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Asymptotic expansions for second-order linear difference equations with a turning point. (English) Zbl 1030.39016

Authors’ summary: A turning-point theory is developed for the second-order difference equation

P n+1 (x)-(A n x+B n )P n (x)+P n-1 (x)=0,n=1,2,3,

where the coefficients A n and B n have asymptotic expansions of the form

A n n -θ s=0 α s n s andB n s=0 β s n s ,

θ0 being a real number. In particular, it is shown how the Airy functions arise in the uniform asymptotic expansions of the solutions to this three-term recurrence relation. As an illustration of the main result, a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the Freud weight exp(-x 4 ), x.

39A11Stability of difference equations (MSC2000)
33C10Bessel and Airy functions, cylinder functions, 0 F 1
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)