Lie algebras and recurrence relations IV: Representations of the Euclidean group and Bessel functions

1996 ◽  
Vol 43 (3) ◽  
pp. 289-316 ◽  
Author(s):  
Philip Feinsilver
Keyword(s):  
Lie Algebras ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Euclidean Group

Lie Algebras and Recurrence Relations II

1989 ◽  
pp. 163-179 ◽  
Author(s):  
Philip Feinsilver
Keyword(s):  
Lie Algebras ◽  
Recurrence Relations

Transformations between second order linear differential equations and a reinterpretation of recurrence relations between Bessel functions with algebraic coefficients

1977 ◽  
Vol 79 (1-2) ◽  
pp. 87-105 ◽  
Author(s):  
John Heading
Keyword(s):  
Differential Equations ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Asymptotic Methods ◽  
Second Order ◽  
Schwarzschild Black Hole ◽  
Matrix Methods ◽  
Large Numbers ◽  
Method Of Solution ◽  
Linear Differential

SynopsisA scheme devised by Chandrasekhar for investigating the transformations between various differential equations of the second order governing perturbations of the Schwarzschild black hole demands further investigation. The transformation between two differential equations in normal form is considered, and a wide survey of the properties of the transformation is given. It is shown how Chandrasekhar's equations fit into the scheme, after which some examples with particular properties are considered. A detailed investigation of Bessel's equation is undertaken using various devices, in particular by employing asymptotic methods for products of Bessel functions, and employing matrix methods for dealing with large numbers of matrix equations which necessitates an interesting method of solution, the results being reinterpretations of the standard recurrence relations for Bessel functions.

scholarly journals The Hermite polynomials and the Bessel functions from a general point of view

2003 ◽  
Vol 2003 (57) ◽  
pp. 3633-3642 ◽  
Author(s):  
G. Dattoli ◽  
H. M. Srivastava ◽  
D. Sacchetti
Keyword(s):  
Difference Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Hermite Polynomials ◽  
Point Of View ◽  
General Point ◽  
Ordinary Case

We introduce new families of Hermite polynomials and of Bessel functions from a point of view involving the use of nonexponential generating functions. We study their relevant recurrence relations and show that they satisfy differential-difference equations which are isospectral to those of the ordinary case. We also indicate the usefulness of some of these new families.

scholarly journals A RECURRENCE RELATION FOR CHARACTERS OF HIGHEST WEIGHT INTEGRABLE MODULES FOR AFFINE LIE ALGEBRAS

2007 ◽  
Vol 09 (02) ◽  
pp. 121-133 ◽  
Author(s):  
WILLIAM J. COOK ◽  
HAISHENG LI ◽  
KAILASH C. MISRA
Keyword(s):  
Recurrence Relation ◽  
Lie Algebras ◽  
Vertex Operator ◽  
Recurrence Relations ◽  
Operator Algebras ◽  
Affine Lie Algebras ◽  
Highest Weight ◽  
Irreducible Modules ◽  
Level 1 ◽  
Integrable Modules

Using certain results for the vertex operator algebras associated with affine Lie algebras, we obtain recurrence relations for the characters of integrable highest weight irreducible modules for an affine Lie algebra. As an application we show that in the simply-laced level 1 case, these recurrence relations give the known characters, whose principal specializations naturally give rise to some multisum Macdonald identities.

scholarly journals Contractions of Lie Algebras and Separation of Variables.

1997 ◽  
Vol 12 (01) ◽  
pp. 53-61 ◽  
Author(s):  
A. A. Izmest'ev ◽  
G. S. Pogosyan ◽  
A. N. Sissakian ◽  
P. Winternitz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lorentz Group ◽  
Homogeneous Spaces ◽  
Separation Of Variables ◽  
Second Order ◽  
Coordinate Systems ◽  
Euclidean Group ◽  
Enveloping Algebra ◽  
Four Levels

The Inönü-Wigner contraction from the Lorentz group O(2,1) to the Euclidean group E(2) is used to relate the separation of variables in the Laplace-Beltrami operators on the two corresponding homogeneous spaces. We consider the contractions on four levels: the Lie algebra, the commuting sets of second order operators in the enveloping algebra o(2,1), the coordinate systems and some eigenfunctions of the Laplace-Beltrami operators.

scholarly journals New realizations of algebras of the Askey–Wilson type in terms of Lie and quantum algebras

2020 ◽  
pp. 2150002
Author(s):  
Nicolas Crampé ◽  
Dounia Shaaban Kabakibo ◽  
Luc Vinet
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Recurrence Relations ◽  
Rational Functions ◽  
The Other ◽  
Quantum Algebras ◽  
Casimir Element ◽  
Usual Representation ◽  
Racah Algebra

The Askey–Wilson algebra is realized in terms of the elements of the quantum algebras [Formula: see text] or [Formula: see text]. A new realization of the Racah algebra in terms of the Lie algebras [Formula: see text] or [Formula: see text] is also given. Details for different specializations are provided. The advantage of these new realizations is that one generator of the Askey–Wilson (or Racah) algebra becomes diagonal in the usual representation of the quantum algebras whereas the second one is tridiagonal. This allows us to recover easily the recurrence relations of the associated orthogonal polynomials of the Askey scheme. These realizations involve rational functions of the Cartan generator of the quantum algebras, where they are linear with respect to the other generators and depend on the Casimir element of the quantum algebras.

scholarly journals MATLAB GUI for computing Bessel functions using continued fractions algorithm

2011 ◽  
Vol 33 (1) ◽  
Author(s):  
E. Hernández ◽  
K. Commeford ◽  
M.J. Pérez-Quiles
Keyword(s):  
User Interface ◽  
Bessel Function ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Engineering Students ◽  
Bessel Functions ◽  
Higher Order ◽  
Graphic User Interface ◽  
Miller’S Algorithm ◽  
The Stability

Higher order Bessel functions are prevalent in physics and engineering and there exist different methods to evaluate them quickly and efficiently. Two of these methods are Miller's algorithm and the continued fractions algorithm. Miller's algorithm uses arbitrary starting values and normalization constants to evaluate Bessel functions. The continued fractions algorithm directly computes each value, keeping the error as small as possible. Both methods respect the stability of the Bessel function recurrence relations. Here we outline both methods and explain why the continued fractions algorithm is more efficient. The goal of this paper is both (1) to introduce the continued fractions algorithm to physics and engineering students and (2) to present a MATLAB GUI (Graphic User Interface) where this method has been used for computing the Semi-integer Bessel Functions and their zeros.

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