@article{Wang2022, 
author = {Su-Dan Wang},
title = {The    q-WZ pairs and divisibility properties of certain polynomials},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {3},
pages = {4115-4124},
keywords = {q-Binomial coefficients, congruences, q-analogues, q-WZ pair},
url = {https://www.sciopen.com/article/10.3934/math.2022227},
doi = {10.3934/math.2022227},
abstract = {Using the    q-WZ (Wilf-Zeilberger) pairs we give divisibility properties of certain polynomials. These results may deemed generalizations of some    q-congruences obtained by Guo earlier, or    q-analogues of some congruences of Sun. For example, we prove that, for    n  ⩾  1 and    0  ⩽  j  ⩽  n, the following two polynomials                                               ∑                      k            =            j                                n                          (        −        1                  )                      k                          [        3        k        −        2        j        +        1        ]                                            [                                                      2                k                −                2                j                            k                                      ]                                                            (            q            ;                          q              2                                      )              k                        (            q            ;                          q              2                                      )                              k                −                j                                      (            −            q            ;            q                          )              n              3                                            (            q            ;            q                          )              k                        (                          q              2                        ;                          q              2                                      )                              k                −                j                                                    ,                                                    ∑                      k            =            j                                n                          (        −        1                  )                      n            −            k                                    q                      (            k            −            j                          )              2                                      [        4        k        +        1        ]                              (            q            ;                          q              2                                      )              k              2                        (            q            ;                          q              2                                      )                              k                +                j                                      (            −            q            ;            q                          )              n              6                                            (                          q              2                        ;                          q              2                                      )              k              2                        (                          q              2                        ;                          q              2                                      )                              k                −                j                                      (            q            ;                          q              2                                      )              j              2                                      .            are divisible by    (  1  +      q    n        )    2    [  2  n  +  1  ]                    [                              2          n                n                    ]            . Here    [  m  ]  =  1  +  q  +  ⋯  +      q          m      −      1        ,  (  a  ;  q      )    m    =  (  1  −  a  )  (  1  −  a  q  )  ⋯  (  1  −  a      q          m      −      1        ), and                      [                    m        k                    ]              =  (      q          m      −      k      +      1        ;  q      )    k        /    (  q  ;  q      )    k  .}
}