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 $Im(τ)>0.$
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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.