Binomial-Weighted Orthogonal Polynomials

This paper discusses a set of polynomials, {φ<subscrpt><italic>r</italic></subscrpt>(<italic>s</italic>)}, orthogonal over a discrete range, with binomial distribution, <italic>b</italic>(<italic>s</italic>; <italic>n</italic>, <italic>p</italic>), as the weighting function. Two recurrence relations are derived. One expresses φ<subscrpt><italic>r</italic></subscrpt> in terms of φ<subscrpt><italic>r</italic>-1</subscrpt> and Δφ<subscrpt><italic>r</italic>-1</subscrpt>, while the other relates φ<subscrpt><italic>r</italic></subscrpt> with φ<subscrpt><italic>r</italic>-1</subscrpt> and φ<subscrpt><italic>r</italic>-2</subscrpt>. It is shown that these polynomials are solutions of a finite difference equation. Also considered are two special cases. The first is the set of Hermite polynomials derived as a limiting case of the binomial-weighted orthogonal polynomials. The second deals with the Poisson distribution used as the weighting function.