On three conjectural congruences of Z.-H. Sun involving Apéry and Apéry-like numbers
On three conjectural congruences of Z.-H. Sun involving Apéry and Apéry-like numbers
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DOI：
10.1007/s00605-023-01838-x
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发表时间：
2023-03
期刊：
影响因子：
--
通讯作者：
Guo-Shuai Mao;An-Bang ZhaoSong
中科院分区：
文献类型：
--
作者：
Guo-Shuai Mao;An-Bang ZhaoSong
In this paper, we mainly prove the following conjectures of Sun Z-H (New congruences involving Apéry-like numbers. Preprint at arXiv:2004.07172v2): Letbe a prime. Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&A'_{2p-1}\equiv A'_1+\frac{200}{3}p^3B_{p-3}\ \left( \mathrm{{mod}}\ p^4\right) ,\\&\quad D_{2p-1}\equiv 64^{2(p-1)}D_1-\frac{44}{3}p^3B_{p-3}\ \left( \mathrm{{mod}}\ p^4\right) , \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A'_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}n+k\\ k\end{array}}\right) $$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) \left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right) $$\end{document} andstands for thenth Bernoulli number. We also give a generalization of a conjecture of Sun involving Franel numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^3$$\end{document}.
In this paper, we mainly prove the following conjectures of Sun Z-H (New congruences involving Apéry-like numbers. Preprint at arXiv:2004.07172v2): Letbe a prime. Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&A'_{2p-1}\equiv A'_1+\frac{200}{3}p^3B_{p-3}\ \left( \mathrm{{mod}}\ p^4\right) ,\\&\quad D_{2p-1}\equiv 64^{2(p-1)}D_1-\frac{44}{3}p^3B_{p-3}\ \left( \mathrm{{mod}}\ p^4\right) , \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A'_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}n+k\\ k\end{array}}\right) $$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) \left( {\begin{array}{c}2n-2k\\ n-k\end{array}}\right) $$\end{document} andstands for thenth Bernoulli number. We also give a generalization of a conjecture of Sun involving Franel numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_n=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^3$$\end{document}.