Long and Ramakrishna generalized the (H.2) supercongruence of Van Hamme to the modulus <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> case. Wei and Wang gave two different q-analogues of this supercongruence for primes <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\equiv 1\pmod {4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The author and Zudilin ever presented a new q-analogue of Van Hamme’s original (H.2) supercongruence for primes <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p\equiv 1\pmod {4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we further extend this q-congruence to the modulus <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Phi _n(q)^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mi>n</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> case, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Phi _n(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the n-th cyclotomic polynomial in q. The main ingredients of our proof are the creative microscoping method, a q-analogue of Watson’s <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$_3F_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>3</mn> <mrow /> </mmultiscripts> <msub> <mi>F</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> summation, and the Chinese remainder theorem for polynomials.

A New q-Analogue of Van Hamme’s (H.2) Supercongruence for Primes $p\equiv 1\pmod {4}$

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2025年12月22日

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Victor J. W. Guo

摘要

Long and Ramakrishna generalized the (H.2) supercongruence of Van Hamme to the modulus $p^3$ $p^{3}$ case. Wei and Wang gave two different q-analogues of this supercongruence for primes $p\equiv 1\pmod {4}$ $p \equiv 1 (mod 4)$ . The author and Zudilin ever presented a new q-analogue of Van Hamme’s original (H.2) supercongruence for primes $p\equiv 1\pmod {4}$ $p \equiv 1 (mod 4)$ . In this paper, we further extend this q-congruence to the modulus $\Phi _n(q)^3$ $Φ_{n} {(q)}^{3}$ case, where $\Phi _n(q)$ $Φ_{n} (q)$ is the n-th cyclotomic polynomial in q. The main ingredients of our proof are the creative microscoping method, a q-analogue of Watson’s $_3F_2$ $_{3} F_{2}$ summation, and the Chinese remainder theorem for polynomials.

A New q-Analogue of Van Hamme’s (H.2) Supercongruence for Primes \(p\equiv 1\pmod {4}\)

摘要