Rational spectral transformations and orthogonal polynomials

Abstract We show that generic rational transformations of the Stieltjes function with polynomial coefficients (ST) can be presented as a finite superposition of four fundamental elementary transforms: Christoffel transform (CT), Geronimus transform (GT) and forward and backward associated transformations A + T , A − T . It is shown that the Laguerre-Hahn polynomials (LHP) on arbitrary nonuniform lattice are covariant with respect to ST (i.e., ST of a LHP yields another LHP), whereas the semi-classical polynomials are covariant with respect to a subclass of linear ST. Some applications of these results to the theory of the semi-classical polynomials are considered.

[1]  J. S. Dehesa,et al.  On orthogonal polynomials with perturbed recurrence relations , 1990 .

[2]  F. Peherstorfer.  Orthogonal and extremal polynomials on several intervals , 1993 .

[3]  J Nuttall,et al.  Asymptotics of diagonal Hermite-Padé polynomials , 1984 .

[4]  T. Chihara,et al.  ON CO-RECURSIVE ORTHOGONAL POLYNOMIALS , 1957 .

[5]  Alphonse P. Magnus.  Special nonuniform lattice ( snul ) orthogonal polynomials on discrete dense sets of points , 1995 .

[6]  Walter Gautschi,et al.  Computational Aspects of Orthogonal Polynomials , 1990 .

[7]  Luc Vinet,et al.  SPECTRAL TRANSFORMATIONS, SELF-SIMILAR REDUCTIONS AND ORTHOGONAL POLYNOMIALS , 1997 .

[8]  H. Bavinck,et al.  Difference Equations for Generalized Meixner Polynomials , 1994 .

[9]  A. Aptekarev,et al.  ASYMPTOTIC PROPERTIES OF POLYNOMIALS ORTHOGONAL ON A SYSTEM OF CONTOURS, AND PERIODIC MOTIONS OF TODA LATTICES , 1986 .

[10]  Ben Silver,et al.  Elements of the theory of elliptic functions , 1990 .

[11]  A. Magnus.  Riccati Acceleration of Jacobi Continued Fractions and Laguerre-hahn Orthogonal Polynomials , 1984 .

[12]  P. Maroni,et al.  Second degree classical forms , 1997 .

[13]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[14]  F. Marcellán,et al.  Operators preserving orthogonality of polynomials , 1996 .

[15]  F. V. Atkinson,et al.  Discrete and Continuous Boundary Problems , 1964 .

[16]  Arieh Iserles,et al.  On polynomials orthogonal with respect to certain Sobolev inner products , 1991 .

[17]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

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

[19]  F. Marcellán,et al.  Orthogonal polynomials associated with some modifications of a linear functional , 1992 .

[20]  L. Vinet,et al.  Difference schrödinger operators with linear and exponential discrete spectra , 1993 .

[21]  Franz Peherstorfer.  Finite perturbations of orthogonal polynomials , 1992 .

[22]  Pascal Maroni,et al.  An introduction to second degree forms , 1995, Adv. Comput. Math..

[23]  Ap. Magnus,et al.  Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials , 1988 .

[24]  J. Wimp,et al.  Some Sequences Arising in the Creation of New Orthogonal Polynomials , 1994 .

[25]  R. Koekoek,et al.  On a difference equation for generalizations of Charlier polynomials , 1995, math/9908144.

[26]  P. Maroni,et al.  Sur l'adjonction d'une masse de Dirac á une forme régulière et semi-classique , 1992 .

[27]  P. Maroni.  Sur la suite de polynômes orthogonaux associée à la formeu=δc+λ(x−c)−1L , 1990 .

[28]  A. Ronveaux,et al.  Upward Extension of the Jacobi Matrix for Orthogonal Polynomials , 1995, math/9510213.