Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras

Abstract

Kravchuk polynomials arise as orthogonal polynomials with respect to the binomial distribution and have numerous applications in harmonic analysis, statistics, coding theory, and quantum probability. The relationship between Kravchuk polynomials and Clifford algebras is multifaceted. In this paper, Kravchuk polynomials are discovered as traces of conjugation operators in Clifford algebras, and appear in Clifford Berezin integrals of Clifford polynomials. Regarding Kravchuk matrices as linear operators on a vector space $V$, the action induced on the Clifford algebra over $V$ is equivalent to blade conjugation, i.e., reflections across sets of orthogonal hyperplanes. Such operators also have a natural interpretation in terms of raising and lowering operators on the algebra. On the other hand, beginning with particular linear operators on the Clifford algebra $\mathcal {C}\ell _Q(V)$, one obtains Kravchuk matrices as operators on the paravector space $V_*$ through a process of operator grade-reduction. Symmetric Kravchuk matrices are recovered as representations of grade-reductions of maps induced by negative-definite quadratic forms on $V$.