Elementary ratio and zero limits for varying recurrences

Abstract

Two theorems of Loureiro and Van Assche on ratio and zero limits provided part of the motivation for this note. Their elementary nature permits direct proofs; Appendix B gives complete proofs of both for the canonical initial conditions, using earlier continued-fraction identities. We prove ratio limits and weak limits of normalized zero measures for varying higher-order three-term recurrences with arbitrary coefficient periods and independent limiting functions for their phases. Joint local coefficient limits give ratio convergence without uniform coefficient bounds. For zero measures, convergence in measure to finite measurable profiles suffices, without boundedness or integrability; the limiting measures may have noncompact support. Coefficients may vanish, and real diagonal terms are allowed for ordinary three-term recurrences. The proofs use interlacing, partial fractions with nonnegative residues, local computations of moments, and stability of zero counts under changes of recurrence coefficients. Truncation removes the coefficient bounds from the zero theorem, whose proof is independent of ratio asymptotics. Uniform bounds give the version with compact support. Power-law limits and their explicit transform formula follow as a special case.