## Abstract

We consider the sequence of polynomials W_{n}(x) defined by the recursion W_{n}(x)=(ax+b)W_{n-1}(x)+dW_{n-2}(x), with initial values W_{0}(x)=1 and W_{1}(x)=t(x-r), where a, b, d, t, r are real numbers, with a, t>0 and d<0. It is known that every polynomial W_{n}(x) is distinct-real-rooted. We find that, as n→∞, the smallest root of the polynomial W_{n}(x) converges decreasingly to a real number, and that the largest root converges increasingly to a real number. Moreover, by using the Dirichlet approximation theorem, we prove that for every integer j≥2, the jth smallest root of the polynomial W_{n}(x) converges as n→∞, and so does the jth largest root. It turns out that these two convergence points are independent of the numbers t, r, and the integer j. We obtain explicit expressions for the above four limit points.

Original language | American English |
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Pages (from-to) | 499-528 |

Number of pages | 30 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 441 |

Issue number | 2 |

DOIs | |

State | Published - 15 Sep 2016 |

## Keywords

- Dirichlet's approximation theorem
- Real-rooted polynomial
- Recurrence
- Root geometry

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics