  The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer
functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial
factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on
the real axis (called critical polynomials). The transforms are identified in
terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2 and S:3/1
binomial coefficient forms. Their `critical polynomial' factors are then
identified as variants of the S:4/1 form, and more compactly in terms of
certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend these
results to a $1$-parameter family of polynomials with zeros only on the
critical line. These polynomials possess the functional equation
$p_n(s;\beta)=\pm p_n(1-s;\beta)$, similar to that for the Riemann xi function.
  It is shown that via manipulation of the binomial factors, these `critical
polynomials' can be simplified to an S:3/2 form, which after normalisation
yields the rational function $q_n(s).$ The denominator of the rational form has
singularities on the negative real axis, and so $q_n(s)$ has the same `critical
zeros' as the `critical polynomial' $p_n(s)$. Moreover as $s\rightarrow \infty$
along the positive real axis, $q_n(s)\rightarrow 1$ from below, mimicking
$1/\zeta(s)$ on the positive real line.
  In the case of the Chebyshev parameters we deduce the simpler S:2/1 binomial
form, and with $\mathcal{C}_n$ the $n$th Catalan number, $s$ an integer, we
show that polynomials $4\mathcal{C}_{n-1}p_{2n}(s)$ and
$\mathcal{C}_{n}p_{2n+1}(s)$ yield integers with only odd prime factors. The
results touch on analytic number theory, special function theory, and
combinatorics.
