Proofs of some conjectures on the reciprocals of the Ramanujan–Gordon identities,The Ramanujan Journal

当前位置： X-MOL 学术 › Ramanujan J. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

Proofs of some conjectures on the reciprocals of the Ramanujan–Gordon identities
The Ramanujan Journal ( IF 0.6 ) Pub Date : 2020-07-16 , DOI: 10.1007/s11139-019-00247-9
Min Bian , Dazhao Tang , Ernest X. W. Xia , Fanggang Xue

Recently, Lin and Wang introduced two special partition functions $RG_1(n)$ and $RG_2(n)$, the generating functions of which are the reciprocals of two identities due to Ramanujan and Gordon. They established several congruences modulo 5 and 7 for $RG_1(n)$ and $RG_2(n)$ and posed four conjectures on congruences modulo 25 for $RG_1(n)$ and $RG_2(n)$ at the end of their paper. In this paper, we confirm the four conjectures given by Lin and Wang by using Ramanujan’s modular equation of fifth degree. Moreover, we also obtain new congruences modulo 25 for $RG_1(n)$ and $RG_2(n)$ based on Newman’s identities. For example, we deduce that for any integer $n\ge 0$,

$$\begin{aligned} RG_1\left( \frac{23375n(3n+1)}{2}+974\right)&\equiv RG_1\left( \frac{23375n(3n+5)}{2}+24349\right) \\&\equiv 0\pmod {25}. \end{aligned}$$

中文翻译：

Ramanujan-Gordon身份倒数的一些猜想的证明

最近，Lin和Wang引入了两个特殊的分区函数\（RG_1（n）\）和\（RG_2（n）\），其生成函数是由于Ramanujan和Gordon而产生的两个身份的倒数。他们建立了几个同余模5,7 \（皂苷Rg1（N）\）和\（RG_2（N）\），并提出了在同余4个猜想模25 \（皂苷Rg1（N）\）和\（RG_2（N） \）在他们的论文末尾。在本文中，我们使用拉曼努詹的第五级模方程来确认林和王给出的四个猜想。此外，我们还获得了\（RG_1（n）\）和\（RG_2（n）\）模25的新同余基于纽曼的身份。例如，我们推论对于任何整数\（n \ ge 0 \），

$$ \ begin {aligned} RG_1 \ left（\ frac {23375n（3n + 1）} {2} +974 \ right）＆\ equiv RG_1 \ left（\ frac {23375n（3n + 5）} {2} + 24349（右）\\＆\ equiv 0 \ pmod {25}。\ end {aligned} $$

更新日期：2020-07-16

点击分享查看原文

点击收藏

阅读更多本刊最新论文