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LINEAR DIFFERENCE EQUATIONS WITH A TRANSITION POINT AT THE ORIGIN

Department of Mathematics, Shenzhen University, Guangdong, P. R. China

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

and

Institute of Computational and Theoretical Studies and Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong

Corresponding author.

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https://doi.org/10.1142/S0219530513500371Cited by:9 (Source: Crossref)

Abstract

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation

where A_n and B_n have asymptotic expansions of the form

with θ ≠ 0 and α₀ ≠ 0 being real numbers, and β₀ = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t₁ = 0, where t = (n + τ₀)^-θx and τ₀ 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 linear difference equation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight x^αexp(-q_mx^m), x > 0, where m is a positive integer, α > -1 and q_m > 0.

Keywords:

AMSC: 41A60, 39A10, 33C45

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