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 Kn(x) is a modified Bessel function of the second kind, yn(x) is the ordinary polynomial, and θn(x) is the reverse polynomial (pg 7 and 34 Grosswald 1978). For example:[1]

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 (2n)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 , 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 (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(α, β)
n
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.[2] The five reverse Bessel Polynomials are obtained by reversing the coefficients. Equivalently, . This results in the following:

See also

References

  1. Wolfram Alpha example
  2. Filaseta, Michael; Trifinov, Ognian (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik. 2002 (550): 125–140. CiteSeerX 10.1.1.6.9538. doi:10.1515/crll.2002.069.
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