Difference equations and quantized discriminants for q-orthogonal polynomials

We show that orthogonal polynomials on generalized q-linear grid have raising and lowering operators and satisfy a second-order q-difference equation. It is shown that for a general class of weight functions on general q-linear grids, the functions of the second kind and the orthogonal polynomials are linear independent solutions of the same second-order q-difference equation. We introduce q-analogues of the discriminant and evaluate the quantized discriminant for general q-orthogonal polynomials in terms of the recursion coefficients. The q-discriminants of the discrete q-Hermite and little q-Jacobi polynomials, as well as a generalized discriminant of the continuous q-Jacobi polynomials, are given explicitly. A q-analogue of the Freud weights is introduced and we derive the second-order q-difference equation they satisfy.

[1]  Diagonalizaton of certain integral operators II , 1993, math/9310221.

[2]  Mourad E. H. Ismail,et al.  q-discriminants and Vertex Operators , 2001, Adv. Appl. Math..

[3]  F. Calogero,et al.  PROPERTIES OF THE ZEROS OF THE CLASSICAL POLYNOMIALS AND OF BESSEL FUNCTIONS , 1978 .

[4]  M. Ismail.  Discriminants and Functions of the Second Kind of Orthogonal Polynomials , 1998 .

[5]  M. Roberts.  Sur quelques théorémés d’algébre , 1861 .

[6]  Christian Krattenthaler,et al.  The major counting of nonintersecting lattice paths and generating functions for tableaux , 1995 .

[7]  Rene F. Swarttouw,et al.  The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Report Fac , 1996, math/9602214.

[8]  Yang Chen,et al.  Ladder operators and differential equations for orthogonal polynomials , 1997 .

[9]  D. Clark,et al.  Estimates of the Hermite and the Freud polynomials , 1990 .

[10]  M. Ismail,et al.  The Associated Ultraspherical Polynomials and their q -Analogues , 1982, Canadian Journal of Mathematics.

[11]  V. B. Uvarov,et al.  Classical Orthogonal Polynomials of a Discrete Variable , 1991 .

[12]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[13]  F. Calogero,et al.  Properties of the zeros of the classical polynomials and of the Bbessel functions , 1979 .

[14]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[16]  Plamen Simeonov,et al.  Difference Equations and Discriminants for Discrete Orthogonal Polynomials , 2005 .

[17]  Mourad E. H. Ismail,et al.  An electrostatics model for zeros of general orthogonal polynomials , 2000 .

[18]  J. Schur,et al.  Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome. , 1931 .

[19]  G. Andrews,et al.  Classical orthogonal polynomials , 1985 .

[20]  William C. Bauldry.  Estimates of asymmetric Freud polynomials on the real line , 1990 .

[21]  M. E. Muldoon,et al.  Reciprocal Power Sums of Differences of Zeros of Special Functions , 1983 .