A new q-supercongruence modulo the fourth power of a cyclotomic polynomial

Employing the q-WZ method, Guo and Wang gave a q-analogue of a supercongruence modulo p4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^4$$\end{document} of Long, where p is a prime greater than 3. Using the method of ‘creative microscoping’ introduced by Guo and Zudilin, we establish a variation of Guo and Wang’s q-supercongruence. As a conclusion, we obtain the following supercongruence: for any prime p⩾5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\geqslant 5$$\end{document} and positive integer r, ∑k=1(pr+1)/24k+1256k2k-2k-12k+2k+12kk2≡prmodpr+3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^{(p^r+1)/2} \frac{4k+1}{256^k}{2k-2\atopwithdelims ()k-1}{2k+2\atopwithdelims ()k+1}{2k\atopwithdelims ()k}^2\equiv p^r \left( \textrm{mod}\,{p^{r+3}}\right) . \end{aligned}$$\end{document}