Linear difference equations with transition points

Wang, Z.; Wong, R.

doi:10.1090/S0025-5718-04-01677-1

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ISSN 1088-6842(online) ISSN 0025-5718(print)

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
Posted: May 25, 2004
MathSciNet review: 2114641
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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

and

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

in an infinite interval containing the transition point

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

is real analytic and strictly positive on

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