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Linear difference equations with transition points
Authors:Z. Wang   R. Wong.
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
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_nsim sum^infty_{s=0}frac{alpha_s}{n^s}, qquadqquad B_nsim sum^infty_{s=0}frac{beta_s}{n^s}nonumber end{displaymath}  

with $alpha_0ne 0$. 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 $w(x)=(1-x)^alpha(1+x)^beta h(x)$, $xin (-1,1)$, where $alpha$, $beta>-1$ and $h$ is real analytic and strictly positive on $[-1, 1]$.

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