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} series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new {}_{12}phi _{11} transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.
Highlights
Ramanujan, in his second letter to Hardy on February 27, 1913, mentioned the following identity ∞ (−1)k (4k + 1) ( )5k k!5 = k=0 )4 (1.1)
The main aim of this paper is to give q-analogues of some known supercongruences, including a partial q-analogue of Long’s supercongruence (1.4)
We provide such a result in Theorem 2.1 in the form of two transformations of truncated basic hypergeometric series
Summary
In his second letter to Hardy on February 27, 1913, mentioned the following identity. The main aim of this paper is to give q-analogues of some known supercongruences, including a partial q-analogue of Long’s supercongruence (1.4) (partial in the sense that the modulo p4 condition is replaced by the weaker condition modulo p3) We provide such a result in Theorem 2.1 in the form of two transformations of truncated basic hypergeometric series. These results are proved by special instances of transformation formulas for basic hypergeometric series. Bs ; q)k (−1)k q(k2) 1+s−r zk , where q = 0 when r > s + 1 Such a series terminates if one of the upper parameters, say, ar , is of the form q−n, where n is a nonnegative integer. We refer the reader to [1,7,16,17,18,19,20,21,22,23,24,27,36,39,45,49,53,54,57,58] for some interesting q -congruences
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