Asymptotics for second-order linear difference equations

Abstract

In the first part of this thesis, we consider the asymptotic solutions of second-order linear difference equations constructed by R. Wong and H. Li (Stud. Appl. Math. 87 (1992), 289-324). By studying a comparison equation, we obtain explicit and numerically computable error bounds of the asymptotic expansions of these solutions. We use the Tricomi-Calitz polynomials as an example and derive an explicit error bound for its asymptotic approximations. In the second part of the thesis, we study the uniform asymptotic expansions for second-order linear difference equations. A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation Pn+1(x) - (Anx + Bn)Pn(x) + Pn-1(x) = 0, where An and Bn have asymptotic expansions of the form An~n-θ ∑∞ s=0 αs/ns, Bn~ ∑∞ s=0 βs/ns, with θ ≠ 0,2 and α0 ≠ 0 being real numbers, and β0 = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t1 = 0, where t = (n + τ0)-θx, with τ 0 being a small shift. In particular, it is shown how Bessel functions Jv and Yv get involved in the uniform asymptotic expansions of the solutions to the three-term recurrence relation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight xα exp(-qmxm), x > 0, where m is a positive integer, α > -1 and qm > 0.

Date of Award	16 Jul 2012
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Sue Cheun Roderick WONG (Supervisor)