Asymptotic expansion of the modified lommel polynomials hn,ν(x) and their zeroszeros

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Author(s)

  • Kei Fung LEE
  • R. Wong

Related Research Unit(s)

Detail(s)

Original languageEnglish
Pages (from-to)3953 - 3964
Journal / PublicationProceedings of the American Mathematical Society
Volume142
Issue number11
Publication statusPublished - Jul 2014

Abstract

The modified Lommel polynomials satisfy the second-order linear difference equation h(n +1),(v)(x) - 2(n + v) x h(n,v)(x)+ h(n-1,v)(x) = 0, n = 0, with initial values h(-1,v)(x) = 0 and h(0,v)(x) = 1, where x is a real variable and v is a fixed positive parameter. An asymptotic expansion, as n - infinity, is derived for these polynomials by using a turning-point theory for three-term recurrence relations developed by Wang and Wong Numer. Math. 91 (2002) and 94 (2003)]. The result holds uniformly in the infinite interval 0 = x infinity, containing the critical value x = 1/N, where N = n + v. Behavior of the zeros of these polynomials is also studied.

Research Area(s)

  • Modified Lommel polynomials, second-order linear difference equations, uniform asymptotic expansions, Airy function