This licentiate consists of two papers treating polynomial sequences defined by linear recurrences. In paper I, we establish necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence {P_i} generated by a three-term recurrence relation P_i(x)+ Q_1(x)P_{i-1}(x) +Q_2(x) P_{i-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. In paper II, we study the root distribution of a sequence of polynomials {P_n(z)} with the rational generating function \sum_{n=0}^{\infty} P_n(z)t^n= \frac{1}{1+ B(z)t^\ell +A(z)t^k} for (k,\ell)=(3,2) and (4,3) where A(z) and B(z) are arbitrary polynomials in z with complex coefficients. We show that the roots of P_n(z) which satisfy A(z)B(z)\neq 0 lie on a real algebraic curve which we describe explicitly.

We examine results concerning the generation of primes in certain types of integer sequences. The sequences discussed all have a connection in that each satisfies a recurrence relation. Mathematicians have speculated over many centuries that these sequences contain an infinite number of prime terms, however no proof has been given as such. We examine a less direct method of showing an infinitude of primes in each sequence by showing that the sequences contain an infinite number of terms with primitive divisors.