Some q-Supercongruences from a Quadratic Transformation by Rahman
Inspired by the recent work on q -congruences and a quadratic transformation formula of Rahman, we provide some new q -supercongruences. By taking parameters specialization in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-hand side as Van Hamme’s (G.2)...
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| Vydáno v: | Resultate der Mathematik Ročník 77; číslo 1 |
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01.02.2022
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| ISSN: | 1422-6383, 1420-9012 |
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| Abstract | Inspired by the recent work on
q
-congruences and a quadratic transformation formula of Rahman, we provide some new
q
-supercongruences. By taking parameters specialization in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-hand side as Van Hamme’s (G.2) supercongruence for
p
≡
1
(
mod
4
)
. We also formulate some related challenging conjectures on supercongruences and
q
-supercongruences. |
|---|---|
| AbstractList | Inspired by the recent work on
q
-congruences and a quadratic transformation formula of Rahman, we provide some new
q
-supercongruences. By taking parameters specialization in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-hand side as Van Hamme’s (G.2) supercongruence for
p
≡
1
(
mod
4
)
. We also formulate some related challenging conjectures on supercongruences and
q
-supercongruences. |
| ArticleNumber | 44 |
| Author | Liu, Yudong Wang, Xiaoxia |
| Author_xml | – sequence: 1 givenname: Yudong surname: Liu fullname: Liu, Yudong organization: Department of Mathematics, Shanghai University – sequence: 2 givenname: Xiaoxia orcidid: 0000-0002-8952-1632 surname: Wang fullname: Wang, Xiaoxia email: xiaoxiawang@shu.edu.cn organization: Department of Mathematics, Shanghai University |
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| Cites_doi | 10.1080/10236198.2019.1622690 10.1007/s00025-020-01272-7 10.1017/CBO9780511526251 10.5802/crmath.35 10.1090/proc/14664 10.1080/10652469.2018.1454448 10.4064/cm7686-11-2018 10.1080/10236198.2021.1900140 10.1007/s00365-020-09524-z 10.1007/s00025-020-01195-3 10.1142/S1793042118501038 10.1090/hmath/009 10.1007/s00013-008-2828-0 10.1090/S0002-9939-08-09389-1 10.1090/proc/14374 10.1142/S1793042119501069 10.1016/j.aim.2019.02.008 10.1016/j.jmaa.2021.125238 10.1007/s11139-018-0096-6 10.1016/j.aam.2020.102003 10.1080/10236198.2020.1862808 10.1007/s10998-021-00432-8 10.1080/10236198.2018.1485669 10.4064/cm133-1-9 10.1007/s00025-019-1126-4 10.1007/s13398-020-00923-2 10.1016/j.jnt.2009.01.013 10.1007/s11856-020-2081-1 10.1142/S1793042121500329 10.1007/s11139-019-00192-7 10.1007/s11139-007-9074-0 10.1007/s00026-019-00461-8 10.1007/s00025-020-01203-6 10.1016/j.jmaa.2019.07.062 |
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| Keywords | Primary 33D15 supercongruences cyclotomic polynomial 11B65 Basic hypergeometric series congruences Secondary 11A07 Rahman’s transformation formula |
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Math.2019346329358391079810.1016/j.aim.2019.02.008 TaurasoRSome q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-analogs of congruences for central binomial sumsColloq. Math.2013133133143313942010.4064/cm133-1-9 GuoVJWSchlosserMJA family of q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-hypergeometric congruences modulo the fourth power of a cyclotomic polynomialIsrael J. Math.2020240821835419315210.1007/s11856-020-2081-1 ChuWq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Series reciprocities and further π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}-formulaKodai Math.201841351253038707011408.33017 Sun, Z.-W.: Two q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-analogues of Euler’s formula ζ(2)=π2/6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (2)=\pi ^2/6$$\end{document}. 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| References_xml | – reference: Sun, Z.-W.: Two q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-analogues of Euler’s formula ζ(2)=π2/6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (2)=\pi ^2/6$$\end{document}. Colloq. Math. 158(2), 313–320 (2019) – reference: ZudilinWRamanujan-type supercongruencesJ. Number Theory2009129818481857252270810.1016/j.jnt.2009.01.013 – reference: GuoVJWLiuJ-Cq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Analogues of two Ramanujan-type formulas for 1/π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\pi $$\end{document}J. Difference Equ. Appl.201824813681373385116710.1080/10236198.2018.1485669 – reference: TaurasoRSome q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-analogs of congruences for central binomial sumsColloq. Math.2013133133143313942010.4064/cm133-1-9 – reference: GuilleraJHypergeometric identities for 10 extended Ramanujan-type seriesRamanujan J.2008152219234237757710.1007/s11139-007-9074-0 – reference: GuoVJWZudilinWRamanujan-type formulae for 1/π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\pi $$\end{document}: q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-analoguesIntegral Transforms Spec. 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q
-congruences and a quadratic transformation formula of Rahman, we provide some new
q
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| Title | Some q-Supercongruences from a Quadratic Transformation by Rahman |
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