We give two q-supercongruences. One is modulo the fifth power of a cyclotomic polynomial, and the other is a q-analogue of the supercongruence: for odd primes p, <Equation ID="Equ29"> <EquationSource Format="TEX">$ \sum _{k=0}^{p-1} \frac{3k+1}{16^k}{2k\atopwithdelims ()k}^3 \equiv p\pmod {p^4}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mfrac> <mrow> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <msup> <mn>16</mn> <mi>k</mi> </msup> </mfrac> <msup> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> </mfenced> <mn>3</mn> </msup> <mo>≡</mo> <mi>p</mi> <mspace width="10.0pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <msup> <mi>p</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>which was first proved by Guillera and Zudilin in the modulus <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> case. Our proof employs Rahman’s and Gasper and Rahman’s quadratic transformations, the creative microscoping method devised by the first author in joint work with Zudilin, along with the Chinese remainder theorem for polynomials.

New q-supercongruences from Rahman’s and Gasper and Rahman’s Transformations

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发表日期：

2026年5月11日

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Victor J. W. Guo,
Yu-Ting Xie

摘要

We give two q-supercongruences. One is modulo the fifth power of a cyclotomic polynomial, and the other is a q-analogue of the supercongruence: for odd primes p, $ \sum _{k=0}^{p-1} \frac{3k+1}{16^k}{2k\atopwithdelims ()k}^3 \equiv p\pmod {p^4}, $ $\sum_{k = 0}^{p - 1} \frac{3 k + 1}{16^{k}} {(\binom{2 k}{k})}^{3} \equiv p (mod p^{4}),$ which was first proved by Guillera and Zudilin in the modulus $p^3$ $p^{3}$ case. Our proof employs Rahman’s and Gasper and Rahman’s quadratic transformations, the creative microscoping method devised by the first author in joint work with Zudilin, along with the Chinese remainder theorem for polynomials.