Zeros of Linear Combinations of Orthogonal Polynomials

Abstract Given a sequence of orthogonal polynomials $$(p_n)_n$$ ( p n ) n with respect to a positive measure in the real line, we study the real zeros of finite combinations of $$K+1$$ K + 1 consecutive orthogonal polynomials of the form $$ q_n(x)=\sum _{j=0}^K\gamma _jp_{n-j}(x),\quad n\ge K, $$ q n ( x ) = ∑ j = 0 K γ j p n - j ( x ) , n ≥ K , where $$\gamma _j$$ γ j , $$j=0,\cdots ,K$$ j = 0 , ⋯ , K , are real numbers with $$\gamma _0=1$$ γ 0 = 1 , $$\gamma _K\not =0$$ γ K ≠ 0 (which do not depend on n ). We prove that for every positive measure $$\mu $$ μ there always exists a sequence of orthogonal polynomials with respect to $$\mu $$ μ such that all the zeros of the polynomial $$q_n$$ q n above are real and simple for $$n\ge n_0$$ n ≥ n 0 , where $$n_0$$ n 0 is a positive integer depending on K and the