Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

• Published in: Constructive approximation Vol. 53; no. 1; pp. 155 - 200
• Main Authors: Guo, Victor J. W., Schlosser, Michael J.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2021; Springer Nature B.V
• Subjects: Analysis; Congruences; Mathematics; Mathematics and Statistics; Numerical Analysis; Polynomials; Transformations (mathematics); Primary 33D15; 33D45; Supercongruences; Basic hypergeometric series; Secondary 11A07; Identities; Linearization; 11F33
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-020-09524-z

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 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.