Bessel polynomials


In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials.
The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is
while the third-degree reverse Bessel polynomial is
The reverse Bessel polynomial is used in the design of Bessel electronic filters.

Properties

Definition in terms of Bessel functions

The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.
where Kn is a, yn is the ordinary polynomial, and θn is the reverse polynomial . For example:

Definition as a hypergeometric function

The Bessel polynomial may also be defined as a confluent hypergeometric function
The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:
from which it follows that it may also be defined as a hypergeometric function:
where n is the Pochhammer symbol.
The inversion for monomials is given by

Generating function

The Bessel polynomials, with index shifted, have the generating function
Differentiating with respect to, cancelling, yields the generating function for the polynomials

Recursion

The Bessel polynomial may also be defined by a recursion formula:
and

Differential equation

The Bessel polynomial obeys the following differential equation:
and

Generalization

Explicit Form

A generalization of the Bessel polynomials have been suggested in literature, as following:
the corresponding reverse polynomials are
For the weighting function
they are orthogonal, for the relation
holds for m ≠ n and c a curve surrounding the 0 point.
They specialize to the Bessel polynomials for α = β = 2, in which situation ρ = exp.

Rodrigues formula for Bessel polynomials

The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :
where a are normalization coefficients.

Associated Bessel polynomials

According to this generalization we have the following generalized differential equation for associated Bessel polynomials:
where. The solutions are,

Particular values

The first five Bessel Polynomials are expressed as:
No Bessel Polynomial can be factored into lower-ordered polynomials with strictly rational coefficients.
The five reverse Bessel Polynomials are obtained by reversing the coefficients.
Equivalently,.
This results in the following: