On a conjecture by Karlin and Szegö

Abstract

In 1961, Karlin and Szegö conjectured : If { P n ( x ) } n = 0 ∞ \{P_n(x)\}_{n=0}^\infty is an orthogonal polynomial system and { P n ′ ( x ) } n = 1 ∞ \{P_n’(x)\}_{n=1}^\infty is a Sturm sequence, then { P n ( x ) } n = 0 ∞ \{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 ∞ \{P_n(x)\}_{n=0}^\infty , { P n ′ ( x ) } n = 1 ∞ \{P_n’(x)\}_{n=1}^\infty is always a Sturm sequence. Thus, in particular, the above conjecture by Karlin and Szegö is false.