Interlacing properties and the Schur-Szegő composition

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Each degree n polynomial in one variable of the form (x+1)(xⁿ⁻¹+c₁xⁿ⁻²+ +cₙ₋₁) is representable in a unique way as a Schur-Szegő composition of n-1 polynomials of the form (x+1)ⁿ⁻¹(x+aᵢ), see Ko1, AlKo and Ko2. Set σⱼ:=∑ 1≤ i₁< <iⱼ≤ n-1aᵢ₁ aᵢⱼ. The eigenvalues of the affine mapping (c₁,,cₙ₋₁) (σ₁,,σₙ₋₁) are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the roots of these polynomials.