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Bessel-zernike discrete variable representation basis.

Charles Cerjan¹

¹Lawrence Livermore National Laboratory, Livermore California 94550, USA.

The Journal of Physical Chemistry. A

|April 21, 2006

Summary

This study demonstrates how Zernike polynomials, a type of Jacobi polynomial, can be used for Bessel function expansions. This method simplifies generating series identities and efficiently evaluating Hankel transforms.

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Area of Science:

Mathematical Physics
Special Functions

Background:

Bessel functions are crucial in various scientific fields.
Orthogonal polynomials offer powerful tools for function approximation and analysis.
Zernike polynomials are a specific class of orthogonal polynomials with unique properties.

Purpose of the Study:

To establish the connection between Bessel discrete variable basis expansions and Zernike polynomials.
To demonstrate the utility of Zernike polynomials for function series expansions.
To explore the application of Zernike polynomials in evaluating Hankel transforms.

Main Methods:

Demonstrating the relationship between Bessel basis expansions and Jacobi polynomials.
Utilizing Zernike polynomials for series expansions of functions over the unit interval.
Applying Zernike expansions to Bessel functions to derive series identities.

Main Results:

A direct connection is shown between Bessel discrete variable basis expansion and Zernike polynomials.
Zernike polynomials provide an alternative and effective method for series expansions.
Expressing Bessel functions using Zernike expansions simplifies the generation of series identities.
Zernike polynomials enable efficient computation of Hankel transforms for specific function types.

Conclusions:

Zernike polynomials offer a valuable framework for analyzing Bessel functions.
The use of Zernike polynomials streamlines the derivation of mathematical identities.
This approach provides an efficient computational method for Hankel transforms.

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