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Papers by Lance Littlejohn

We reconsider the problem of classifying all classical orthogo- nal polynomial sequences which ar... more We reconsider the problem of classifying all classical orthogo- nal polynomial sequences which are solutions to a second-order differential equation of the form '2.x/y 00 .x/C'1.x/y 0 .x/D n y.x/: We first obtain new (algebraic) necessary and sufficient conditions on the coef- ficients '1.x/ and'2.x/ for the above differential equation to have orthogonal polynomial solutions. Using this result, we then

European Journal of Combinatorics, 2015

For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of ... more 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.

We provide the mathematical foundation for the XmX_mXm-Jacobi spectral theory. Namely, we present a... more We provide the mathematical foundation for the XmX_mXm-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional XmX_mXm-Jacobi orthogonal polynomials as eigenfunctions. This proves that those polynomials are indeed eigenfunctions of the self-adjoint operator (rather than just formal eigenfunctions). Further, we prove the completeness of the exceptional XmX_mXm-Jacobi orthogonal polynomials (of degrees m,m+1,m+2,...m, m+1, m+2, ...m,m+1,m+2,...) in the Lebesgue--Hilbert space with the appropriate weight. In particular, the self-adjoint operator has no other spectrum.

Special Functions, 2000

ABSTRACT Bochner-Krall orthogonal polynomials are orthogonal polynomials which arise as eigenfunc... more ABSTRACT Bochner-Krall orthogonal polynomials are orthogonal polynomials which arise as eigenfunctions of a differential equation of spectral type with polynomial coefficients: L N [y](x)=∑ i=1 N l i (x)y (i) (x)=λ n y(x)· We discuss recent progress on Bochner-Krall orthogonal polynomials including Magnus’ conjecture

Differential and Integral Equations

Journal of Mathematical Analysis and Applications, 2015

The exceptional X1-Jacobi differential expression is a second-order ordinary differential express... more The exceptional X1-Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by Gómez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions P (α,β) n ∞ n=1 called the exceptional X1-Jacobi polynomials. There is no exceptional X1-Jacobi polynomial of degree zero. These polynomials form a complete orthogonal set in the weighted Hilbert space L 2 ((−1, 1); w α,β ), where w 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 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

$Research paper thumbnail of Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math>\alpha =\beta =-1$$$

Annali di Matematica Pura ed Applicata, 2012

In this paper, we consider the second-order differential expression

Linear Algebra and its Applications

In an earlier paper, Kwon, Littlejohn and Yoon characterized symmetric Sobolev bilinear forms and... more In an earlier paper, Kwon, Littlejohn and Yoon characterized symmetric Sobolev bilinear forms and showed that they have, like symmetric matrices, a diagonal representation. In this paper, we present a new proof of one of their main results by interpreting the coefficients in the diagonal representation of a Sobolev-type bilinear form from a combinatorial point of view. We view this as an improvement over the original proof which relied on mathematical induction.

The Two-Year College Mathematics Journal, 1980

Transactions of the American Mathematical Society, 1995

... inner product 1 f°° (M) re -tn / P(x)q(x)xae~xdx + Mp(0)q(0) + Np'(0)q'(0), I(a+ 1)... more ... inner product 1 f°° (M) re -tn / P(x)q(x)xae~xdx + Mp(0)q(0) + Np'(0)q'(0), I(a+ 1) Jo where a > -1, M > 0, and N > 0. On the other hand, R. Koekoek [3, 4 ... r-k+\ Rik-x(o, r) - Y cJP(2k + 27 - 1, 2; + 1 )B§¡$(o, x) 7=0 r-k+i = (U2k-X(T) - T2k-X(x') - S2k-X(o)) - Y CjP(2k + 2j-\,2j+\) 7=0 x(U^(x ...

Studies in Applied Mathematics, 2014

For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic for... more For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the classical Stirling numbers of the second kind. Thereby a supplement of the asymptotic analysis for these numbers is established.