Interlacing of zeros of orthogonal polynomials

Remarks

For part 1, let $y_1, \dots, y_m$ be the zeros of odd multiplicity of $p_n$ in the interior of $I$, the points where $p_n$ changes sign. Suppose $m < n$. Then $p_n(x) \prod_j (x - y_j)$ does not change sign on $I$ and is not zero $\mu$-almost everywhere, so its integral against $\mu$ is not zero. But $\prod_j (x - y_j)$ has degree $m < n$, and $p_n$ is orthogonal to all polynomials of lower degree, a contradiction. Part 2 follows from the three-term recurrence through the Christoffel–Darboux formula. If the leading coefficients are positive, it gives $p_{n+1}'(x) p_n(x) - p_{n+1}(x) p_n'(x) > 0$ for all real $x$. Evaluating at consecutive zeros of $p_{n+1}$ shows that $p_n$ changes sign between them.

Example: for the Legendre polynomials, $p_1 = x$ has the zero $0$, and $p_2 = \frac12(3x^2 - 1)$ has the zeros $\pm 1/\sqrt{3}$. Indeed $-1/\sqrt{3} < 0 < 1/\sqrt{3}$, and all lie in $(-1, 1)$.

Going deeper

For a student

The zeros of $p_n$ are the eigenvalues of the $n \times n$ truncated Jacobi matrix, the symmetric tridiagonal matrix built from the recurrence coefficients. In this language, interlacing is the Cauchy interlacing theorem for a matrix and its principal submatrix. The zeros of $p_n$ are the nodes of Gaussian quadrature for $\mu$, and the fact that they are real, simple and inside $I$ is what makes that quadrature work. As $n \to \infty$, the counting measures of the zeros, divided by $n$, describe the asymptotic distribution of the zeros, a central topic of the theory.