Proof of a q-supercongruence conjectured by Guo and Schlosser

作者：Long Li, Su-Dan Wang · 发表于：RACSAM · 年份：2020 · DOI：10.1007/s13398-020-00923-2 · 被引用次数：62 · 研究领域：Mathematics

In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer n>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>1$$\end{document} and M=(n+1)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=(n+1)/2$$\end{document} or n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document}, ∑k=0M[4k-1]q2[4k-1]2(q-2;q4)k4(q4;q4)k4q4k≡(2q+2q-1-1)[n]q24(mod[n]q24Φn(q2)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv (2q+2q^{-1}-1)[n]_{q^2}^4\pmod {[n]_{q^2}^4\Phi _n(q^2)}, \end{aligned}$$\end{document}where [n]=[n]q=(1-qn)/(1-q),(a;q)0=1,(a;q)k=(1-a)(1-aq)⋯(1-aqk-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{was...