Orthogonal Polynomials and Their Derivatives, II

Semiclassical Multiple Orthogonal Polynomials and the Properties of Jacobi–Bessel Polynomials

“…The study of orthogonal polynomials with respect to semiclassical functionals (2), (7), (5), and (6), started more than a hundred years ago with the work of Laguerre [17]. In spite of its long history and a number of powerful modern results (see, for example [18,24,26]) one cannot say that the theory of semiclassical OP enjoys the same level of development and completeness as the theory of classical polynomials.…”

Complex Path Integral Representation for Semiclassical Linear Functionals

“…For s>0, examples have been given in [2,5,6]; the whole class of semiclassical functionals which are positive definite on the real line is given in [3]. A. P. Magnus in [9] solved the problem for``generic semiclassical'' orthogonal polynomials which correspond to regular solutions of (1) in case (A) provided that the zeros of the polynomial , are distinct.…”

A characterization of some q-orthogonal polynomials

“…(1) They are eigenfunctions of a second order Sturm-Liouville differential equation (2) Their derivatives also from a sequence of orthogonal polynomials (3) They have a Rodrigues type formula. (4) They satisfy a first order differential equation of the following form, π(x)P n (x) = (α n x + β n )P n (x) + γ n P n−1 (x), (1.1) An elementary proof of property 4 was carried out in [1].…”