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A detailed description of the binomial theorem and an application to permutation binomials over finite fields

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Abstract

In this paper, we present a detailed description of the binomial theorem and obtain some new classes of combinatorial identities. As an application, we discuss a class of permutation binomials over finite fields \({\mathbb {F}}_q\), which is of the form \(x^{\mu +\nu }+2x^{\mu }\), where \(q\equiv 1\pmod {3}\) and \((\nu , q-1)=\frac{q-1}{3}\).

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Acknowledgements

The authors sincerely thank the referees for providing them suggestions and constructive comments which led to the improvement of this article.

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Authors and Affiliations

  1. School of Mathematical Science, South China Normal University, Guangzhou, 510640, Guangdong, People’s Republic of China

    Zhilin Zhang

  2. Mathematics and Computational Science, Hunan First Normal University, Changsha, 410205, Hunan, People’s Republic of China

    Lang Tang

  3. Liangtian Town Third Primary School, Guangzhou, 510550, Guangdong, People’s Republic of China

    Ningjing Huang

Authors
  1. Zhilin Zhang
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  2. Lang Tang
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  3. Ningjing Huang
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Correspondence to Zhilin Zhang, Lang Tang or Ningjing Huang.

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Zhilin Zhang: was supported by the National Natural Science Foundation of China (No: 12001204) and the China Postdoctoral Science Foundation (No: 2019M662945)

Lang Tang: was supported by the Hunan Provincial Natural Science Foundation (No: 2020JJ5096) and the National Natural Science Foundation of China (No: 11871206)

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Zhang, Z., Tang, L. & Huang, N. A detailed description of the binomial theorem and an application to permutation binomials over finite fields. J. Appl. Math. Comput. 68, 177–198 (2022). https://doi.org/10.1007/s12190-021-01525-w

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  • DOI: https://doi.org/10.1007/s12190-021-01525-w

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