论文标题
限制分区和Q派对分数
The Restricted Partition and q-Partial Fractions
论文作者
论文摘要
限制分区功能$ p_ {n}（n）$将$ n $的分区计入最多$ n $零件。在十九世纪，西尔维斯特（Sylvester）表明，这些分区可以表示为$ k $ - 周期性的准级别元素（$ 1 \ leq k \ leq n $），他称为波浪。现在众所周知，人们可以使用特殊类型的部分分数分解（所谓的$ q $ - 零件分数）轻松地执行波浪分解。在本文中，我们表明，这些$ Q $ - 优点的系数可以表示为Ramanujan总和的线性组合。特别是，我们首次展示了退化的伯努利数字，退化的欧拉数和Ramanujan总和的特殊概括，我们将其称为高斯 - 拉马尼亚·总和，在某些波的公式中。这些系数不仅提供了$ p_ {n}（n）$的良好近似值，而且还可以用于获得良好的界限。此外，我们为这些总和提供了组合含义。我们的部分分数方法是基于对多项式环的$ i $ addic完成的投影运算符，其中$ i $是循环多项式产生的理想。
The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Ramanujan sums. In particular, we show, for the first time, an appearance of the degenerate Bernoulli numbers, the degenerate Euler numbers and a special generalization of the Ramanujan sums, which we term as a Gaussian-Ramanujan sum, in the formulae for certain waves. These coefficients not only provide a good approximation of $p_{N}(n)$ but they can also be used for obtaining good bounds. Further, we provide a combinatorial meaning to these sums. Our approach for partial fractions is based on a projection operator on the $I$-adic completion of the ring of polynomials, where $I$ is an ideal generated by the Cyclotomic polynomial.
