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

Relative t-designs in the n-dimensional hypercube Qn are equivalent to weighted regular t-wise balanced designs, which generalize combinatorial t-(n, k, ?) designs by allowing multiple block sizes as well weights. Partly motivated recent study on tight Euclidean two concentric spheres, this paper we discuss relative supported shells. We show under a mild condition that such t-design induces structure of coherent configuration with fibers. Moreover, from deduce polynomial family Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is main tool establish these results. By explicitly evaluating behavior zeros when they degenerate Hermite an appropriate limit process, prove theorem gives partial evidence non-trivial shells rare for large t.