Congruences Involving Hecke-Rogers Type Series and Modular Forms

Congruences Involving Hecke-Rogers Type Series and Modular Forms

Year:    2023

Author:    Guo-Shuai Mao, Yan Liu

Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 147–155

Abstract

In this paper, we prove two supercongruences of Hecke-Rogers type series and Modular forms conjectured by Chan, Cooper and Sica, such as, if
$$z_2=\sum_{m=-∞}^{\infty}\sum_{n=-\infty}^{\infty} q^{m^2+n^2},   x_2=\frac{\eta^{12}(2\tau)}{z_2^6}$$
and
$$z_2=\sum_{n=0}^{\infty}f_{2,n}x_2^n,$$ 
then
$$f_{2,pn}\equiv f_{2,n}  (mod \ p^2) \ \ when \ \  p\equiv 1(mod \  4),$$ where
$$\eta(\tau)=q^{\frac{1}{24}} \Pi_{n=1}^{\infty}(1-q^n),$$
and $q=exp(2πiτ)$ with $Im(τ)>0.$

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v56n2.23.03

Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 147–155

Published online:    2023-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    9

Keywords:    Supercongruences modular forms Hecke-Rogers type series $p$-adic Gamma function.

Author Details

Guo-Shuai Mao

Yan Liu