TY - JOUR
T1 - Congruences Involving Hecke-Rogers Type Series and Modular Forms
AU - Mao , Guo-Shuai
AU - Liu , Yan
JO - Journal of Mathematical Study
VL - 2
SP - 147
EP - 155
PY - 2023
DA - 2023/06
SN - 56
DO - http://doi.org/10.4208/jms.v56n2.23.03
UR - https://global-sci.org/intro/article_detail/jms/21825.html
KW - Supercongruences, modular forms, Hecke-Rogers type series, $p$-adic Gamma function.
AB - <p style="text-align: justify;">In this paper, we prove two supercongruences of Hecke-Rogers type series
and Modular forms conjectured by Chan, Cooper and Sica, such as, if <br/>$$z_2=\sum_{m=-∞}^{\infty}\sum_{n=-\infty}^{\infty} q^{m^2+n^2},&nbsp; &nbsp;x_2=\frac{\eta^{12}(2\tau)}{z_2^6}$$<br/>and <br/>$$z_2=\sum_{n=0}^{\infty}f_{2,n}x_2^n,$$&nbsp;<br/>then <br/>$$f_{2,pn}\equiv f_{2,n}&nbsp; (mod \ p^2) \ \ when \ \&nbsp; p\equiv 1(mod \&nbsp; 4),$$ where <br/>$$\eta(\tau)=q^{\frac{1}{24}} \Pi_{n=1}^{\infty}(1-q^n),$$<br/>and $q=exp(2πiτ)$ with $Im(τ)&gt;0.$</p>