Linear difference equations with a transition point at the origin

Li Hua Cao, Yutian LI*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation Pn+1(x) ? (Anx + Bn)Pn(x) + Pn?1(x) = 0, where An and Bn have asymptotic expansions of the form An ∼ n-o ∞ θ s=o αs/ns, Bn ∼ ∼ n-o ∞ θ s=o βs/ns with θ ≠ 0 and A0 ≠ 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)x and τ0 is a small shift. In particular, it is shown how the Bessel functions Jν and Yν 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 Aexp(-qmxm), x > 0, where m is a positive integer, A > -1 and qm > 0.

Original languageEnglish
Pages (from-to)75-106
Number of pages32
JournalAnalysis and Applications
Volume12
Issue number1
DOIs
Publication statusPublished - Jan 2014

Scopus Subject Areas

  • Analysis
  • Applied Mathematics

User-Defined Keywords

  • Bessel functions
  • Difference equation
  • Orthogonal polynomials
  • Three-term recurrence relations
  • Transition point
  • Uniform asymptotic expansions

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