scholarly journals Ratio asymptotics for biorthogonal matrix polynomials with unbounded recurrence coefficients

2020 ◽  
pp. 51-51
Author(s):  
Amílcar Branquinho ◽  
Juan Garca-Ardila ◽  
Francisco Marcellán
Keyword(s):  
Recurrence Relation ◽  
Recurrence Coefficients ◽  
Matrix Polynomials ◽  
Ratio Asymptotics ◽  
Matrix Coefficients ◽  
Biorthogonal Polynomials ◽  
Matrix Biorthogonal Polynomials ◽  
Three Term Recurrence Relation

In this paper we study matrix biorthogonal polynomials sequences that satisfy a nonsymmetric three term recurrence relation with unbounded matrix coefficients. The outer ratio asymptotics for this family of matrix biorthogonal polynomials is derived under quite general assumptions. Some illustrative examples are considered.

scholarly journals An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 534
Author(s):  
Lino G. Garza ◽  
Luis E. Garza ◽  
Edmundo J. Huertas
Keyword(s):  
Difference Equation ◽  
Recurrence Relation ◽  
Positive Measure ◽  
Asymptotic Properties ◽  
Nonlinear Difference Equation ◽  
Sobolev Type ◽  
Real Numbers ◽  
Recurrence Coefficients ◽  
Algebraic Properties ◽  
Three Term Recurrence Relation

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(0)q′(0), where p,q are polynomials, M0, M1 are non-negative real numbers and μ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dμ(x)=e−x4dx.

scholarly journals An ubiquitous three-term recurrence relation

2021 ◽  
Vol 62 (3) ◽  
pp. 032106
Author(s):  
Paolo Amore ◽  
Francisco M. Fernández
Keyword(s):  
Recurrence Relation ◽  
Three Term Recurrence Relation

Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Numerical Methods ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Rational Function ◽  
Recurrence Relations ◽  
Recurrence Coefficients ◽  
Basic Task ◽  
Sobolev Orthogonal Polynomials ◽  
Gauss Type ◽  
Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

scholarly journals On hypergeometric generalized negative binomial distribution

2002 ◽  
Vol 29 (12) ◽  
pp. 727-736 ◽  
Author(s):  
M. E. Ghitany ◽  
S. A. Al-Awadhi ◽  
S. L. Kalla
Keyword(s):  
Recurrence Relation ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Field Data ◽  
Goodness Of Fit ◽  
Negative Binomial ◽  
Generalized Negative Binomial Distribution ◽  
The Right ◽  
Three Term Recurrence Relation

It is shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right, and leptokurtic. Also, a three-term recurrence relation for computing probabilities from the considered distribution is given. Application of the distribution to entomological field data is given and its goodness-of-fit is demonstrated.

scholarly journals On the three term recurrence relation and BDF methods

2020 ◽  
Author(s):  
Larissa Ferreira Marques ◽  
Vanessa Botta ◽  
Messias Meneguette
Keyword(s):  
Recurrence Relation ◽  
Bdf Methods ◽  
Three Term Recurrence Relation

scholarly journals ELEMENTARY OSCILLATION CRITERIA FOR A THREE TERM RECURRENCE RELATION WITH OSCILLATORY COEFFICIENT SEQUENCE

1998 ◽  
Vol 29 (3) ◽  
pp. 227-232
Author(s):  
GUANG ZHANG ◽  
SUI-SUN CHENG
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Qualitative Properties ◽  
Oscillation Criteria ◽  
Three Term Recurrence Relation

Qualitative properties of recurrence relations with coefficients taking on both positive and negative values are difficult to obtain since mathematical tools are scarce. In this note we start from scratch and obtain a number of oscillation criteria for one such relation : $x_{n+1}-x_n+p_nx_{n-r}\le 0$.

scholarly journals Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I

2020 ◽  
Author(s):  
Grzegorz Świderski ◽  
Bartosz Trojan
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Behaviour ◽  
Recurrence Relation ◽  
Scaling Limits ◽  
Christoffel Functions ◽  
Asymptotically Periodic ◽  
Three Term Recurrence Relation

Abstract For Jacobi parameters belonging to one of three classes: asymptotically periodic, periodically modulated, and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel–Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports.

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