On Legendre numbers

The Legendre-Stirling numbers

The Legendre-Stirling numbers are the coeffi cients 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 Littlejoh...

Delannoy numbers and Legendre polytopes

We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x − 1)/2 in the n-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpr...

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 secondorder 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...

Links We Almost Missed Between Delannoy Numbers and Legendre Polynomials

On Polar Legendre Polynomials

We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuou...