Nonnegative solutions of a nonlinear recurrence

AbstractOrthonormal polynomials with weight ¦τ¦ϱ 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