The zeros of linear combinations of translates of polynomials

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 λ ∈ I, all zeros of f (x + 2ic) + 2λf (x) + f (x – 2ic) are also real.

Sturm's Theorem for Equations with Delayed Argument

Sturm's type theorems on separation of zeros of solutions are proved for the second order linear differential equations with delayed argument.