q-Analogues of two "divergent" Ramanujan-type supercongruences

TL;DR

This paper establishes q-analogues of two divergent Ramanujan-type supercongruences using the q-WZ method, extending previous results and confirming one of Sun's related congruences while proposing another.

Contribution

It introduces novel q-analogues of specific Ramanujan-type supercongruences and related congruences, expanding the understanding of these mathematical structures.

Findings

Proved q-analogues of two divergent Ramanujan-type supercongruences.

Confirmed one of Sun's related congruences.

Proposed a conjectural q-analogue of another related congruence.

Abstract

Guillera and Zudilin proved three "divergent" Ramanujan-type supercongruences by means of the Wilf-Zeilberger algorithmic technique. In this paper, we prove $q$-analogues of two of them via the $q$-WZ method. Additionally, we give $q$-analogues of two related congruence of Sun, one is confirmed and the other is conjectural.