Asymptotic behavior of solutions of three-term Poincaré difference equations
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions of three-term Poincaré difference equations
zbMATH Open document ID: 0888.39006
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-11-06T00:00:00Z
full work available at URL: CONT27-1.html
review text: The author considers the difference equation \(Z_n= b_nZ_{n-1} +a_n Z_{n-2}\) with complex coefficients \(a_n\to a\), \(b_n\to b\) for \(n\to\infty\) and \(a_n\neq 0\). The solutions \(x_1\), \(x_2\) of \(x^2= ax+b\) shall satisfy \(|x_1|<|x_2|\). Under the assumption that \(\sum(a_n-a)\) and \(\sum(b_n-b)\) are only conditionally convergent, conditions are given such that \(\lim_{n\to\infty} Z_nx_2^{-n}=Z\) exists, and the error \(|Z-Z_nx_2^{-n} |\) is estimated. The results are applied to ten examples, in particular to recurrence equations arising from hypergeometric functions and continued fractions, respectively.
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 1051532
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: three-term Poincaré difference equations
zbMATH Keywords: recurrence equations
zbMATH Keywords: hypergeometric functions
zbMATH Keywords: continued fractions
author: Lisa Lorentzen
reviewed by: Lothar Berg
MaRDI profile type: Publication
OpenAlex ID: W2072814369
cites work: Q5558293
cites work: Q5629915
cites work: Q3995284
cites work: Analytic theory of singular difference equations
cites work: Q5824493
cites work: Recursion formulae for generalized hypergeometric functions
cites work: Associated Continuous Hahn Polynomials
cites work: Exceptional q-Askey-Wilson Polynomials and Continued Fractions
cites work: Watson’s Basic Analogue of Ramanujan’s Entry $40$ and Its Generalization
cites work: Q4764038
cites work: Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II: Associated big \(q\)-Jacobi polynomials
cites work: Two Families of Associated Wilson Polynomials
cites work: Two families of orthogonal polynomials related to Jacobi polynomials
cites work: q-Hermite Polynomials, Biorthogonal Rational Functions, and q-Beta Integrals
cites work: Generalized orthogonality and continued fractions
cites work: The Associated Askey-Wilson Polynomials
cites work: Q4853655
cites work: Q3889331
cites work: Q5804312
cites work: Continued fractions with applications
cites work: The special functions and their approximations. Vol. I, II
cites work: Convergence and Analytic Continuation for a Class of Regular C-Fractions
cites work: Q3473567
cites work: Q3781508
cites work: Some Continued Fractions of Ramanujan and Meixner-Pollaczek Polynomials
cites work: Q4009422
cites work: Wilson polynomials and some continued fractions of Ramanujan
cites work: Associated Wilson polynomials
cites work: Q4732836
cites work: Meromorphic Extension of Analytic Continued Fractions Across Their Divergence Line with Applications to Orthogonal Polynomials
cites work: Solution formulas for linear difference equations with applications to continued fractions
cites work: Q3496443
cites work: Q3961719
cites work: Asymptotic Behavior of Solutions of Poincare Difference Equations
cites work: Recursion Formulae of Hypergeometric Functions
cites work: Q3332204
cites work: Some Explicit Padé Approximants for the Function ${{\Phi '} / \Phi }$ and a Related Quadrature Formula Involving Bessel Functions
cites work: Explicit Formulas for the Associated Jacobi Polynomials and Some Applications
cites work: Pollaczek polynomials and Padé approximants: Some closed-form expressions
cites work: Resurrecting the asymptotics of linear recurrences
DOI: 10.1216/RMJM/1181071957
Recommended article: Q2998492
Recommended article: Q4232913
Recommended article: Q3769338
Recommended article: Q5749744
Recommended article: Asymptotic Behavior of Solutions of Poincare Difference Equations
Recommended article: Q3372574
Recommended article: Asymptotic Behavior of a Poincaré Difference Equation
Recommended article: Asymptotic properties of solutions of third order difference equations
Recommended article: Q4735462
Recommended article: Q3684437
description: scientific article; zbMATH DE number 1051532