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Linear difference equations with transition points. (English) Zbl 1083.41022
Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation $$y_{n+1}(x)-(A_nx +B_n)y_n(x)+y_{n-1}(x)=0,$$ where $A_n$ and $B_n$ have power series expansions of the form $$A_n \sim\sum^\infty_{s=0}\frac {\alpha_s} {n^s},\qquad B_n\sim\sum^\infty_{s=0}\frac {\beta_s} {n^s}$$ with $\alpha_0\ne 0$. The 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, the authors 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 $[-1,1]$.

41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
39A10Additive difference equations
33C45Orthogonal polynomials and functions of hypergeometric type
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