论文标题
Lipschitz求和公式的应用和Raabe余弦变换的概括
Applications of Lipschitz summation formula and a generalization of Raabe's cosine transform
论文作者
论文摘要
事实证明，一般求和公式在分析，数理论和其他数学分支中非常有用。 Lipschitz总和是其中之一。在本文中，我们通过提供了一种概括Ramanujan的新转换公式来提供其应用程序。 Ramanujan的结果反过来是对Sl $ _2（\ Mathbb {Z}）$上的Eisenstein系列$ e_k（z）$的模块化转换的概括，其中$ z \ to-1/z，z \ in \ mathbb {h} $。我们结果的证明涉及包含Cauchy主值积分的精致分析。凯萨瓦尼（Kesarwani）的最新结果的简单证明给出了$ \ sum_ {n = 1}^{\ infty}σ_{2m}（n）e^{ - ny} $的非模块化转换。为了获得这种转变，我们自然会遇到Raabe余弦变换的新概括，该属性也得到了证明。作为这种结果的必然，我们对赖特的渐近估算值进行了概括，以概述正整数$ n $的平面分区数量的生成功能。
General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan's result, in turn, is a generalization of the modular transformation of Eisenstein series $E_k(z)$ on SL$_2(\mathbb{Z})$, where $z\to-1/z, z\in\mathbb{H}$. The proof of our result involves delicate analysis containing Cauchy Principal Value integrals. A simpler proof of a recent result of ours with Kesarwani giving a non-modular transformation for $\sum_{n=1}^{\infty}σ_{2m}(n)e^{-ny}$ is also derived using the Lipschitz summation formula. In the pursuit of obtaining this transformation, we naturally encounter a new generalization of Raabe's cosine transform whose several properties are also demonstrated. As a corollary of this result, we get a generalization of Wright's asymptotic estimate for the generating function of the number of plane partitions of a positive integer $n$.
