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

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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.


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