Sums of squares of Krawtchouk polynomials, Catalan numbers, and some algebras over the Boolean lattice

Writing the values of Krawtchouk polynomials as matrices, we consider weighted partial sums along columns. For the general case, we find an identity that, in the symmetric case yields a formula for such partial sums. Complete sums of squares along columns involve "Super Catalan" numbers. We look as well for particular values (matrix entries) involving the Catalan numbers. Properties considered and developed in this work are applied to calculations of various dimensions that describe the structure of some *-algebras over the Boolean lattice based on inclusion/superset relations expressed algebraically using zeons [zero-square elements].