Two q-congruences on Double Basic Hypergeometric Sums

Abstract

In this paper, we establish two q-congruences concerning double basic hypergeometric sums modulo the cube of a cyclotomic polynomial. In particular, we prove a conjecture put forward by Fang and Guo [Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 2022, 116(1): Paper No. 25, 7 pp.]: for any odd integer n > 1, the following q-congruence holds modulo Φ__n(q)³,

$\sum_{k=0}^{(n-1)/2}(-1)^k[4k+1]\frac{(q;q^2)_k^4(q^2;q^4)k}{(q^2;q^2)_k^4(q^4;q^4)k}q^k\sum_{j=1}^k\left(\begin{array}{c}\frac{q^{2j-1}}{[2j-1]^2}\end{array}-\frac{q^{2j}}{[2j]^2}\right)\\\equiv\begin{cases}[n]\frac{(q^2;q^4)^2_{n-1)/4}}{(q^4;q^4)^2_{n-1)/4}}\Sigma_{j=1}^{n-1)/2}\frac{(-1)^{j-1}g^{2j}}{[2j]^2},\;{\rm{if}}\;n\equiv1\;(\rm{mod}\;4),\\\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}q^{(n+1)/2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;n\equiv3\;(\rm{mod}\;4)\end{cases}$

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Keywords

q-congruence / congruence / basic hypergeometric series / double series / 33D15 / 11A07 / 11B65

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