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

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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