Divided difference operators and classical orthogonal polynomials

Skew Divided Difference Operators and Schubert Polynomials

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with posit...

Differential and Difference Equations for Products of Classical Orthogonal Polynomials

Difference equations for discrete classical multiple orthogonal polynomials

For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a (r + 1)th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal p...

Q-classical Orthogonal Polynomials: a General Difference Calculus Approach

It is well known that the classical families of orthogonal polynomials are characterized as the polynomial eigenfunctions of a second order homogenous linear differential/difference hypergeometric operator with polynomial coefficients. In this paper we present a study of the classical orthogonal polynomials sequences, in short classical OPS, in a more general framework by using the differential...

Classical Orthogonal Polynomials: a General Difference Calculus Approach

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator with polynomial coefficients. In this paper we present a study of classical orthogonal polynomials in a more general framework by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified...