J. Austral. Math. Soc.  72 (2002), 109-117
The zeros of linear combinations of translates of polynomials

Peter Walker
  College of Arts and Science
  American University of Sharjah
  P.O. 26666 Sharjah
  United Arab Emirates
  peterw@aus.ac.ae


Abstract
We investigate the location and separation of zeros of certain three-term linear combination of translates of polynomials. In particular, we find an interval of the form  I = [ - 1, 1 + h ],  h > 0 such that for a polynomial  f, all of whose zeros are real, and $\lambda \in I$, all zeros of $f(x+2ic)+2\lambda f(x)+f(x-2ic)$ are also real.
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