Nonnegative solutions of a nonlinear recurrence

Abstract

Orthonormal polynomials with weight |τ|ρ{variant} exp(-τ4) have leading coefficients with recurrence properties which motivate the more general equations ξm(ξm - 1 + ξm + ξm + 1) = γm2, m = 1, 2,..., where ξo is a fixed nonnegative value and γ1, γ2,... are positive constants. For this broader problem, the existence of a nonnegative solution is proved and criteria are found for its uniqueness. Then, for the motivating problem, an asymptotic expansion of its unique nonnegative solution is obtained and a fast computational algorithm, with error estimates, is given. © 1983.