Difference Equations and Discriminants for Discrete Orthogonal Polynomials

Abstract

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.

Haydock’s recursive solution of self-adjoint problems. Discrete spectrum

“…where the coefficients A n (x) and B n (x) may in general be nonpolynomial functions [35][36][37][38], and D x is a divided-difference operator (discrete derivative) that maps Π n [x], the linear space of polynomials in x over C with degree at most n ∈ Z ≥0 , into Π n−1 [x] [35][36][37]. Note in passing that D x p n (x) being a polynomial of degree n − 1 can in general be only represented as (cf.…”

“…The other one results by expressing p n−1 from the fundamental TTRR (6) and substituting it back into the original structure relation (14). The resulting pair of structure relations (i) leads directly to a pair of mutually adjoint raising and lowering ladder operators [38], (ii) implies that orthogonal polynomials satisfy in general a second-order difference equation (cf. Sec.…”

“…4 of Ref. [38]), and (iii) allows one to introduce a discrete analogue of the Bethe Ansatz equations (cf. Sec.…”

“…[37]), and any OPS orthogonal with respect to a discrete measure supported on equidistant points (Theorem 1.1 of Ref. [38]). [In the semi-classical case, the function E(x) itself satisfies a first order difference equation with polynomial coefficients (Theorem 1 of Ref.…”

On zeros of discrete orthogonal polynomials

“…Although we deduce our results from a general statement we will deal mainly with special functions rather than with orthogonal polynomials per se. Nevertheless it is worth noticing a recent result stating that polynomials orthogonal on a finite or half-infinite set of consecutive integers satisfy a second order linear difference equation [2,3].…”