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 K_n is a modified Bessel function of the second kind, y_n is the ordinary polynomial, and θ_n is the reverse polynomial. For example:

Definition as a hypergeometric function

The Bessel polynomial may also be defined as a confluent hypergeometric function
A similar expression holds true for the generalized Bessel polynomials :
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.

Generating function

The Bessel polynomials, with index shifted, have the generating function
Differentiating with respect to, cancelling, yields the generating function for the polynomials
Similar generating function exists for the polynomials as well:
Upon setting, one has the following representation for the exponential function:

Recursion

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

Differential equation

The Bessel polynomial obeys the following differential equation:
and

Orthogonality

The Bessel polynomials are orthogonal with respect to the weight integrated over the unit circle of the complex plane. In other words, if,
They are also orthogonal with respect to a real weight, provided it is a hyperfunction.

Generalization

Explicit form

A generalization of the Bessel polynomials have been suggested in literature, as following:
the corresponding reverse polynomials are
The explicit coefficients of the polynomials are:
Consequently, the polynomials can explicitly be written as follows:
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.

Inverse Formulae

Powers of are expressed in terms of the generalized Bessel polynomials from the inverse connection formulae which have applications in change of basis to these polynomials.
where and.
Similarly, for the reverse generalized Bessel polynomials
where and.

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,

Zeros

If one denotes the zeros of as, and that of the by, then the following estimates exist:
and
for all. Moreover, all these zeros have negative real part.
Sharper results can be said if one resorts to more powerful theorems regarding the estimates of zeros of polynomials.
One result is the following:

Particular values

The Bessel polynomials up to are
No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.
The reverse Bessel polynomials are obtained by reversing the coefficients.
Equivalently,.
This results in the following: