Some q-supercongruences for truncated 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 Wadim 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 φ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new 12 φ 11 transformation formula is further utilized to obtain a generalization of Rogers' linearization formula for the continuous q-ultraspherical polynomials.if p ≡ 3 (mod 4).(1.2)Here and throughout the paper, p always denotes an odd prime and Γ p (x) is the p-adic Gamma function. The congruence (1.2) was later proved by McCarthy and Osburn [30] through a combination of ordinary and Gaussian hypergeometric series. Recently, the congruence (1.2) for p ≡ 3 (mod 4) and p > 3 was further generalized by Liu [27] to the modulus p 4 case.

A new series for π3and related congruences

“…(Actually, Wang stated his results only in the language of hypergeometric series.) These two congruences extend a conjecture of Guo and M. J. Schlosser [21]. We are also able to prove some other variants of Ramanujan-…”

A Common q-Analogue of Two Supercongruences

“…It should be mentioned that a different q-analogue of (1.9) was given in [12, Theorem 4.9] with r = −1, d = 2 and a = 1 (see also [10,Section 5]). Both Theorems 1.1 and 1.2 are particular cases of a more general result, which we state and prove in the next section.…”

q-Analogues of Dwork-type supercongruences

“…In recent years, q-analogues of congruences and supercongruences have been investigated by many authors (see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16]21,25,27]). In particular, using the q-WZ method [29] the author and Wang [14] gave a q-analogue of (1.1): for odd n,…”