Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points

On Discrete Orthogonal Polynomials of Several Variables

Let V be a set of points in R. Define a linear functional L on the space of polynomials, Lf = ∑ x∈V f(x)ρ(x), where ρ is a nonzero function on V . The structure of discrete orthogonal polynomials of several variables with respect to the bilinear form 〈f, g〉 = L(fg) is studied. For a given V , the subspace of polynomials that will generate orthogonal polynomials on V is identified. One result sh...

Orthogonal polynomials on a bi-lattice

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N + 1 − β. We combine both lattices to obtain the bi-lattice N ∪ (N + 1 − β) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients which satisfy a non-linear system of recurrence equations, which we can identify as a limiting case of an (asymm...

Discrete entropies of orthogonal polynomials

Let pn, n ∈ N, be the nth orthonormal polynomial on R, whose zeros are λ j , j = 1, . . . , n. Then for each j = 1, . . . , n, ~ Ψj def = ( Ψ1j , . . . ,Ψ 2 nj ) with Ψij = p 2 i−1(λ (n) j ) ( n−1 ∑ k=0 pk(λ (n) j ) ) −1 , i = 1, . . . , n, defines a discrete probability distribution. The Shannon entropy of the sequence {pn} is consequently defined as Sn,j def = − n ∑ i=1 Ψij log ( Ψij ) . In t...

On Generating Orthogonal Polynomials for Discrete Measures

Integrable systems on the lattice and orthogonal polynomials of discrete variable

Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.