On a congruence involving $q$-Catalan numbers

Abstract

On a congruence involving q-Catalan numbers

Another Family of q-Congruences Modulo the Square of a Cyclotomic Polynomial

“…Here and throughout the paper, we say that a rational function A(q) is congruent to another rational function B(q) modulo a polynomial P (q), denoted by A(q) ≡ B(q) (mod P (q)), if the numerator of the reduced form of A(q) − B(q) is divisible by P (q) in the polynomial ring Z[q]. For some other recent progress on q-congruences, we refer the reader to [3][4][5][6][7][8][9][10][11][14][15][16]19,20,22,27,28,30,31].…”

Further q-analogues of the (G.2) Supercongruence of Van Hamme

“…For some other recent progress on congruences and q-congruences, see [2][3][4][5][6][7][9][10][11][12]18,19,21,[24][25][26][27][28]. Especially, Guo and Zudlin [9] introduced the 'creative microscoping' method which is useful for proving q-congruences.…”

Some q-supercongruences modulo the square and cube of a cyclotomic polynomial

“…In recent years, q-congruences and q-supercongruences have been established by different authors (see, for example, [5][6][7][8][9][10][11][12][13][15][16][17][18][19][20][21]23,27,[30][31][32]34]). In particular, the present authors [9] proved that, for any odd integer d ≥ 5, 2 ), if n ≡ −1 (mod d), 0 (mod n (q) 3 ), if n ≡ −1/2 (mod d).…”