# Perturbation analysis for three term recurrences

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence

$$J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), \qquad J^{(\alpha,\beta)}_0 = \gamma_0,\quad J_1^{(\alpha,\beta)}(x) = \gamma_1x+\alpha_1.$$

I'm interested in the stability of the three term recurrence under small perturbations of its coefficients. For instance, if $\tilde{X}$ denotes the quantity $X$ perturbed by $\epsilon|X|$ where $|\epsilon|\ll 1$, and $\tilde{J}^{(\alpha,\beta)}_n$ satisfies

$$\tilde{J}_{n+1}^{(\alpha,\beta)}(x) = (\tilde{A}_n+\tilde{B}_n x)\tilde{J}^{(\alpha,\beta)}_n + \tilde{C}_n\tilde{J}^{(\alpha,\beta)}(x), \qquad \tilde{J}^{(\alpha,\beta)}_0 = \tilde{\gamma}_0,\quad \tilde{J}_1^{(\alpha,\beta)}(x) = \tilde{\gamma}_1x+\tilde{\alpha}_1,$$

then is anything known about the magnitude of

$$| J_{n}^{(\alpha,\beta)}(x) - \tilde{J}_{n}^{(\alpha,\beta)}(x)|$$

or

$$|| J_{n}^{(\alpha,\beta)}(x) - \tilde{J}_{n}^{(\alpha,\beta)}(x) ||_2?$$

Numerical observations on the Chebyshev polynomial $T_k(x) = \cos(k\cos^{-1}(x))$, $x\in[-1,1]$ suggests that

$$||J_{n}^{(\alpha,\beta)}(x) - \tilde{J}_{n}^{(\alpha,\beta)}(x)||_2 \leq \epsilon\mathcal{O}( n^2).$$

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