Document Zbl 1462.33006

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W.</a>; <a href="/authors/schlosser.michael-j" title="Author Profile">Schlosser, Michael J.</a></div> <h2 class="title"> <strong>Some $q$-supercongruences from transformation formulas for basic hypergeometric series.</strong> <i>(English)</i> <a class="label nowrap" href="/1462.33006">Zbl 1462.33006</a> </h2> <div class="source"> <a href="/serials/1091" title="Journal Profile">Constr. Approx.</a> <a href="/?q=in%3A460496" title="Articles in this Issue">53, No. 1, 155-200 (2021)</a>. </div> <div class="abstract">The main achievements of this vast paper are the $q$-analogues of four supercongruences from, respectively:<br class="zbmathjax-paragraph">1) <span class="zbmathjax-textit">L. Long</span> [Pac. J. Math. 249, No. 2, 405–418 (2011; <a href="/1215.33002">Zbl 1215.33002</a>)], here modulo $p^3$ instead of the original modulo $p^4$; <br class="zbmathjax-paragraph">2) <span class="zbmathjax-textit">E. Mortenson</span> [Proc. Am. Math. Soc. 136, No. 12, 4321–4328 (2008; <a href="/1171.11061">Zbl 1171.11061</a>)]; <br class="zbmathjax-paragraph">3) <span class="zbmathjax-textit">L. Long</span> and <span class="zbmathjax-textit">R. Ramakrishna</span> [Adv. Math. 290, 773–808 (2016; <a href="/1336.33018">Zbl 1336.33018</a>)], here in a simplified version; <br class="zbmathjax-paragraph">4) <span class="zbmathjax-textit">V. J. W. Guo</span> and <span class="zbmathjax-textit">W. Zudilin</span> [Adv. Math. 346, 329–358 (2019; <a href="/1464.11028">Zbl 1464.11028</a>)], i.e., the same article where the creative microscoping method (CMM) was introduced.<br class="zbmathjax-paragraph">The authors remark the difference between the first pair of theorems (1-2), proved via the CMM, and the second pair (3-4), obtained in a similar, but not identical, way.<br class="zbmathjax-paragraph">As auxiliary results, the paper establishes many other $q$-congruences employing the hypergeometric formulas of <span class="zbmathjax-textit">G. Gasper</span> [Trans. Am. Math. Soc. 312, No. 1, 257–277 (1989; <a href="/0664.33010">Zbl 0664.33010</a>)], of <span class="zbmathjax-textit">G. Gasper</span> and <span class="zbmathjax-textit">M. Rahman</span> [Can. J. Math. 42, No. 1, 1–27 (1990; <a href="/0707.33009">Zbl 0707.33009</a>)], of <span class="zbmathjax-textit">M. Rahman</span> [Can. J. Math. 45, No. 2, 394–411 (1993; <a href="/0774.33012">Zbl 0774.33012</a>)], and some classic ones (Watson’s ${}_8\phi _7$ transformation, $q$-Dixon sum, Sears-Slater’s transformation), together with basic tools like the recurrence by induction, Fermat’s Little Theorem, $q$-Chu-Vandermonde identity, and a wide variety of $q$-shifted factorials’ properties.<br class="zbmathjax-paragraph">Inspired by a formula related to two ${}_{14}\phi _{13}$ series given by <span class="zbmathjax-textit">R. Langer</span> et al. [SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper No. 55, 20 p. (2009; <a href="/1205.33027">Zbl 1205.33027</a>)], the authors find also a ${}_{12}\phi _{11}$ transformation supplied into two multiples of nonterminating ${}_4\phi _3$ series; they apply it to prove two new $q$-supercongruences and to generalize the linearization of the continuous $q$-ultraspherical polynomials provided by <span class="zbmathjax-textit">L. J. Rogers</span> [Proc. Lond. Math. Soc. 26, 15–32 (1895; <a href="/26.0289.01">JFM 26.0289.01</a>)].<br class="zbmathjax-paragraph">In addition to the Pochhammer symbols, the authors recall the $n$th cyclotomic polynomials, and the $q$-Catalan numbers studied by <span class="zbmathjax-textit">J. Fürlinger</span> and <span class="zbmathjax-textit">J. Hofbauer</span> [J. Comb. Theory, Ser. A 40, 248–264 (1985; <a href="/0581.05006">Zbl 0581.05006</a>)].