scholarly journals Orthogonal polynomials with a resolvent-type generating function

2008 ◽  
Vol 360 (08) ◽  
pp. 4125-4143 ◽  
Author(s):  
Michael Anshelevich
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function

scholarly journals The Tensor Pade´-Type Approximant with Application in Computing Tensor Exponential Function

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Chuanqing Gu ◽  
Yong Liu
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Exponential Function ◽  
Efficient Algorithm ◽  
Linear Functional ◽  
Function Form ◽  
Numerical Examples ◽  
First Time ◽  
Formal Orthogonal Polynomials

Tensor exponential function is an important function that is widely used. In this paper, tensor Pade´-type approximant (TPTA) is defined by introducing a generalized linear functional for the first time. The expression of TPTA is provided with the generating function form. Moreover, by means of formal orthogonal polynomials, we propose an efficient algorithm for computing TPTA. As an application, the TPTA for computing the tensor exponential function is presented. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.

scholarly journals A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

2011 ◽  
Vol 25 (1) ◽  
pp. 21-35 ◽  
Author(s):  
Juan J. Moreno-Balcázar ◽  
Teresa E. Pérez ◽  
Miguel A. Piñar
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function

scholarly journals A Triple Lacunary Generating Function for Hermite Polynomials

10.37236/1927 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Pallavi Jayawant
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Combinatorial Proof ◽  
Classical Orthogonal Polynomials ◽  
Combinatorial Objects ◽  
Combinatorial Interpretations

Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's approach to generating functions for the Hermite polynomials to obtain a triple lacunary generating function. We define renormalized Hermite polynomials $h_n(u)$ by $$\sum_{n= 0}^\infty h_n(u) {z^n\over n!}=e^{uz+{z^2\!/2}}.$$ and give a combinatorial proof of the following generating function: $$ \sum_{n= 0}^\infty h_{3n}(u) {{z^n\over n!}}= {e^{(w-u)(3u-w)/6}\over\sqrt{1-6wz}} \sum_{n= 0}^\infty {{(6n)!\over 2^{3n}(3n)!(1-6wz)^{3n}} {z^{2n}\over(2n)!}}, $$ where $w=(1-\sqrt{1-12uz})/6z=uC(3uz)$ and $C(x)=(1-\sqrt{1-4x})/(2x)$ is the Catalan generating function. We also give an umbral proof of this generating function.

Generating function: multiple orthogonal polynomials

2016 ◽  
Vol 10 ◽  
pp. 761-772
Author(s):  
W. Carballosa ◽  
J. C. Hernandez-Gomez ◽  
L. R. Pineiro ◽  
Jose M. Sigarreta
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Multiple Orthogonal Polynomials

Generating functions and semi-classical orthogonal polynomials

1994 ◽  
Vol 124 (5) ◽  
pp. 1003-1011 ◽  
Author(s):  
Pascal Maroni ◽  
Jeannette Van Iseghem
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Special Form ◽  
Classical Orthogonal Polynomials ◽  
Order Linear ◽  
First Order

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.

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