Hahn Polynomials, Discrete Harmonics, and t-Designs
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It is shown that certain Hahn polynomials and their q-analogues play in combinatorics a similar role as Gegenbauer polynomials in real Euclidean geometry. The concept of harmonic function on a fiber of a regular lattice is introduced; then, Hahn polynomials are associated with harmonic spaces of fixed degrees. An algebraic theory of t-designs is presented in this framework, and a linear programming bound is derived for the cardinality of a design.
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