scholarly journals From orthogonal polynomials on the unit circle to functional equations via generating functions

2015 ◽  
Vol 368 (6) ◽  
pp. 4027-4063 ◽  
Author(s):  
María José Cantero ◽  
Arieh Iserles
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Generating Functions ◽  
Functional Equations

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

Multiple Orthogonal Polynomials on the Unit Circle

2007 ◽  
Vol 28 (2) ◽  
pp. 173-197 ◽  
Author(s):  
Judit Minguez Ceniceros ◽  
Walter Van Assche
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Multiple Orthogonal Polynomials

Zeros of a family of hypergeometric para-orthogonal polynomials on the unit circle

2013 ◽  
Vol 286 (17-18) ◽  
pp. 1778-1791 ◽  
Author(s):  
Dimitar K. Dimitrov ◽  
A. Sri Ranga
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle

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.

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 THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

2010 ◽  
Vol 21 (02) ◽  
pp. 145-155 ◽  
Author(s):  
P. ROMÁN ◽  
S. SIMONDI
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Hypergeometric Functions ◽  
Spherical Functions ◽  
Hypergeometric Equation ◽  
Necessary Condition ◽  
Hypergeometric Differential Equation ◽  
The Matrix ◽  
Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

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