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Asymptotics of linear recurrences
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In this article, asymptotic results for second-order linear difference equations containing a parameter (or, equivalently, three-term recurrence relations) are reviewed for solutions to these equations when the parameter is fixed or varying in an interval containing a turning point or a transition point.Abstract:
In this paper, we review the asymptotic results that are now available for second-order linear difference equations containing a parameter (or, equivalently, three-term recurrence relations) These include asymptotic expansions for solutions to these equations when the parameter is fixed or varying in an interval containing a turning point or a transition point Also presented is a method for deriving asymptotic approximations for solutions when the initial values are given These results are particularly useful when a given system of orthogonal polynomials (i) does not satisfy any second-order differential equation, (ii) does not have any integral representation, and (iii) is not even associated with a unique (or any) weight functionread more
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References
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Journal ArticleDOI
Resurrecting the asymptotics of linear recurrences
Jet Wimp,Doron Zeilberger +1 more
TL;DR: In this paper, the asymptotics of the solutions of linear recurrence equations have been studied for combinatorists and computer scientists, and the authors present a theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in their toolkit.
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Analytic theory of singular difference equations
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General theory of linear difference equations
TL;DR: In this article, it was shown that rational functions of e27r8-lX are a part of the monodromic group constants of an ordinary linear differential equation and that there exists a purely Riemannian theory of linear differellce equations.
Journal ArticleDOI
Asymptotic expansions for second-order linear difference equations
Roderick Wong,H. Li +1 more
TL;DR: In this article, formal series solutions are obtained for the difference equation y(n+2)+a(n)y(n+)n+1)+b(n), where a n and b n have asymptotic expansions of the form a n ∼∑ ∞ s = 0 a s n s and b b n √ ∞s = 0 b n n s, for large values of n, and b 0 ≠ 0.
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