Asymptotic behavior of polynomials satisfying three-term recurrence relations

Abstract

Sufficient conditions are given to ensure that lim n→∞B 2 z n+1 =e γz for all zϵC , where the B n ( z) are defined by three-term recurrence relations B −1(z):=0,B 0(z):=1, B n(z):=(1+β nz)B n−1(z) + α nz λB n−2(z), n ⩾ 1 and hence are nth denominators of continued fractions α 1z lambda; 1+β 1z + α 2z λ 1+β 2z + α 3z λ 1+β 3z +⋯, λ=1 or2,0≠α nϵC,β nϵC . Here γ :=(2 − λ) α + β, where α:=lim α n and β:= lim β n . In addition to proving uniform convergence on compact subsets of C , we obtain explicit information on the order of convergence of the sequences {B n( z (n+ 1))} and { p k ( n)} n=1 ∞, where ∑ k=0 d n P k ( n) z k := B n ( z/( n + 1)) Important types of continued fractions subsumed under the above class include regular C-fractions, general T-fractions, and associated continued fractions, all three of which have their approximants in Padé tables. Since J-fractions are essentially equivalent to associated continued fractions, many of our results describe the asymptotic behavior of orthogonal polynomial sequences.