论文标题
与环流多项式相关的某些部分分数身份
Some Partial Fraction Identities associated with the Cyclotomic Polynomials
论文作者
论文摘要
我们为理性功能建立了一些部分分数身份，其分母是环形多项式的隐式产物。为了实现这一目标，我们首先开发了一种由Heviside掩盖方法启发的部分分数分解的一般代数方法。因此，我们将我们的方法称为扩展的掩盖方法。使用我们的方法，我们获得了某些生成功能的$ Q $ - 零件分数的直接公式。作为我们公式的直接结果，可以计算伪多项式时间中的sylvester numerant，frobenius数和ehrhart多项式。此外，我们提供了一个框架，用于对傅立叶 - 迪德金总和及其相关的rademacher互惠定理的概括，从而扩展了Carlitz，Zagier和Gessel的结果。通过进行傅立叶分析，我们证明了我们的扩展掩盖方法简单地解释了互惠法背后的机制。
We establish some partial fraction identities for rational functions whose denominators are implicit products of the cyclotomic polynomials. To achieve this, we first develop a general algebraic approach for partial fraction decomposition inspired by the Heaviside's cover-up method. We thus call our method the Extended Cover-Up Method. Using our method we obtain direct formulas for $q$-partial fractions for certain generating functions. As a direct consequence of our formulas one can compute the Sylvester denumerants, the Frobenius number and the Ehrhart polynomials in pseudo-polynomial time. Further, we provide a framework for a generalization of the Fourier-Dedekind sum and their associated Rademacher reciprocity theorem extending the results of Carlitz, Zagier and Gessel. By performing a Fourier analysis we demonstrate that our extended cover-up method explains in simple terms the mechanism behind the reciprocity law.
