This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence {P-i}(i=1)(infinity) generated by a three-term recurrence relation P-i(x) + Q(1)(x)P-i (1)(x) + Q(2)(x)Pi-2(x) = 0 with the standard initial conditions P-0(x) = 1, P-1(x) = 0, where Q(1)(x) and Q(2)(x) are arbitrary real polynomials.

This paper discusses the location of zeros of polynomials in a polynomial sequence {P_n(z)} generated by a three-term recurrence relation of the form P_n(z)+B(z)P_{n−1}(z)+A(z)P_{n−k}(z)=0 with k>2 and the standard initial conditions P_0(z)=1,P_{−1}(z)=…=P_{−k+1}(z)=0, where A(z) and B(z) are arbitrary coprime real polynomials. We show that there always exist polynomials in {Pn(z)} with non-real zeros.

A conjecture of Khang Tran  claims that for an arbitrary pair of polynomials A(z) and B(z), every zero of every polynomial in the sequence {P_n(z)} satisfying the three-term recurrence relation of length k

P_n(z) + B(z)P_{n−1}(z) + A(z)P_{n−k}(z) = 0

with the standard initial conditions P_0(z) = 1, P_{−1}(z) = · · · = P_{−k+1}(z) = 0 which is not a zero of A(z) lies on the real (semi)-algebraic curve C  given by

Im(( B^k(z)/ A(z)) = 0 and 0 ≤ (−1)^k ≤ Re(( B^k(z)/ A(z)) ≤ k^k (k − 1)^{k−1}. In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by

P_n(z) + B(z)P_{n−l}(z) + A(z)P_{n−k}(z) = 0, with coprime k and l and the same standard initial conditions as above, every root of P_n(z) which is not a zero of A(z)B(z) belongs to the real algebraic curve C_{l,k} given by

Im(( B^k(z)/ A(z)) = 0.

Below we establish the conditions guaranteeing the reality of all the zeros of polynomials P_n(z) in the polynomial sequence {P_n(z)} satisfying a five-term recurrence relation

P_n(z) = zP_{n−1}(z) + αP_{n−2}(z) + βP_{n−3}(z) + γP_{n−4}(z),

with the standard initial conditions P_0(z) = 1, P_{−1}(z) = P_{−2}(z) = P_{−3}(z) = 0, where α, β, γ are real coefficients, γ ≠ 0 and z is a complex variable. We interprete this sequence of polynomials as principal minors of an appropriate banded Teoplitz matrix whose associated Laurent polynomial b(z) is holomorphic in C \ {0}. We show that when either all the critical points in the complex plane of b(z) are real; or only two are real together with a pair of complex conjugate critical points and some extra conditions on the parameters, the set b^{−1}(R) contains a Jordan curve with 0 in its interior and in some cases a nonsimple curve enclosing 0. The presence of the said curves is necessary and sufficient for every polynomial in the sequence {P_n(z)} to be hyperbolic.