Lance L. Littlejohn

A combinatorial interpretation of the Legendre-Stirling numbers

The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical second-order Legendre differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown by Littlejohn and Wellman, are the coefficients of integral powers of the Laguerre differential expression. An open question regarding the Legendre-Stirling numbers has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.

The Legendre-Stirling numbers

The Legendre-Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre-Stirling numbers. In this paper, we establish several properties of the Legendre-Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.

The Jacobi-Stirling numbers

The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations, thereby extending and supplementing known recent contributions to the literature.

On the asymptotic normality of the Legendre-Stirling numbers of the second kind

For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.

The spectral analysis of three families of exceptional Laguerre polynomials

The Bochner Classification Theorem (1929) characterizes the polynomial sequences { p n } n = 0 ∞ , with deg p n = n that simultaneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders. Indeed, up to a complex linear change of variable, only the classical Hermite, Laguerre, and Jacobi polynomials, with certain restrictions on the polynomial parameters, satisfy these conditions. In 2009, Gomez-Ullate, Kamran, and Milson found that for sequences { p n } n = 1 ∞ , deg p n = n (without the constant polynomial), the only such sequences satisfying these conditions are the exceptional X 1 -Laguerre and X 1 -Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials { p n } n ? N 0 ? A , where A is a finite subset of the non-negative integers N 0 and where deg p n = n for all n ? N 0 ? A . We call such a sequence an exceptional polynomial sequence of codimension | A | , where the latter denotes the cardinality of A . All exceptional sequences with a non singular weight, found to date, have the remarkable feature that they form a complete orthogonal set in their natural Hilbert space setting.Among the exceptional sets already known are two types of exceptional Laguerre polynomials, called the Type I and Type II exceptional Laguerre polynomials, each omitting m polynomials. In this paper, we briefly discuss these polynomials and construct the self-adjoint operators generated by their corresponding second-order differential expressions in the appropriate Hilbert spaces. In addition, we present a novel derivation of the Type III family of exceptional Laguerre polynomials along with a detailed disquisition of its properties. We include several representations of these polynomials, orthogonality, norms, completeness, the location of their local extrema and roots, root asymptotics, as well as a complete spectral study of the second-order Type III exceptional Laguerre differential expression.

A spectral study of the second-order exceptional X1-Jacobi differential expression and a related non-classical Jacobi differential expression

Abstract The exceptional X 1 -Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by Gomez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions { P ˆ n ( α , β ) } n = 1 ∞ called the exceptional X 1 -Jacobi polynomials. There is no exceptional X 1 -Jacobi polynomial of degree zero. These polynomials form a complete orthogonal set in the weighted Hilbert space L 2 ( ( − 1 , 1 ) ; w ˆ α , β ) , where w ˆ α , β is a positive rational weight function related to the classical Jacobi weight. Among other conditions placed on the parameters α and β, it is required that α , β > 0 . In this paper, we develop the spectral theory of this expression in L 2 ( ( − 1 , 1 ) ; w ˆ α , β ) . We also consider the spectral analysis of the ‘extreme’ non-exceptional case, namely when α = 0 . In this case, the polynomial solutions are the non-classical Jacobi polynomials { P n ( − 2 , β ) } n = 2 ∞ . We study the corresponding Jacobi differential expression in several Hilbert spaces, including their natural L 2 setting and a certain Sobolev space S where the full sequence { P n ( − 2 , β ) } n = 0 ∞ is studied and a careful spectral analysis of the Jacobi expression is carried out.

Differential operator for discrete Gegenbauer–Sobolev orthogonal polynomials: Eigenvalues and asymptotics

We consider the following discrete Sobolev inner product involving the Gegenbauer weight

On the classification of differential equations having orthogonal polynomial solutions

(f,g)_S:=\int_{-1}^1f(x)g(x)(1-x^2)^{\alpha}dx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big],