Linear difference equations with transition points |
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Authors: | Z. Wang R. Wong. |
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Affiliation: | Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, P. O. Box 71010, Wuhan 430071, Peoples Republic of China ; Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong |
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Abstract: | Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation where and have power series expansions of the form with . Our results hold uniformly for in an infinite interval containing the transition point given by . As an illustration, we present an asymptotic expansion for the monic polynomials which are orthogonal with respect to the modified Jacobi weight , , where , and is real analytic and strictly positive on . |
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Keywords: | Difference equation transition points three-term recurrence relation orthogonal polynomials |
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