** 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 *k*³2 be an integer,
while let *G*_{j}(x)=a_{j} \in
**C **(2-*k*£*j*£0)
and *px+q, G*_{1}*(x)=rx+s* be given polynomials of *x*
with complex coefficients, where

*pr¹*0. For *n*³2
the sequence {*G*_{n}(x)} is defined by the following recursion
of order *k*
*G*_{n}(x)=(px+q)G_{n-}_{1}*(x)+eG*_{n-k}(x),
where *e=*1 or *e=*-1.
We prove that the absolute values of the zeros of
polynomials *G*_{n}(x)(n³1)
have a common upper bound, which depends
only on *a*_{j} (2-*k*
£j£*0),
p,q,r *and* s. *Namely, if *G*_{n}(z)=0 for a* z*\in **C**
with some *n*³1 then
*|z|*£{1\over{|*pr*|}}(max(*|ps-rq|+|p|
*å^{0}_{j=2-k}|a_{j}|,
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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