On Co-polynomials and d-Orthogonality

This paper deals with the theory of d-orthogonal polynomials and it aims to find out some sufficient conditions for the zeros of the above polynomials to be real and distinct using Darboux factorization together with the properties of totally positive matrices. We shall show that such condition exists and it requires the recurrence coefficients to be strictly positive. The so-called co-polynomials are deeply investigated and they are explicitly expressed in terms of the basic solutions. Some of them are used to determine the entries of the matrices in LU and UL decomposition of Hessenberg matrix. Moreover, some Casorati determinants with co-polynomials entries are considered. © The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2023.