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A12054">Enzo Bonacci (Latina)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7326825">Cited in <strong>7</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A7326825">Cited in <strong>68</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A33D15" title="MSC2020">33D15</a> </td> <td class="space"> Basic hypergeometric functions in one variable, ${}_r\phi_s$ </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05A10" title="MSC2020">05A10</a> </td> <td class="space"> Factorials, binomial coefficients, combinatorial functions </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05A30" title="MSC2020">05A30</a> </td> <td class="space"> $q$-calculus and related topics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11A07" title="MSC2020">11A07</a> </td> <td class="space"> Congruences; primitive roots; residue systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11F33" title="MSC2020">11F33</a> </td> <td class="space"> Congruences for modular and $p$-adic modular forms </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A33D45" title="MSC2020">33D45</a> </td> <td class="space"> Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ahypergeometric+series">hypergeometric series</a>; <a href="/?q=ut%3A%5C%28q%5C%29-analogues">$q$-analogues</a>; <a href="/?q=ut%3Asupercongruences">supercongruences</a>; <a href="/?q=ut%3A%5C%28q%5C%29-shifted+factorials">$q$-shifted factorials</a>; <a href="/?q=ut%3Aidentities">identities</a>; <a href="/?q=ut%3Acreative+microscoping+method">creative microscoping method</a>; <a href="/?q=ut%3Alinearization">linearization</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1215.33002">Zbl 1215.33002</a>; <a href="/1171.11061">Zbl 1171.11061</a>; <a href="/1336.33018">Zbl 1336.33018</a>; <a href="/0664.33010">Zbl 0664.33010</a>; <a href="/0707.33009">Zbl 0707.33009</a>; <a href="/0774.33012">Zbl 0774.33012</a>; <a href="/1205.33027">Zbl 1205.33027</a>; <a href="/0581.05006">Zbl 0581.05006</a>; <a href="/1464.11028">Zbl 1464.11028</a>; <a href="/26.0289.01">JFM 26.0289.01</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1462.33006" data-ciurl="/ci/07326825" data-biburl="/bibtex/07326825.bib" data-amsurl="/amsrefs/07326825.bib" data-xmlurl="/xml/07326825.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07326825.pdf" title="Zbl 1462.33006 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00365-020-09524-z" aria-label="DOI for “Some $q$-supercongruences from transformation formulas for basic hypergeometric series”" title="10.1007/s00365-020-09524-z">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1812.06324"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Andrews, GE, $q$-Analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher, Discrete Math., 204, 15-25 (1999) · <a href="/0937.05014" class="nowrap">Zbl 0937.05014</a></td> </tr><tr> <td>[2]</td> <td class="space">Andrews, GE; Askey, RA; Roy, R., Special functions, Encyclopedia of Mathematics and Its Applications (1999), Cambridge: Cambridge University Press, Cambridge · <a href="/0920.33001" class="nowrap">Zbl 0920.33001</a></td> </tr><tr> <td>[3]</td> <td class="space">Ahlgren, S.; Ono, K., Gaussian hypergeometric series evaluation and Apéry number congruences, J. 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Soc., 140, 765-777 (2012) · <a href="/1276.11027" class="nowrap">Zbl 1276.11027</a></td> </tr><tr> <td>[16]</td> <td class="space">Guo, VJW, A $q$-analogue of a Ramanujan-type supercongruence involving central binomial coefficients, J. Math. Anal. Appl., 458, 590-600 (2018) · <a href="/1373.05025" class="nowrap">Zbl 1373.05025</a></td> </tr><tr> <td>[17]</td> <td class="space">Guo, V.J.W.: A $q$-analogue of the (A.2) supercongruence of Van Hamme for primes $p\equiv 1~(mod \; 4)$. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, Art. 123 (2020) · <a href="/1439.33006" class="nowrap">Zbl 1439.33006</a></td> </tr><tr> <td>[18]</td> <td class="space">Guo, V.J.W.: $q$-Analogues of Dwork-type supercongruences. J. Math. Anal. 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