[Paper Review] $q$-Supercongruences from Jackson's $_8ϕ_7$ summation and Watson's $_8ϕ_7$ transformation

Chuanan Wei|arXiv (Cornell University)|Apr 14, 2021

Advanced Mathematical Identities21 references4 citations

TL;DR

This paper establishes new $q$-supercongruences modulo the fifth and sixth powers of cyclotomic polynomials using Jackson’s $_8 heta_7$ summation, Watson’s $_8 heta_7$ transformation, the creative microscoping method, and the Chinese remainder theorem for coprime polynomials. It provides $q$-analogues of Long and Ramakrishna’s supercongruence formula, extending them to higher moduli and including double-series identities with precise $q$-deformations of classical results.

ABSTRACT

$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's $_8ϕ_7$ summation, Watson's $_8ϕ_7$ transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a $q$-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] and two $q$-supercongruences involving double series.

Motivation & Objective

• To establish $q$-supercongruences modulo high powers of cyclotomic polynomials $\Phi_n(q)^5$ and $\Phi_n(q)^6$, which are rare in the literature.
• To extend Long and Ramakrishna’s $p$-adic supercongruence for the hypergeometric sum $\sum (6k+1)(1/3)_k^6/k!^6$ to a $q$-analogue with higher modulus.
• To provide $q$-deformations of classical hypergeometric supercongruences using advanced $q$-series techniques and $q$-integer identities.
• To prove two new $q$-supercongruences involving double series and $q$-shifted factorials, valid under specific congruence conditions on $n$ modulo 3.
• To generalize previous conjectures by Guo and Schlosser on $q$-supercongruences by proving them with stronger modulus and explicit $q$-deformed expressions.

Proposed method

• Utilizes Jackson’s $_8\phi_7$ summation and Watson’s $_8\phi_7$ transformation to generate $q$-hypergeometric identities with parameters.
• Applies the creative microscoping method to refine $q$-supercongruences by introducing auxiliary parameters and analyzing limits as parameters approach 1.
• Employs the Chinese remainder theorem for coprime polynomials to combine congruences modulo $[n]$ and $\Phi_n(q)^k$ into stronger results modulo $[n]\Phi_n(q)^k$.
• Uses $q$-integer identities and $q$-shifted factorial notation to express $q$-analogues of classical hypergeometric sums with precise modular behavior.
• Applies L’Hôpital’s rule in the $q$-setting to handle singular limits as parameters $a \to 1$, enabling the derivation of $q$-supercongruences in the $a \to 1$ limit.
• Derives $q$-supercongruences for sums of the form $\sum [6k+1] \frac{(q;q^3)_k^6}{(q^3;q^3)_k^6} q^{3k}$ modulo $[n]\Phi_n(q)^4$ or $[n]\Phi_n(q)^5$, depending on $n \mod 3$.

Experimental results

Research questions

• RQ1 Can $q$-supercongruences modulo $\Phi_n(q)^5$ and $\Phi_n(q)^6$ be established for $q$-hypergeometric sums related to the classical supercongruence (1.2) of Long and Ramakrishna?
• RQ2 How can the creative microscoping method be combined with $q$-hypergeometric transformations to derive $q$-analogues of higher-order supercongruences?
• RQ3 What is the precise $q$-deformation of the sum $\sum (6k+1)(1/3)_k^6/k!^6$ modulo $p^{s+4}$ and $p^{s+5}$, and how does it reflect the $p$-adic structure?
• RQ4 Can double-series $q$-supercongruences be constructed and proven using $q$-analogue of the Chinese remainder theorem and $q$-series identities?
• RQ5 What is the role of the $q$-integer $[n]$ and the cyclotomic polynomial $\Phi_n(q)$ in controlling the modulus of $q$-supercongruences for $n \equiv 1,2 \pmod{3}$?

Key findings

• Theorem 1.1 establishes a $q$-supercongruence modulo $[n]\Phi_n(q)^4$ for $n \equiv 1 \pmod{3}$, with a correction term involving $[n]^2$ and sums of $q^{3j}/[3j]^2$ and $q^{3j-1}/[3j-1]^2$.
• Theorem 1.2 proves a $q$-supercongruence modulo $[n]\Phi_n(q)^5$ for $n \equiv 2 \pmod{3}$, with the right-hand side proportional to $5[2n] \frac{(q^2;q^3)_{(2n-1)/3}^3}{(q^3;q^3)_{(2n-1)/3}^3}$.
• Corollary 1.3 gives a $p$-adic supercongruence modulo $p^{s+4}$ for $p^s \equiv 1 \pmod{3}$, with a correction term involving a sum over $j$ of $\left(\frac{1}{(3j-1)^2} - \frac{1}{(3j)^2}\right)$.
• Corollary 1.4 provides a $p$-adic supercongruence modulo $p^{s+5}$ for $p^s \equiv 2 \pmod{3}$, with the value proportional to $10p^s \frac{(2/3)_{(2p^s-1)/3}^3}{(1)_{(2p^s-1)/3}^3}$.
• Theorems 3.1 and 3.2 generalize the results to double-series $q$-supercongruences with parameters $a$, $b$, $c$, and $d$, valid for $r = \pm 1$ and $n \equiv -1 \pmod{d}$ or $n \equiv 1 \pmod{d}$.
• The paper proves two conjectures of Guo and Schlosser by establishing $q$-supercongruences modulo $[n]\Phi_n(q)^4$ and $[n]\Phi_n(q)^5$, respectively, using the $q$-analogue of the Chinese remainder theorem and limit techniques.