Two-point completion of zero interlacing and Wronskian-type bounds, with applications to Jacobi and other orthogonal polynomials

Abstract

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using suitable mixed recurrence relations, we identify two additional points, called the completion points, required to obtain completed interlacing and establish general results for polynomials with real zeros. We also show that completed interlacing yields Wronskian-type bounds for the original polynomial pairs. The completed-interlacing results are applied to Jacobi, Meixner-Pollaczek, and Pseudo-Jacobi polynomials. In the Jacobi case, we improve earlier results by determining two explicit additional points that complete the interlacing of $P_n^{(\alpha,\beta)}$ and $P_{n+1}^{(\alpha+1,\beta+1)}$. We also analyze the locations of these points and show that the common-zero cases are non-generic in the Jacobi parameter domain. For this doubly-shifted Jacobi pair, the Wronskian-type bounds yield cross-family Tur\'an-type inequalities and lead to an associated Stieltjes representation and complete-monotonicity consequences. For Meixner-Pollaczek polynomials, we address an open question concerning consecutive degrees with the parameter increased by one, and we obtain analogous completed interlacing results for Pseudo-Jacobi polynomials.