Stieltjes interlacing of zeros of Laguerre polynomials from different sequences

Stieltjes’ Theorem (cf. [11]) proves that if {pn}n=0 is an orthogonal sequence, then between any two consecutive zeros of pk there is at least one zero of pn for all positive integers k, k < n; a property called Stieltjes interlacing. We prove that Stieltjes interlacing extends across different sequences of Laguerre polynomials Ln, α > −1. In particular, we show that Stieltjes interlacing holds between the zeros of L n−1 and L α n+1, α > −1, when t ∈ {1, . . . , 4} but not in general when t > 4 or t < 0 and provide numerical examples to illustrate the breakdown of interlacing. We conjecture that Stieltjes interlacing holds between the zeros of L n−1 and those of L a n+1 for 0 < t < 4. More generally, we show that Stieltjes interlacing occurs between the zeros of Ln+1 and the zeros of the kth derivative of Ln, as well as with the zeros of L α+k+t n−k for t ∈ {1, 2} and k ∈ {1, 2, . . . , n−1}. In each case, we identify associated polynomials, analogous to the de Boor-Saff polynomials (cf. [3], [7]), that are completely determined by the coefficients in a mixed three term recurrence relation, whose zeros complete the interlacing process.