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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