A New q-Extension of the (H.2) Congruence of Van Hamme for Primes p≡1(mod4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 1\pmod {4}$$\end{document}

Long and Ramakrishna (Adv. Math. 290:773–808, 2016) gave the following extension of Van Hamme’s (H.2) congruence: [graphic not available: see fulltext]where p is an odd prime, (x)k=x(x+1)⋯(x+k-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x)_k=x(x+1)\cdots (x+k-1)$$\end{document} denotes the Pochhammer symbol, and Γp(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _p(x)$$\end{document} is Morita’s p-adic Gamma function. In this paper, different from the one obtained by Wei [Results Math. 76 (2021), Art. 92], we provide a new q-analogue of the above congruence for any prime p≡1(mod4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 1\pmod {4}$$\end{document}. Meanwhile, we also confirm a conjectural q-congruence of Guo (Results Math. 76:109, 2021).