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We give a new and elementary proof of Jackson’s terminating -analogue of Dixon’s identity by using recurrences and induction.
For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as
for p≡ 3 (mod 4),
for p≡ 2 (mod 3),
where and . We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula.
Let denote the th cyclotomic polynomial in . Recently, Guo, Schlosser and Zudilin proved that for any integer with ,
where . In this note, we give a generalization of the above -congruence to the modulus case. Meanwhile, we give a corresponding -congruence modulo for . Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a summation formula.
We give several different -analogues of the following two congruences of Z.-W. Sun:
where is an odd prime, is a positive integer, and is the Jacobi symbol. The proofs of them require the use of some curious -series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.
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