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q -analogues of two supercongruences of Z.-W. Sun

Cheng-Yang Gu, Victor J. W. Guo (2020)

Czechoslovak Mathematical Journal

We give several different q -analogues of the following two congruences of Z.-W. Sun: k = 0 ( p r - 1 ) / 2 1 8 k 2 k k 2 p r ( mod p 2 ) and k = 0 ( p r - 1 ) / 2 1 16 k 2 k k 3 p r ( mod p 2 ) , where p is an odd prime, r is a positive integer, and ( m n ) is the Jacobi symbol. The proofs of them require the use of some curious q -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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