Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.

[1]  UNITARY MATRIX MODELS AND PAINLEVÉ III , 1996, hep-th/9609214.

[2]  Leon M. Hall,et al.  Special Functions , 1998 .

[3]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .

[4]  K. Życzkowski,et al.  Random unitary matrices , 1994 .

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

[6]  Mourad E. H. Ismail,et al.  Difference equations and quantized discriminants for q-orthogonal polynomials , 2003, Adv. Appl. Math..

[7]  P. Forrester,et al.  Electrostatics and the zeros of the classical polynomials , 1986 .

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

[9]  D. Shevitz,et al.  Unitary-matrix models as exactly solvable string theories. , 1990, Physical review letters.

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

[11]  Ira M. Gessel,et al.  Symmetric functions and P-recursiveness , 1990, J. Comb. Theory, Ser. A.

[12]  I︠a︡. L. Geronimus.  Polynomials orthogonal on a circle and their applications , 1954 .

[13]  G. L..  Collected Papers , 1912, Nature.

[14]  Craig A. Tracy,et al.  Random Unitary Matrices, Permutations and Painlevé , 1999 .

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

[16]  F. Alberto Grünbaum,et al.  Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials , 1998 .

[17]  Tom H. Koornwinder,et al.  Orthogonal Polynomials With Weight Function (1 - x) α ( l + x) β + M δ(x + 1) + Nδ(x - 1) , 1984, Canadian Mathematical Bulletin.

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

[19]  Paul Neval,et al.  Ge´za Freud, orthogonal polynomials and Christoffel functions. A case study , 1986 .

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

[21]  Discriminants and Functions of the Second Kind of Orthogonal Polynomials , 1998 .

[22]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[23]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[24]  P. I. Pastro.  Orthogonal polynomials and some q-beta integrals of Ramanujan☆ , 1985 .

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