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Some properties of polynomials orthogonal over the set 〈1, 2, ...N

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Summary

Using identities being discrete counterparts of those which are statisfied by the Legendre polynomials, the author proves that if the polynomials 〈Ψ m (N) (t)〉 (m=0, 1, ..., N−1) form an orthogonal set over the set 〈1, 2, ..., N〉 with equal weight attached to its elements, then

$$\left| {\Psi _m^{(N)} (t)} \right|< \left| {\Psi _m^{(N)} (1)} \right| = \left| {\Psi _m^{(N)} (N)} \right| (t = 2,3,...,N - 1)$$

when m(m+1)⩽N−1. This result is then extended to a wide class of Hahn polynomials.

References

  1. [1]

    R. Askey -G. Gasper,Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients, Proc. Camb. Phil. Soc.,70 (1971), pp. 243–255.

  2. [2]

    G. Gasper,Projection formulas for orthogonal polynomials of a discrete variable, J. Math. Anal. Appl.,45 (1974), pp. 176–198.

  3. [3]

    E. Plesczyńska,Further applications of the T* test to time series analysis, Zastos. Mat.,12 (1971), pp. 33–61.

  4. [4]

    E. Pleszczyńska,Trend estimation problems in time series analysis, Dissertationes Math., vol.104 (1973).

  5. [5]

    J. Shohat,Théorie générale des polynômes orthogonaux de Tchebichef, Mém. Sci. Math., vol.68, Paris (1934).

  6. [6]

    G. Szegö,Orthogonal Polynomials, Amer. Math. Soc. Colloquium Pub., vol.23, 2nd ed., New York (1959).

  7. [7]

    E. T. Whittaker - G. N. Watson,A Course of Modern Analysis, 4th ed., Cambridge University Press (1927).

  8. [8]

    M. W. Wilson,On the Hahn polynomials, SIAM J. Math. Anal.,1 (1970), pp. 131–139.

  9. [9]

    M. W. Wilson,Convergence properties of discrete analogs of orthogonal polynomials, Computing,5 (1970), pp. 1–5.

  10. [10]

    S. K. Zaremba,Tests for the presence of trends in linear processes, Dissertationes Math., vol.94 (1972).

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Entrata in Redazione il 16 ottobre 1973.

Sponsored by the United States Army under Contract no. DA-31-124-ARO-D-462.

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Zaremba, S.K. Some properties of polynomials orthogonal over the set 〈1, 2, ...N〉. Annali di Matematica 105, 333–345 (1975) doi:10.1007/BF02414937

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Keywords

  • Equal Weight
  • Wide Class
  • Legendre Polynomial
  • Discrete Counterpart
  • Hahn Polynomial