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.

中文翻译：

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来自基本超几何级数变换公式的一些q-超同余

使用基本超几何级数的变换公式，结合各种技术，如适当组合项和创造性显微镜，获得了几个新的 q 超同余，这是第一作者最近与 Zudilin 合作开发的一种方法。更具体地说，本文的结果包括由 Long、Long 和 Ramakrishna 建立的超同余的 q -类比（指 p -adic 恒等式对于 p 的某个更高的幂仍然有效），以及其他几个 q -超同余。使用的六个基本超几何变换公式是 Watson 变换、Rahman 的二次变换、Gasper 和 Rahman 的三次变换、Gasper 和 Rahman 的四次变换、Ismail、Rahman 和 Suslov 的双级数变换，以及用于非终止性很好的 $${}_{12}\phi _{11}$$ 12 ϕ 11 系列的新转换公式。此外，还使用了非终止 q -Dixon 求和公式。进一步利用新的 $${}_{12}\phi _{11}$$ 12 ϕ 11 变换公式的特例来获得 Rogers 对连续 q 超球面多项式的线性化公式的推广。