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Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ areequivalent to weighted regular $t$-wise balanced designs, which generalizecombinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes aswell as weights. Partly motivated by the recent study on tight Euclidean$t$-designs on two concentric spheres, in this paper we discuss tight relative$t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mildcondition that such a relative $t$-design induces the structure of a coherentconfiguration with two fibers. Moreover, from this structure we deduce that apolynomial from the family of the Hahn hypergeometric orthogonal polynomialsmust have only integral simple zeros. The Terwilliger algebra is the main toolto establish these results. By explicitly evaluating the behavior of the zerosof the Hahn polynomials when they degenerate to the Hermite polynomials underan appropriate limit process, we prove a theorem which gives a partial evidencethat the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported ontwo shells are rare for large $t$.

Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials