Jacobi–Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression

Abstract

We develop the left-definite analysis associated with the self-adjoint Jacobi operator A ( , ) k , generated from the classical secondorder Jacobi differential expressionin the Hilbert space L 2 , (−1, 1) := L 2 ((−1, 1); w , (t)), where w , (t) = (1 − t) (1 + t) , that has the Jacobi polynomials {P ( , ) m } ∞ m=0 as eigenfunctions; here, , > − 1 and k is a fixed, non-negative constant. More specifically, for each n ∈ N, we explicitly determine the unique left-definite Hilbert-Sobolev space W ( , ) n,k (−1, 1) and the corresponding unique left-definite selfadjoint operator B ( , )

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

“…The study required the introduction of the new set of q-Jacobi-Stirling numbers, which we have characterised and provided a combinatorial interpretation. Clearly, by allowing q → 1 we recover all the classical polynomial related results found in [2,7,8,11,26,27,30] from the viewpoint of both the algebra of these operators and the combinatorial interpretations of the connection coefficients.…”

“…The results here obtained are the q-version to those in [26] (and also in [7,8]), as well as to those in [2,6,11,12,30,32], since we provide here combinatorial interpretations to the arisen coefficients and eigenvalues. This study has the merit of addressing all the q-classical polynomial sequences as whole in a coherent framework that brings together generalisations of the q-differential equation (1.2) and associated combinatorial interpretations.…”

“…In fact, as observed in [30,Theorem 1], there exists a polynomial P so that any even order differential operator having classical polynomials as eigenfunctions can be written as P (L). Thus, if O = 2n k=0 a k (x)D k , where a k (x) are polynomials of degree at most k, then a natural question is to state the connection between the polynomials a k (x) and the polynomial coefficients Φ and Ψ in the differential operator L. This question has been addressed in [26] and partly in [7,8]. More precisely, explicit expressions for any integral power of L were obtained for each of the classical polynomial families, orthogonal with respect to a positivedefinite measure in a series of papers by Everitt et al (see [7,8] and the references therein).…”

A combinatorial interpretation of the Legendre-Stirling numbers

“…However, as the authors show in [9] (see also [3], [6], [7], and [8]), it is possible to compute these spaces and inner products for several well-known self-adjoint operators for each positive integer r.…”

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

“…In this case, the nth degree Jacobi polynomial y = P α,β n (x) is a solution of the equation l α,β [y](x) = (n(n + α + β + 1) + k) y(x) (n ∈ N 0 ); details of the properties of these polynomials can be found in [7,16]. The rightdefinite spectral analysis has been studied in [1,9]. Through the Glazman-Krein-Naimark (GKN) theory it has been known that there exists a self-adjoint operator A (α,β) generated from the Jacobi differential expression in the Hilbert space L 2 ((−1, 1); (1−x) α (1+x) β ) having the Jacobi polynomials as a complete set of eigenfunctions.…”