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