INTERPOLATION OF CHEBYSHEV POLYNOMIALS AND INTERACTING FOCK SPACES

2006 ◽  
Vol 09 (03) ◽  
pp. 361-371 ◽  
Author(s):  
IZUMI KUBO ◽  
HUI-HSIUNG KUO ◽  
SUAT NAMLI
Keyword(s):  
Orthogonal Polynomials ◽  
Anderson Model ◽  
Chebyshev Polynomials ◽  
Field Operator ◽  
Parameter Family ◽  
Probability Measures ◽  
Fock Spaces ◽  
Arcsine Distribution ◽  
Multiplicative Renormalization ◽  
Renormalization Method

We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.

scholarly journals GENERATING FUNCTIONS OF CAUCHY–STIELTJES TYPE FOR ORTHOGONAL POLYNOMIALS

2009 ◽  
Vol 12 (01) ◽  
pp. 91-98 ◽  
Author(s):  
MAREK BOŻEJKO ◽  
NIZAR DEMNI
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Short Proof ◽  
Probabilistic Interpretation ◽  
Infinitely Divisible ◽  
The Family ◽  
Multiplicative Renormalization ◽  
Renormalization Method ◽  
Stieltjes Type

We give a free probabilistic interpretation of the multiplicative renormalization method. As a byproduct, we give a short proof of the Asai–Kubo–Kuo problem on the characterization of the family of measures for which this method applies with h(x) = (1 - x)-1 which turns out to be the free Meixner family. We also give a representation of the Voiculescu transform of all free Meixner laws (even in the non-freely infinitely divisible case).

CLASSES OF MRM-APPLICABLE MEASURES FOR THE POWER FUNCTION OF GENERAL ORDER

2013 ◽  
Vol 16 (04) ◽  
pp. 1350028
Author(s):  
IZUMI KUBO ◽  
HUI-HSIUNG KUO ◽  
SUAT NAMLI
Keyword(s):  
Orthogonal Polynomials ◽  
Power Function ◽  
Generating Functions ◽  
The Other ◽  
Large Classes ◽  
Beta Distributions ◽  
General Order ◽  
Multiplicative Renormalization ◽  
Renormalization Method ◽  
Special Classes

The authors have previously studied multiplicative renormalization method (MRM) for generating functions of orthogonal polynomials. In particular, they have determined all MRM-applicable measures for renormalizing functions h(x) = ex, h(x) = (1 - x)-κ, κ = 1/2, 1, 2. For the cases h(x) = ex and (1 - x)-1, there are very large classes of MRM-applicable measures. For the other two cases κ = 1/2, 2, MRM-applicable measures belong to special classes of a certain kind of beta distributions. In this paper, we determine all MRM-applicable measures for h(x) = (1 - x)-κ with κ ≠ 0, 1, 1/2.

scholarly journals Identification of the theory of orthogonal polynomials in d-indeterminates with the theory of 3-diagonal symmetric interacting Fock spaces on ℂd

2017 ◽  
Vol 20 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Luigi Accardi ◽  
Abdessatar Barhoumi ◽  
Ameur Dhahri
Keyword(s):  
Orthogonal Polynomials ◽  
Polynomial Algebra ◽  
Random Variables ◽  
Positive Definite ◽  
Probability Measures ◽  
Fock Spaces ◽  
Positive Definite Kernels ◽  
Algebraic Classification ◽  
The Real

The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on [Formula: see text] with moments of any order and more generally of states on the polynomial algebra on [Formula: see text]. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.

Interacting Fock Spaces Related to the Anderson Model

1998 ◽  
Vol 01 (02) ◽  
pp. 247-283 ◽  
Author(s):  
Yun-Gang Lu
Keyword(s):  
Integral Equation ◽  
Distribution Function ◽  
Anderson Model ◽  
Fock Space ◽  
Field Operator ◽  
Vacuum Expectation ◽  
Fock Spaces ◽  
Test Functions ◽  
Test Function ◽  
New Type

A new type of interacting Boltzmannian Fock space, emerging from the stochastic limit of the Anderson model, is investigated. We describe the structure of the space and the form, assumed in this case, by the principles of factorization and of total connection. Using these principles, the vacuum expectation of any product of creation and annihilation operators can be calculated. By means of these results, for any test function, a system of diffrence equations satisfied by the moments of the field operator and an integral equation satisfied by their generating function is deduced. In many interesting cases this equation is solved and the vacuum distribution function of the field operator (even its density) is explicitly determined. This evidentiates a new phenomenon which cannot take place in the usual Fock spaces (and did not appear in the simplest examples of interacting Fock spaces): by taking different test functions, the vacuum distribution of the field operator does not change only parametrically, but radically. In particular we find the semi-circle, the reciprocal-semi-circle (or Arcsine), the double-beta,…, and many other distributions.

scholarly journals Interacting Fock Spaces and Gaussianization of Probability Measures

1998 ◽  
Vol 01 (04) ◽  
pp. 663-670 ◽  
Author(s):  
Luigi Accardi ◽  
Marek Bożejko
Keyword(s):  
Probability Measure ◽  
Orthogonal Polynomials ◽  
Fock Space ◽  
Field Operator ◽  
Multiplication Operator ◽  
Probability Measures ◽  
Number Operator ◽  
Infinitely Divisible ◽  
Future Paper ◽  
Interacting Fock Space

We prove that any probability measure on ℝ, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus (in the nonsymmetric case) a function of the number operator. This follows from a canonical isomorphism between the L2-space of the measure and the interacting Fock space in which the number vectors go into the orthogonal polynomials of the measure and the modified field operator into the multiplication operator by the x-coordinate. A corollary of this is that all the momenta of such a measure are expressible in terms of the Szëgo–Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only noncrossing pair partitions (and singletons, in the nonsymmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call Gaussianization. Finally we define, in terms of the Szëgo–Jacobi parameters, a new convolution among probability measures which we call universal because any probability measure (with all moments) is infinitely divisible with respect to this convolution. All these results will be extended to the case of many (in fact infinitely many) variables in a future paper.

scholarly journals Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Waleed M. Abd-Elhameed ◽  
Youssri H. Youssri
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Connection Coefficients ◽  
Current Article ◽  
In Virtue Of ◽  
Connection Formulas

AbstractThe principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

scholarly journals Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 354 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Jiří Patera
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Bell Polynomials ◽  
The Third ◽  
Weight Lattice ◽  
Hybrid Character ◽  
Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

scholarly journals On the connection between tridiagonal matrices, Chebyshev polynomials, and Fibonacci numbers

2020 ◽  
Vol 12 (2) ◽  
pp. 280-286
Author(s):  
Carlos M. da Fonseca
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Tridiagonal Matrices

AbstractIn this note, we recall several connections between the determinant of some tridiagonal matrices and the orthogonal polynomials allowing the relation between Chebyshev polynomials of second kind and Fibonacci numbers. With basic transformations, we are able to recover some recent results on this matter, bringing them into one place.

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