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 (Krall & Frink, 1948)
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials (See Grosswald 1978, Berg 2000).
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"("x") is a , "y""n"("x") is the ordinary polynomial, and "θ""n"("x") is the reverse polynomial (pg 7 and 34 Grosswald 1978). For example:
Definition as a hypergeometric function.
The Bessel polynomial may also be defined as a confluent hypergeometric function (Dita, 2006)
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 (−2"n")"n" is the Pochhammer symbol (rising factorial).
The inversion for monomials is given by
Generating function.
The Bessel polynomials, with index shifted, have the generating function
Differentiating with respect to formula_14, cancelling formula_15, yields the generating function for the polynomials formula_16
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 (Krall, Fink), 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("x") = exp(−2 / "x").
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 formula_32. 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, formula_35.
This results in the following: