New bounds for Hahn and Krawichouk polynomials

Abstract

For the Hahn and Krawtchouk polynomials orthogonal on the set $\{0, \ldots,N\}$ new identities for the sum of squares are derived which generalize the trigonometric identity for the Chebyshev polynomials of the first and second kind. These results are applied in order to obtain conditions (on the degree of the polynomials) such that the polynomials are bounded (on the interval $[0,N]$) by their values at the points $0$ and $N$. As special cases we obtain a discrete analogue of the trigonometric identity and bounds for the discrete Chebyshev polynomials of the first and second kind.