Binomial Polynomials mimicking Riemann's Zeta Function

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 ${}_3{F}_2(1)$ 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 $s\to \mathrm{\infty }$ along the positive real axis, $q_n(s)\to 1$ from below, mimicking $1\mathrm{/}\zeta (s)$ 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.