论文标题
某些兰伯特系列的明确转换
Explicit transformations of certain Lambert series
论文作者
论文摘要
我们称之为\ emph {master Identity}的确切转换是第一次获得$ \ sum_ {n = 1}^{\ infty}σ_{a}（a}（a}（n）e^{ - ny} $的$ a \ in \ in \ nathbb {c} $（c} $（y Mathbb）和re $ $（y $ $（y））。当$ a $是一个非零的整数时，新的模块化型转换是其特殊情况。在这些转换中明确看到了模块化的精确障碍。其中包括Ramanujan著名公式（2M+1）$的新颖伴侣。源自主身份的$ a = 0 $情况的Wigert-Bellman身份也被得出。当$ a $是一个奇怪的整数时，Eisenstein系列在$ \ textup {sl} _ {2} \ left（\ Mathbb {z} \ right）$上的众所周知的模块化转换，Dedekind Eta的功能以及Ramanujan的功能以及Ramanujan的for ym $ Qm（2MM+1）$ sentitive nessitive。后者的身份本身是使用吉南（Guinand）的Vorono \“ {\ dotlessi}的求和公式和N.〜S.〜Koshliakov进行的整体评估，涉及修改后的Bessel函数$K_ν（Z）$的整体评估。修改后的Bessel函数的两种变量概括。
An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}σ_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a non-zero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan's famous formula for $ζ(2m+1)$. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. When $a$ is an odd integer, the well-known modular transformations of the Eisenstein series on $\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's formula for $ζ(2m+1)$ are derived from the master identity. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function $K_ν(z)$. Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new modular-type transformation involving the sums-of-squares function $r_k(n)$. Some results on functions self-reciprocal in the Watson kernel are also obtained.
