Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations

Classical 2-orthogonal polynomials and differential equations

We construct the linear differential equations of third order satisfied by the classical 2orthogonal polynomials. We show that these differential equations have the following form: R4,n(x)P (3) n+3(x)+R3,n(x)P ′′ n+3(x)+R2,n(x)P ′ n+3(x)+R1,n(x)Pn+3(x)=0, where the coefficients {Rk,n(x)}k=1,4 are polynomials whose degrees are, respectively, less than or equal to 4, 3, 2, and 1. We also show tha...

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...

Self-adjoint differential equations for classical orthogonal polynomials

This paper deals with spectral type differential equations of the self-adjoint differential operator, 2r order: L(2r)[Y ](x)= 1 (x) d dxr ( (x) r (x) dY (x) dxr ) = rnY (x). If (x) is the weight function and (x) is a second degree polynomial function, then the corresponding classical orthogonal polynomials, {Qn(x)}∞n=0, are shown to satisfy this differential equation when rn is given by rn = r−...

Connection Coefficients of Orthogonal Polynomials with Applications to Classical Orthogonal Polynomials

New criteria for nonnegativity of connection coefficients between to systems of orthogonal polynomials are given. The results apply to classical orthogonal polynomials.