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Linear difference equations with transition points

Author(s): Z. Wang; R. Wong.
Journal: Math. Comp. 74 (2005), 629-653.
MSC (2000): Primary 41A60, 39A10, 33C45
Posted: May 25, 2004
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Abstract: Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation

\begin{displaymath}y_{n+1}(x)-(A_nx+B_n)y_n(x)+y_{n-1}(x)=0, \nonumber \end{displaymath}  

where $A_n$ and $B_n$ have power series expansions of the form

\begin{displaymath}A_n\sim \sum^\infty_{s=0}\frac{\alpha_s}{n^s}, \qquad\qquad B_n\sim \sum^\infty_{s=0}\frac{\beta_s}{n^s}\nonumber \end{displaymath}  

with $\alpha_0\ne 0$. Our results hold uniformly for $x$ in an infinite interval containing the transition point $x_+$given by $\alpha_0 x_++\beta_0=2$. As an illustration, we present an asymptotic expansion for the monic polynomials $\pi_n(x)$ which are orthogonal with respect to the modified Jacobi weight $w(x)=(1-x)^\alpha(1+x)^\beta h(x)$, $x\in (-1,1)$, where $\alpha$, $\beta>-1$ and $h$ is real analytic and strictly positive on $[-1, 1]$.

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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: 10.1090/S0025-5718-04-01677-1
PII: S 0025-5718(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
Posted: May 25, 2004
Additional Notes: The work of this author was partially supported by the Research Grant Council of Hong Kong under Project 9040522
Copyright of article: Copyright 2004, American Mathematical Society

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