On conjectures of Stenger in the theory of orthogonal polynomials

Abstract

Abstract The conjectures in the title deal with the zeros xj $x_{j}$, j=1,2,…,n $j=1,2, \ldots ,n$, of an orthogonal polynomial of degree n>1 $n>1$ relative to a nonnegative weight function w on an interval [a,b] $[a,b]$ and with the respective elementary Lagrange interpolation polynomials ℓk(n) $\ell _{k} ^{(n)}$ of degree n−1 $n-1$ taking on the value 1 at the zero xk $x_{k}$ and the value 0 at all the other zeros xj $x_{j}$. They involve matrices of order n whose elements are integrals of ℓk(n) $\ell _{k}^{(n)}$, either over the interval [a,xj] $[a,x_{j}]$ or the interval [xj,b] $[x_{j},b]$, possibly containing w as a weight function. The claim is that all eigenvalues of these matrices lie in the open right half of the complex plane. This is proven to be true for Legendre polynomials and a special Jacobi polynomial. Ample evidence for the validity of the claim is provided for a variety of other classical, and nonclassical, weight functions when the integrals are weighted, but not necessarily otherwise. Even in the case of weighted integrals, however, the conjecture is found by computation to be false for a piecewise constant positive weight function. Connections are mentioned with the theory of collocation Runge–Kutta methods in ordinary differential equations.

Keywords

• Zeros of orthogonal polynomials
• Lagrange interpolation
• Matrix eigenvalues
• Conjectured location of eigenvalues in the complex plane