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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 n x+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 s=0 α s n s ,B n s=0 β s n s

with α 0 0. The results hold uniformly for x in an infinite interval containing the transition point x + given by α 0 x + +β 0 =2. As an illustration, the authors present an asymptotic expansion for the monic polynomials π n (x) which are orthogonal with respect to the modified Jacobi weight w(x)=(1-x) α (1+x) β h(x), x(-1,1), where α,β>-1 and h is real analytic and strictly positive on [-1,1].

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