Numerical methods for scientists and engineers (2nd ed.)
Numerical methods for scientists and engineers (2nd ed.)
Introduction to numerical analysis: 2nd edition
Introduction to numerical analysis: 2nd edition
Discrete Orthonormal Sequences
Journal of the ACM (JACM)
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This paper discusses a set of polynomials, {&phgr;r(s)}, orthogonal over a discrete range, with binomial distribution, b(s; n, p), as the weighting function. Two recurrence relations are derived. One expresses &phgr;r in terms of &phgr;r-1 and &Dgr;&phgr;r-1, while the other relates &phgr;r with &phgr;r-1 and &phgr;r-2. 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.