We work with sequences of integrals that we call SCE integrals. We establish their expressions in terms of five families of polynomials. We demonstrate the relations between these integrals and we focus on one of the three integrals: the determination of the family of polynomials noted $e_n$ $(n \in \mathbb{N})$. We show that these polynomials are hypergeometric. From this property, the NU method can be applied to this family. We determine the Rodrigues formula. These polynomials have properties that distinguish them from classical hypergeometric polynomials. We state and demonstrate the theorem adapted to the determination of the $e_n$ generating function. Finally, the sequence of polynomials studied is expressed in terms of associated Laguerre polynomials with negative upper indices.