Linearly related orthogonal polynomials: the vision of J. Petronilho

In a recent volume dedicated to the memory of the Portuguese mathematician J. Petronilho, the third volume of the Coimbra Mathematical Texts published by Springer, a work was presented containing a series of results on what the authors called the general inverse problem for linearly related orthogonal polynomials.The purpose of the present note is twofold: first, to show that what the authors of that paper refer to as the general inverse problem does not, in fact, correspond to what should properly bear that designation, since Petronilho had already studied far more general relations; and second, to demonstrate that a number of their results are immediate consequences of the proof of a theorem established by J. Petronilho in 2006, which already encapsulates the very essence of any subsequent development that might be pursued in this direction.