Extensions of discrete classical orthogonal polynomials beyond the orthogonality

Abstract

MSC: 33C45 42C05 34B24Keywords: Classical orthogonal polynomials Inner product involving difference operators Non-standard orthogonality a b s t r a c t It is well-known that the family of Hahn polynomials {h α,β n (x; N)} n≥0 is orthogonal with respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can be characterized by a ∆-Sobolev orthogonality for every n and present a factorization for Hahn polynomials for a degree higher than N.We also present analogous results for dual Hahn, Krawtchouk, and Racah polynomials and give the limit relations among them for all n ∈ N 0 . Furthermore, in order to get these results for the Krawtchouk polynomials we will obtain a more general property of orthogonality for Meixner polynomials.

Orthogonality of q-polynomials for non-standard parameters

“…Notice that, although Theorem 2.2 seems to be new, it has been used implicitly in [9] where the operator T k is the forward difference operator or another one related with , and it is applied to Racah, Hahn, dual Hahn, and Krawtchouk polynomials. In fact, this result can be applied to the Laguerre polynomials L (−N ) n , Gegenbauer polynomials C (−N +1/2) n , and Jacobi polynomials P (α,β) n with one parameter being a negative integer, by taking the operator T k as the derivative operator.…”

“…In fact, the part F t AG in (2) is obtained as the matrix form of (5) for ( p n ) N n=0 with an adequate change of basis, and as a consequence this part includes non-explicit terms. On the other hand, the measure µ d in (4) is totally explicit for the families studied in [9].…”

“…The aim of this paper is to develop the ideas from [9] in order to obtain a general tool which provides a non-standard orthogonality property of the form (4) for any sequence of classical orthogonal polynomials with some vanishing coefficient γ n . The tool is a degenerate version of the Favard theorem, developed in Section 2, which is also valid when there exist more than one index n so that γ n = 0 and even when the family of orthogonal polynomials is not classical.…”

On the zeros of Meixner polynomials

“…The asymptotic distribution of the zeros of M n (x; β, c), β > 0, c < 0 as n → ∞, (after the necessary rescaling) could possibly be proved using the complex orthogonality (cf. [12]) and potential theoretical methods or a Riemann-Hilbert approach, as was done in the case of Jacobi polynomials for non-standard parameters (cf. [27], [29]).…”

“…[24, (1.7.2)]) and therefore the zeros of the polynomials {M n (x; β, c)} ∞ n=0 are distinct and in (−∞, −β). A non-Hermitian orthogonality with respect to a complex weight function for Meixner polynomials M n (x; β, c) when β, c ∈ C, c / ∈ [0, ∞) and β / ∈ Z − is discussed in [12,Proposition 9]. The standard orthogonality for a finite number of Meixner polynomials M n (x; β, c) when c < 0 and β is equal to a negative integer, say β = −N , N ∈ N is that of the Krawtchouk polynomials defined by (cf.…”