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<p>By making use of Watson’s <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1912_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(_8\phi _7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>8</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>7</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> transformation formula, we prove two families of <i>q</i>-congruences modulo the square of a cyclotomic polynomial. As conclusions, we confirm a supercongruence conjecture of the author and Wei, and also partially confirm another supercongruence conjecture of theirs. We put forward several conjectures on <i>q</i>-congruences for further study.</p>

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Two Families of q-Congruences From Watson’s \(_8\phi _7\) Transformation

  • 期刊论文

  • 发表日期：

    2025年6月19日

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  • Victor J. W. Guo

摘要

By making use of Watson’s \(_8\phi _7\) 8 ϕ 7 transformation formula, we prove two families of q-congruences modulo the square of a cyclotomic polynomial. As conclusions, we confirm a supercongruence conjecture of the author and Wei, and also partially confirm another supercongruence conjecture of theirs. We put forward several conjectures on q-congruences for further study.