Binomial-Weighted Orthogonal Polynomials

ABSTRACT

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.

REFERENCES

• 1: FORSYTHE, G .E . Generation and use of orthogonal polynomials for data-fittlng with a digital computer. J. Soc. Ind. Appl. Math. 5, 2 (June 1957), 74-88.
• 2: Richard W Hamming, Numerical methods for scientists and engineers (2nd ed.), Dover Publications, Inc., New York, NY, 1986
• 3: Begnaud Francis Hildebrand, Introduction to numerical analysis: 2nd edition, Dover Publications, Inc., New York, NY, 1987
• 4: Paul W. Broome, Discrete Orthonormal Sequences, Journal of the ACM (JACM), v.12 n.2, p.151-168, April 1965 [doi>10.1145/321264.321265]
• 5: MILNE, W.E. Numerical Calculus. Princeton U. Press, Princeton, N. J., 1949.
• 6: FELLER, W. An Introduction to Probability Theory and Its Applications. Wiley, New York, 1957.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
          Subjects: Polynomials, methods for

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on polynomials

General Terms:
Measurement, Performance, Theory