A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equationwhere A n and B n have asymptotic expansions of the formwith θ = 0 and α 0 = 0 being real numbers, and β 0 = ±2. Our result hold uniformly for the scaled variable t in an infinite interval containing the transition point t 1 = 0, where t = (n + τ 0 ) −θ x and τ 0 is a small shift. In particular, it is shown how the Bessel functions J ν and Y ν get involved in the uniform asymptotic expansions of the solutions to the above threeterm recurrence relation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight x α exp(−q m x m ), x > 0, where m is a positive integer, α > −1 and q m > 0.

Asymptotic approximations for the continuous Hahn polynomials and their zeros as the degree grows to infinity are established via their three-term recurrence relation. The methods are based on the uniform asymptotic expansions for difference equations developed by Wang and Wong (Numer. Math., 2003) and the matching technique in the complex plane developed by Wang (J. Approx. Theory, 2014).

In this paper, we study the asymptotic behavior of the Wilson polynomials [Formula: see text] as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable [Formula: see text] is fixed and (ii) when the variable is rescaled as [Formula: see text] with [Formula: see text]. Case (ii) has two subcases, namely, (a) zero-free zone ([Formula: see text]) and (b) oscillatory region [Formula: see text]. Corresponding results are also obtained in these cases (iii) when [Formula: see text] lies in a neighborhood of the transition point [Formula: see text], and (iv) when [Formula: see text] is in the neighborhood of the transition point [Formula: see text]. The expansions in the last two cases hold uniformly in [Formula: see text]. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations.

Two new anthraquinone glycosides, named 1-methyl-8-hydroxyl-9,10-anthraquinone-3-O-β-D-(6'-O-cinnamoyl)glucopyranoside (1) and rhein-8-O-β-D-[6'-O-(3''-methoxyl malonyl)]glucopyranoside (2), have been isolated from the roots of Rheum palmatum, together with seven known compounds, rhein-8-O-β-D-glucopyranoside (3), physcion-8-O-β-D-glucopyranoside (4), chrysophanol-8-O-β-D-glucopyranoside (5), aleo-emodin-8-O-β-D-glucopyranoside (6), emodin-8-O-β-D-glucopyranoside (7), aleo-emodin-ω-O-β-D-glucopyranoside (8), and emodin-1-O-β-D-glucopyranoside (9). Their structures were elucidated on the basis of chemical and spectral analysis.

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