Upward extension of the Jacobi matrix for orthogonal polynomials (Q1925092)

It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation. The coefficients in this relation can be put together in a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. In the paper, new systems of orthogonal polynomials are introduced and investigated, which result from a Jacobi matrix that is shifted downward by adding \(r\) new rows and columns. Thus, the new definition in some sense is a counterpart to the definition of associated orthogonal polynomials. The \(r\) new rows and columns contain \(2r\) new parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. An explicit expression of the new orthogonal polynomials is given in terms of the original orthogonal polynomials, their associated polynomials, and the \(2r\) new parameters. If the original polynomials are classical ones, then the new ones satisfy a fourth-order differential equation. Furthermore, it is shown how the orthogonalizing measure of the new orthogonal polynomials can be obtained. Details of this are worked out for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.