A $q$-microscope for supercongruences

Abstract

By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$\sum _{n=0}^{\infty }\frac{\left(\genfrac{}{}{0ex}{}{4n}{2n}\right){\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}^2}{2^{8n}{3}^{2n}}(8n+1)=\frac{2\sqrt{3}}{\pi },$$ of the two supercongruences $$S(p-1)\equiv p\left(\frac{-3}{p}\right)(\mathrm{mod}{p}^3)\text{and}S\left(\frac{p-1}{2}\right)\equiv p\left(\frac{-3}{p}\right)(\mathrm{mod}{p}^3),$$ valid for all primes $p>3$, where $S(N)$ denotes the truncation of the infinite sum at the $N$-th place and $(\frac{-3}{\cdot })$ stands for the quadratic character modulo $3$.

26 pages

References in corpus (2)

• Supercongruences for rigid hypergeometric Calabi--Yau threefolds
• Ramanujan-type formulae for $1/\pi$: $q$-analogues

Cited by in corpus (47)

• Congruences on sums of $q$-binomial coefficients
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• On a $q$-deformation of modular forms
• Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial
• Supercongruences for rigid hypergeometric Calabi--Yau threefolds
• Some new $q$-congruences for truncated basic hypergeometric series
• A common $q$-analogue of two supercongruences
• Dold sequences, periodic points, and dynamics
• Symbolic summation methods and hypergeometric supercongruences
• Proof of a conjecture of Adamchuk
• $q$-Supercongruences from Jackson's ${}_8{\varphi }_7$ summation and Watson's ${}_8{\varphi }_7$ transformation
• Congruence properties for the trinomial coefficients
• Proof of two supercongruences conjectured by Z.-W. Sun
• Proof of two supercongruences by the Wilf-Zeilberger method
• Two $q$-supercongruences from Watson's transformation
• Proof of some supercongruences via the Wilf-Zeilberger method
• Supercongruences of multiple harmonic $q$-sums and generalized finite/symmetric multiple zeta values
• ($q$-)Supercongruences hit again
• Some $q$-supercongruences modulo the fourth power of a cyclotomic polynomial
• A family of $q$-congruences modulo the square of a cyclotomic polynomial
• Some congruences related to a congruence of Van Hamme
• $q$-Supercongruences from the $q$-Saalschütz identity
• q-analogues of the G.2 Supercongruence of Van Hamme
• $q$-Supercongruences with parameters
• On a conjecture related to integer-valued polynomials
• Some supercongruences arising from symbolic summation
• On two supercongruences of double binomial sums
• On supercongruences for truncated sums of squares of basic hypergeometric series
• Proof of a supercongruence conjectured by Sun through a $q$-microscope
• The united proofs for three $q$-extensions of Dougall's ${}_2{H}_2$ summation formula
• On two congruences involving Franel numbers
• A variation of $q$-Wolstenholme's theorem
• Divisibility results concerning truncated hypergeometric series
• On a congruence involving $q$-Catalan numbers
• A symmetric formula for hypergeometric series
• A further $q$-analogue of Van Hamme's (H.2) supercongruence for $p\equiv 1(\mathrm{mod}4)$
• A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$
• Proof of a q-supercongruence conjectured by Guo and Schlosser
• Some variations of a "divergent" Ramanujan-type $q$-supercongruence
• Some q-supercongruences from Rahman's summation formula
• Factors of certain sums involving central q-binomial coefficients
• Some $q$-congruences involving central $q$-binomial coefficients
• Streamlined WZ method proofs of Van Hamme supercongruences
• $N$-tuple sum analogues for Ramanujan-type congruences
• Transformation and summation formulas for basic hypergeometric series associated with the circumference ratio