论文标题
一个同时递归的系统：LED-Shannon-Ollerton身份的概括
A System of Four simultaneous Recursions: Generalization of the Ledin-Shannon-Ollerton Identity
论文作者
论文摘要
本文进一步概括了香农和奥勒顿的最新结果，他们复活了由于LEDIN引起的旧身份。本文将LED-Shannon-Ollerton概括为所有金属序列。结果为第一个$ n $整数的产品的产品和金属序列的第一个$ n $成员提供了封闭的公式。本文的三个关键创新是i）将概括性的证明减少到解决4个同时递归系统的解决方案； ii）使用移位操作证明多项式的平等； iii）由满足4个同时递归的四个多项式家族的系数产生的新OEI序列。
This paper further generalizes a recent result of Shannon and Ollerton who resurrected an old identity due to Ledin. This paper generalizes the Ledin-Shannon-Ollerton result to all the metallic sequences. The results give closed formulas for the sum of products of powers of the first $n$ integers with the first $n$ members of the metallic sequence. Three key innovations of this paper are i) reducing the proof of the generalization to the solution of a system of 4 simultaneous recursions; ii) use of the shift operation to prove equality of polynomials; and iii) new OEIS sequences arising from the coefficients of the four polynomial families satisfying the 4 simultaneous recursions.
