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<p>We prove a family of <i>q</i>-supercongruences modulo the fourth power of a cyclotomic polynomial. Our results may be regarded as a uniform generalization of a <i>q</i>-supercongruence by Wei, two <i>q</i>-supercongruences by Liu and Wang, and two previous <i>q</i>-supercongruences by the author himself. Our proof makes use of the method of “creative microscoping” introduced by the author and Zudilin, Watson’s <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/13398_2026_1866_IEq3_HTML.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="120" Type="Linedraw" Width="26" /> </InlineMediaObject> </InlineEquation> transformation formula, and the Chinese remainder theorem for coprime polynomials.</p>

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A family of q-supercongruences from Watson’s transformation

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Victor J. W. Guo

摘要

We prove a family of q-supercongruences modulo the fourth power of a cyclotomic polynomial. Our results may be regarded as a uniform generalization of a q-supercongruence by Wei, two q-supercongruences by Liu and Wang, and two previous q-supercongruences by the author himself. Our proof makes use of the method of “creative microscoping” introduced by the author and Zudilin, Watson’s transformation formula, and the Chinese remainder theorem for coprime polynomials.