Linear difference equations with transition points

Authors:
Z. Wang and R. Wong

Journal:
Math. Comp. **74** (2005), 629-653

MSC (2000):
Primary 41A60, 39A10, 33C45

DOI:
https://doi.org/10.1090/S0025-5718-04-01677-1

Published electronically:
May 25, 2004

MathSciNet review:
2114641

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Z. Wang**

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

Email:
mcwang@cityu.edu.hk

**R. Wong**

Affiliation:
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Email:
mawong@cityu.edu.hk

DOI:
https://doi.org/10.1090/S0025-5718-04-01677-1

Keywords:
Difference equation,
transition points,
three-term recurrence relation,
orthogonal polynomials

Received by editor(s):
April 2, 2003

Received by editor(s) in revised form:
October 6, 2003

Published electronically:
May 25, 2004

Additional Notes:
The work of this author was partially supported by the Research Grant Council of Hong Kong under Project 9040522

Article copyright:
© Copyright 2004
American Mathematical Society