Polynomial bases

1.1 Well known facts

Xiu and Karniadakis (2002) mention the following “Well known facts”:

All orthogonal polynomials $\{Q_n(x)\}$ satisfy a three-term recurrence relation $$-x{Q}_n(x)=A_n{Q}_{n+1}(x)-(A_n+C_n)Q_n(x)+C_n{Q}_{n-1}(x),n\ge 1$$ where $A_n,C_n\ne 0$ and $C_n/A_{n-1}>0.$ Together with $Q_{-1}(x)=0$ and $Q_0(x)=1,$ all $Q_n(x)$ can be determined by the recurrence relation.

It is well known that continuous orthogonal polynomials satisfy the second-order differential equation $$s(x)y^{\prime \prime }+\tau (x)y^{\prime }+\lambda y=0$$ where $s(x)$ and $\tau (x)$ are polynomials of at most second and first degree, respectively, and $$\lambda ={\lambda }_n=-n{\tau }^{\prime }-\frac{1}{2}n(n-1)s^{\prime \prime }$$ are the eigenvalues of the differential equation; the orthogonal polynomials $y(x)=$ $y_n(x)$ are the eigenfunctions.

• Convex hull of zeros

  There’s a well-known theorem in complex analysis that says that if p is a polynomial, then the zeros of its derivative p′ lie inside the convex hull of the zeros of p.