用二项式和逆算子变换递归序列
Transforming Recurrent Sequences by Using the Binomial and Invert Operators
摘要：
本文研究了二项式算子和逆（内插）算子在线性递归序列集合上的作用。我们证明，这些运营商保持这一套，我们确定他们如何改变的特征多项式。我们表明，这些运营商，与其他两个基本运营商（基本上是左，右移位）的帮助下，可以转换任何脉冲序列（从$(0,...,0,1)$开始的线性递归序列）到任何其他脉冲序列，通过两个过程，我们称之为\emph{construction}和\emph{deconstruction}。最后，我们给出了多项式序列和金字塔数的一些应用。证明了$r$ -斐波那契数数是$(r-1)$--bonacci数的Bell多项式变换.
英文摘要：
In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from $(0,...,0,1)$) into any other impulse sequence, by two processes that we call \emph{construction} and \emph{deconstruction}. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that $r$--bonacci numbers are a Bell polynomial transform of the $(r-1)$--bonacci numbers.
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