关于线性递归序列代数结构的几点注记
Some notes on the algebraic structure of linear recurrent sequences
摘要：
可以在所有线性递归序列的集合上定义几种运算，例如二项式卷积（Hurwitz乘积）或多项式卷积（牛顿乘积）。利用初等技巧，我们证明了这个集合配备的逐项和和上述产品是R-代数，给定任何交换环$R$的单位元。此外，我们还明确地给出了任意两个线性递归序列的Hurwitz积和Newton积的特征多项式。最后，我们还研究了这些$R-$代数是否同构，同时考虑了使用Hadamard积和卷积积得到的R-代数。
英文摘要：
Several operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products are R-algebras, given any commutative ring $R$ with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these $R-$algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.
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