Victor J. W. Guo and Michael J. Schlosser

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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