On a conjecture by Karlin and Szego

Abstract

In 1961, Karlin and Szego conjectured : If {P-n(x)}(infinity)(n=0) is an orthogonal polynomial system and {P'(n)(x)}(infinity)(n=1) is a Sturm sequence, then {P-n(x)}(infinity)(n=0) 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)}(infinity)(n=0), {P'(n)(x)}(infinity)(n=1) is always a Sturm sequence. Thus, in particular, the above conjecture by Karlin and Szego is false.