$和$(\tilde H_R,*)$之间的一种新的同构，其中$+$是分量和，$*$是卷积积（或柯西积），$\tilde H_R$是以$1_R$开始的序列集.我们还定义了一个递归变换的$H_R$，一起同构，允许突出新的关系，一些研究的整数序列。此外，这些连接允许引入连接到D'Arcais数和Ramanujan tau函数的多项式族。通过这种方式，我们还推导出贝尔多项式、除数函数和拉马努金τ函数的关系。最后，我们强调柯西和狄利克雷产品之间的连接。
英文摘要：
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.
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