We prove that any set of polynomials orthogonal with respect to a discrete measure supported on equidistant points contained in a half line satisfy a second order difference equation. We also give a discrete analogue of the discriminant and give a general formula for the discrete discriminant of a discrete orthogonal polynomial. As an application we give explicit evaluations of the discrete discriminants of the Meixner and the Hahn polynomials. A difference analogue of the Bethe Ansatz equations is also mentioned.
It is proven that every sequence from the Askey scheme of hypergeomeric polynomials satisfies differential or difference equations of first order of the form T pn(x) = An(x)p n−1 (x) − Bn(x)pn(x), where T is a linear degree reducing operator, which leeds to the fact that these polynomial sets satisfy relation of the form p n (x) = An(x)p n−1 (x) − Bn(x)pn(x).
In this paper we construct the set of Charlier polynomials, Krawtchouk polynomials and Hahn polynomials starting from their weight functions. We also compute their norms and moments.
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