Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

Abstract

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

New q-supercongruences related to Ramanujan-type supercongruences

“…Also, Swisher [17] showed a refinement of van Hamme's (G.2): for any prime p ≡ 1 (mod 4), During the past few years, there has been an increasing attention to the issue of finding q-analogues of congruences and supercongruences. Some progress on q-congruences can be found in [2,3,7,10,15,16,19,20]. Particularly, using the q-WZ method [24], Guo and Wang [4] showed that (1.1) possesses the following q-analogue: for odd integers n,…”

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).…”

“…In recent years, q -congruences and q -supercongruences have been established by different authors (see, for example, [ 5 – 13 , 15 – 21 , 23 , 27 , 30 – 32 , 34 ]). In particular, the present authors [ 9 ] proved that, for any odd integer , Here and in what follows, we adopt the standard q -notation: is the q -integer ; is the q -shifted factorial , with the compact notation used for their products; and denotes the n -th cyclotomic polynomial in q , which may be defined as where is an n -th primitive root of unity.…”

A family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial

“…During the past few years, many congruences and supercongruences were generalized to the q-setting by various authors (see, for instance, [4-18, 23, 24, 28,30]). In particular, the authors [15,Thm. 2.3] proposed the following partial q-analogue of Long and Ramakrishna's supercongruence (1.3):…”

“…The authors [15,Conjectures 12.10 and 12.11] also proposed the following conjectures, the first one generalizing the q-congruence (1.4) for n ≡ 2 (mod 3).…”