Bessel polynomials
Encyclopedia
In mathematics
, the Bessel polynomials are an orthogonal
sequence of polynomial
s. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series (Krall & Fink, 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
.
s from which the polynomial draws its name.
where
is a modified Bessel function of the second kind and 
is the reverse polynomial (pag 7 and 34 Grosswald 1978).
(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
is the Pochhammer symbol
(rising factorial).
and
and
the corresponding reverse polynomials are
For the weighting function
they are orthogonal, for the relation
holds for
and 
a curve surrounding the 0 point.
They specialize to the Bessel polynomials for
, in which situation 
.
where
are normalization coefficients.
where
. The solutions are,
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Bessel polynomials are an orthogonal
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
sequence of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series (Krall & Fink, 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
Bessel filter
In electronics and signal processing, a Bessel filter is a type of linear filter with a maximally flat group delay . Bessel filters are often used in audio crossover systems...
.
Definition in terms of Bessel functions
The Bessel polynomial may also be defined using Bessel functionBessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s from which the polynomial draws its name.
where
is a modified Bessel function of the second kind and is the reverse polynomial (pag 7 and 34 Grosswald 1978).Definition as a hypergeometric function
The Bessel polynomial may also be defined as a confluent hypergeometric functionConfluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...
(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
is the Pochhammer symbolPochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...
(rising factorial).
Generating function
The Bessel polynomials have the generating functionRecursion
The Bessel polynomial may also be defined by a recursion formula:and
Differential equation
The Bessel polynomial obeys the following differential equation:and
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
and a curve surrounding the 0 point.They specialize to the Bessel polynomials for
, in which situation .Rodrigues formula for Bessel polynomials
The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :where
are normalization coefficients.Associated Bessel polynomials
According to this generalization we have the following generalized associated Bessel polynomials differential equation:where
. The solutions are,Particular values