Stieltjes’ theorem for classical discrete orthogonal polynomials

We write “classical orthogonal polynomials on the real line” rather than simply “classical orthogonal polynomials” because, for instance, from the algebraic point of view of Maroni (see Ref. 9), the Jacobi polynomials exist and are “classical” even when $−α,−β,−α−β+1∉N$ (see Refs. 17, Chaps. 8 and 9, for a recent survey on the subject). Moreover, the Bessel polynomials are classical in the same sense as the other three systems. As Maroni says, “comme dans le roman d’Alexandre Dumas, les trois mousquetaires étaient quatre en réalité.”

For a “continuous” case, as the Askey–Wilson polynomials, Askey and Wilson used a consequence of Markov’s theorem, which goes back to Szegő (see Ref. 22, Theorem 6.12.2), to study the monotonicity of zeros of these polynomials (see Ref. 2, Sec. 7).

In first edition of “Special Functions of Mathematical Physics” (see Ref. 13), the author only consider the case x(s) = s. The second edition (see Ref. 15) was significantly enriched with the Eq. (1.4); although A. A. Samarskii’s preface is the same in both editions.

If a and b are finite, (1.7) can be written in the form ω(a)a(a) = 0 and ω(b)a(b) = 0. If a = −∞ and/or b = ∞, then (1.7) must hold for each k in the limiting sense.

Recall that for a continuous case on the grid (VI) as the Askey–Wilson polynomials, the monotonicity of their zeros was studied in Ref. 2, Sec. 7.