Zeros of quasi-orthogonal Jacobi polynomials (Q288685)

A classical theorem of Stieltjes states that, if \(\{ p_n \}_{n=0}^{\infty}\) is a sequence of orthogonal polynomials, then the zeros of \(p_k\) and \(p_n\), \(k<n\), are interlacing in the sense that each open interval of the form \(] - \infty , z_1 [\), \(] z_1,z_2 [\), \(\ldots\), \(] z_{k-1},z_k [\), \( ] z_k,\infty [\), where \(z_1 < z_2< \ldots < z_k\) are the zeros of \(p_k\), contains at least one zero of \(p_n\). Moreover, recently, Beardon has proved that, if \(\{ p_n \}_{n=0}^{\infty}\) is a sequence of orthogonal polynomials with \(p_m\) and \(p_n\) having no common zeros for \(m \not= n\), then there exist real polynomials \(S_{n-m-1}\) of degree \(n-m-1\) whose simple real zeros, together with the zeros of \(p_m\), interlace with the zeros of \(p_n\) for \(m<n\). The polynomials \(S_{n-m-1}\) were first observed, albeit, in a rather different context by de Boor and Saff in 1986. An important feature of these de Boor-Saff polynomials is that they are completely determined by the coefficients in the three-term recurrence relation satisfied by the orthogonal sequence \(\{ p_n \}_{n=0}^{\infty}\). The question arises as to whether Stieltjes interlacing occurs between the zeros of two polynomials \(p_n\) and \(q_k\), \(k<n-1\), from different orthogonal sequences \(\{ p_n \}_{n=0}^{\infty}\) and \(\{ q_n \}_{n=0}^{\infty}\), and whether polynomials analogous to the de Boor-Saff polynomials exist in this more general situation. Among the classical orthogonal families of Gegenbauer, Laguerre and Jacobi polynomials, natural choices of different orthogonal sequences are those corresponding to different values of the appropriate parameter(s) and also the (orthogonal) sequences of their derivatives. In this paper, it is proved that Stieltjes interlacing extends across different sequences of Laguerre polynomials \(L_n^{\alpha}\), \(\alpha >-1\). In particular, it is shown that Stieltjes interlacing holds between the zeros of \(L_{n-1}^{\alpha+t}\) and \(L_{n+1}^{\alpha}\), \(n \in \mathbb{N}\), \(\alpha>-1\), \(t \in \{ 1,2,3,4 \}\) but not in general when \(t>4\) or \(t<0\) and provide numerical examples to illustrate the breakdown of interlacing. The conjecture is made that Stieltjes interlacing holds between the zeros of \(L_{n-1}^{\alpha+t}\) and those of \(L_{n+1}^{\alpha}\) for \(0<t<4\). More generally, it is shown that Stieltjes interlacing occurs between the zeros of \(L_{n+1}^{\alpha}\) and the zeros of the \(k\)-th derivative of \(L_n^{\alpha}\), as well as with the zeros of \(L_{n-k}^{\alpha+k+t}\) for \(t \in \{ 1,2 \}\) and \(k \in \{ 1,2,\ldots,n-1 \}\). In each case, associated polynomials analogous to the de Boor-Saff polynomials, which are completely determined by the coefficients in a mixed three-term recurrence relation, are identified, whose zeros complete the interlacing process.