Stieltjes interlacing of zeros of Laguerre polynomials from different sequences

Stieltjes’ Theorem (cf. Szegö (1959) [10]) proves that if {pn}n=0∞ is an orthogonal sequence, then between any two consecutive zeros of pkpk there is at least one zero of pnpn for all positive integers kk, k−1α>−1. In particular, we show that Stieltjes interlacing holds between the zeros of Ln−1α+t and Ln+1α, α>−1α>−1, when t∈{1,…,4}t∈{1,…,4} but not in general when t>4t>4 or t<0t<0 and provide numerical examples to illustrate the breakdown of interlacing. We conjecture that Stieltjes interlacing holds between the zeros of Ln−1α+t and those of Ln+1α for 0