scholarly journals A pair of biorthogonal polynomials for the Szegö-Hermite weight function

1988 ◽  
Vol 11 (4) ◽  
pp. 763-767 ◽  
Author(s):  
N. K. Thakare ◽  
M. C. Madhekar
Keyword(s):  
Positive Integer ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Polynomial Sequences ◽  
Biorthogonal Polynomials

A pair of polynomial sequences{Snμ(x;k)}and{Tmμ(x;k)}whereSnμ(x;k)is of degreeninxkandTmμ(x;k)is of degreeminx, is constructed. It is shown that this pair is biorthogonal with respect to the Szegö-Hermite weight function|x|2μexp(−x2),(μ>−1/2)over the interval(−∞,∞)in the sense that∫−∞∞|x|2μexp(−x2)Snμ(x;k)Tmμ(x;k)dx=0,   ifm≠n                    ≠0,   ifm=nwherem,n=0,1,2,…andkis an odd positive integer.Generating functions, mixed recurrence relations for both these sets are obtained. Fork=1, both the above sets get reduced to the orthogonal polynomials introduced by professor Szegö.

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 Polynomial sequences generated by infinite Hessenberg matrices

2017 ◽  
Vol 5 (1) ◽  
pp. 64-72 ◽  
Author(s):  
Luis Verde-Star
Keyword(s):  
Orthogonal Polynomial ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hessenberg Matrix ◽  
Polynomial Sequences ◽  
Lower Triangular ◽  
Companion Matrices ◽  
Matrix Similarity ◽  
Invertible Matrices ◽  
Construction Algorithms

Abstract We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

scholarly journals Generalization of Konhauser Polynomials

2020 ◽  
Vol 2 (11) ◽  
pp. 8-14
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Biorthogonal System ◽  
Integral Representations ◽  
New Class ◽  
Generating Relations ◽  
Biorthogonal Polynomials

In this paper, we are showing study of biorthogonal polynomials associated with generalization of Laguere polynomials of Srivastava and Singhal [14]. It happens to generalized Konhauser. here we are trying to obtain the generating functions, recurrence relations, biorthogonality relations, integral representations and also bilinear and bilateral generating relations for the new class of biorthogonal system.

scholarly journals Numerical construction of the generalized Hermite polynomials

2003 ◽  
pp. 49-63
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Stable Method ◽  
Numerical Construction ◽  
Three Term Recurrence Relation

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

scholarly journals On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 955
Author(s):  
Abey S. Kelil ◽  
Alta S. Jooste ◽  
Appanah R. Appadu
Keyword(s):  
Integral Representation ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Finite Order ◽  
Recurrence Relations ◽  
Pollaczek Polynomials ◽  
Perturbed Polynomials ◽  
Recursive Relations ◽  
Fisher's Information

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.

scholarly journals A class of orthogonal polynomials of a new type

1983 ◽  
Vol 6 (1) ◽  
pp. 171-180
Author(s):  
M. Dutta ◽  
Kanchan Prabha Manocha
Keyword(s):  
Positive Integer ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Generating Function ◽  
Recurrence Relations ◽  
Variable Parameter ◽  
Negative Integer ◽  
Special Cases ◽  
New Type ◽  
Fixed Positive Integer

The purpose of this paper is to study a class of polynomialsFnm(x,λ,ν)associated with a class ofmdifferential equations, whereλis fixed,nis a variable parameter,mis fixed positive integer andνis a non-negative integer<m. We also obtain all the important properties of this class of polynomials including Rodrigues' formulas, generating function, and recurrence relations. Several special cases of interest are obtained from our analysis.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

SYMMETRIC FUNCTIONS OF BINARY PRODUCTS OF TRIBONACCI LUCAS NUMBERS AND ORTHOGONAL POLYNOMIALS

2021 ◽  
Vol 21 (2) ◽  
pp. 461-478
Author(s):  
HIND MERZOUK ◽  
ALI BOUSSAYOUD ◽  
MOURAD CHELGHAM
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Symmetric Functions ◽  
Lucas Numbers

In this paper, we will recover the new generating functions of some products of Tribonacci Lucas numbers and orthogonal polynomials. The technic used her is based on the theory of the so called symmetric functions.

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