Some q-supercongruences from transformation formulas for basic hypergeometric series
(46 pages)
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
12Φ11 series. Also, the nonterminating
q-Dixon summation formula is used. A special case of the new
12Φ11 transformation formula is further utilized to
obtain a generalization of Rogers' linearization formula for the
continuous q-ultraspherical polynomials.
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