Orthogonal polynomials: their growth relative to their sums

Abstract Assume that a n → a 2 and b n → b . Let the polynomials {pn} be defined by the recurrence relation xpn(x) = an + 1pn + 1(x) + bnpn(x) + anpn − 1(x), n = 0, 1, 2, …, where p0 = const > 0 and p−1 = 0. It is proved that for every 0 n→∞ lim |p n (x)| p ∑ k=O n−1 ||p n (x)| p uniformly for xϵ [b − a, b + a]. The uniform convergence actually holds not only when the initial condition p0 and p−1 are restricted to polynomials but also for almost all kinds of initial functions p0 and p−1. A number of applications to orthogonal polynomials are also studied.