Riv.Mat.Univ.Parma (6) 1 (1998)


On a bound for the zeros of polynomials defined by special linear recurrences of order K

Pages: 173-180
Received: 14 October 1998   
Mathematics Subject Classification (2000): 11B39 - 12D10

Abstract: Let k³2 be an integer, while let Gj(x)=aj \in C (2-k£j£0) and px+q, G1(x)=rx+s be given polynomials of x with complex coefficients, where

pr¹0. For n³2 the sequence {Gn(x)} is defined by the following recursion of order k
Gn(x)=(px+q)Gn-1(x)+eGn-k(x), where e=1 or e=-1.
We prove that the absolute values of the zeros of polynomials Gn(x)(n³1) have a common upper bound, which depends only on aj (2-k £j£0), p,q,r and s. Namely, if Gn(z)=0 for a z\in C with some n³1 then
|z|£{1\over{|pr|}}(max(|ps-rq|+|p| å0j=2-k|aj|, 2|r|)+|rq|).
This result extends and generalizes some earlier results presented in [5],[6] [7] for the case k=2.

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