Some Inverse Problems for d-Orthogonal Polynomials

This paper is devoted to the study of the generalized inverse problem of the left product of a d–dimensional vector form by a polynomial. The objective is to find the regularity conditions of the vector linear form ${mathcal{V}}$ defined by ${mathcal{U} = mathcal{RV}}$ , where ${mathcal{R}}$ is a d × d matrix polynomial. In such a case, the d–OPS {Q _n } _{n ≥ 0} corresponding to ${mathcal{V}}$ is d–quasi– orthogonal of order l with respect to ${mathcal{U}}$ . Secondly, we study the inverse problem: Given a d -OPS P _{n n ≥ 0} with respect to ${mathcal{U}}$ , characterize the parameters ${{a^{(i)}_{n}}{^{dl}_{i=1}}}$ such that the sequence $${Q_{n+dl} = P_{n+dl} + sum _{i=1}^{dl} a_{n+dl}^{(i)}P_{n+dl-i},quad ngeq 0}$$ , is d–orthogonal with respect to some regular vector linear form ${mathcal{V}}$ . As an immediate consequence, find the explicit relation between ${mathcal{U}}$ and ${mathcal{V}}$ .