On a conjecture by Karlin and Szegö

Abstract:

In 1961, Karlin and Szegö conjectured : If $\{P_n(x)\}_{n=0}^\infty$ is an orthogonal polynomial system and $\{P_n’(x)\}_{n=1}^\infty$ is a Sturm sequence, then $\{P_n(x)\}_{n=0}^\infty$ is essentially (that is, after a linear change of variable) a classical orthogonal polynomial system of Jacobi, Laguerre, or Hermite. Here, we prove that for any orthogonal polynomial system $\{P_n(x)\}_{n=0}^\infty$, $\{P_n’(x)\}_{n=1}^\infty$ is always a Sturm sequence. Thus, in particular, the above conjecture by Karlin and Szegö is false.