Grassl,M.Özbudak,F.Özkaya,B.Temür,B.G.2024-07-052024-07-05202401936-244710.1007/s12095-024-00705-22-s2.0-85185476761https://doi.org/10.1007/s12095-024-00705-2https://hdl.handle.net/20.500.14411/4167In this paper, we address an open problem posed by Bai and Xia in [2]. We study polynomials of the form f(x)=x4q+1+λ1x5q+λ2xq+4 over the finite field F5k, which are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on λ1,λ2∈F5k so that f(x) is a permutation monomial, binomial, or trinomial of F52k. © The Author(s) 2024.eninfo:eu-repo/semantics/openAccess11T7112E10Absolutely irreducibleFinite fieldsMSC 11T06Permutation polynomialsArticleQ2Q3164825841