  Let $\epsilon\in \{-1,1\}$. A sequence of prime numbers $p_1, p_2, p_3, ...$,
such that $p_i=2p_{i-1}+\epsilon$ for all $i$, is called a {\it Cunningham
chain} of the first or second kind, depending on whether $\epsilon =1$ or -1
respectively. If $k$ is the smallest positive integer such that $2p_k+\epsilon$
is composite, then we say the chain has length $k$. Although such chains are
necessarily finite, it is conjectured that for every positive integer $k$,
there are infinitely many Cunningham chains of length $k$. A sequence of
polynomials $f_1(x), f_2(x), ...$, such that $f_i(x)\in \Z[x]$, $f_1(x)$ has
positive leading coefficient, $f_i(x)$ is irreducible in $\Q[x]$, and
$f_i(x)=xf_{i-1}(x)+\epsilon$ for all $i$, is defined to be a {\it polynomial
Cunningham chain} of the first or second kind, depending on whether $\epsilon
=1$ or -1 respectively. If $k$ is the least positive integer such that
$f_{k+1}(x)$ is reducible over $\Q$, then we say the chain has length $k$. In
this article, for chains of each kind, we explicitly give infinitely many
polynomials $f_1(x)$, such that $f_{k+1}(x)$ is the only term in the sequence
$\{f_i(x)\}_{i=1}^{\infty}$ that is reducible. As a first corollary, we deduce
that there exist infinitely many polynomial Cunningham chains of length $k$ of
both kinds, and as a second corollary, we have that, unlike the situation in
the integers, there exist infinitely many polynomial Cunningham chains of
infinite length of both kinds.
