On supercongruences for truncated sums of squares of basic hypergeometric series

Some q-supercongruences on double and triple sums

“…In [2], El Bachraoui introduced an elegant lemma. Now, we are going to show a generalization of this lemma, which plays a crucial role in the derivation of our theorems.…”

“…Applying the 'creative microscoping' method introduced in [10], El Bachraoui [2] observed several supercongruences from the square of truncated basic hypergeometric series. Later, Li [12] gave some new q-supercongruences.…”

$q$-Supercongruences from squares of basic hypergeometric series

“…El Bachraoui [2] employed the creative microscoping method to prove a few new qsupercongruences, such as n−1 k=0 k j=0 c q (j)c q (k − j) ≡ q[n] 2 (mod [n]Φ n (q) 2 ), (1.2) where c q (k) stands for the k-th term on the left-hand side of (1.1) and gcd(n, 6) = 1. Note that the left-hand side of (1.2) may be deemed a truncated form of the square of the basic hypergeometric series ∞ k=0 c q (k).…”

“…Finally, in Section 6, we put forward eight open problems. Besides the creative microscoping method and a lemma of El Bachraoui [2], we shall also employ the Chinese remainder theorem for relatively prime polynomials to prove Theorems 1.3 and 1.4.…”

On the divisibility of sums of even powers of $q$-binomial coefficients

“…In the past few years, congruences for q-binomial coefficients as well as basic hypergeometric series attracted many experts' attention (see, for example, [1,2,[4][5][6][7][10][11][12][13][14][17][18][19]).…”