Congruences Involving Hecke-Rogers Type Series and Modular Forms

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 $Im(τ)>0.$