Congruences Involving Hecke-Rogers Type Series and Modular Forms (Journal of Mathematical Study, 2023)
In this paper, we prove two supercongruences of Hecke-Rogers type seriesand Modular forms conjectured by Chan, Cooper and Sica, such as, if $$z_2=\sum_{m=-∞}^{∞}\sum_{n=-∞}^{∞} q^{m^2+n^2}, x_2=\frac{η^{12}(2τ)}{z_2^6}$$and $$z_2=\sum_{n=0}^{∞}f_{2,n}x_2^n,$$ then $$f_{2,pn}\equiv f_{2,n} (mod \ p^2) \ \ when \ \ p\equiv 1(mod \ 4),$$ where $$η(τ)=q^{\frac{1}{24}} Π_{n=1}^{∞}(1-q^n),$$and $q=exp(2πiτ)$ with $Im(τ)>0.$
Keywords: Supercongruences; modular forms; Hecke-Rogers type series; $p$-adic Gamma function.