# Bessel polynomials

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 & Fink, 1948)

:$y_n\left(x\right)=sum_\left\{k=0\right\}^nfrac\left\{\left(n+k\right)!\right\}\left\{\left(n-k\right)!k!\right\},left\left(frac\left\{x\right\}\left\{2\right\} ight\right)^k$

Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials (See Grosswald 1978, Berg 2000).

:$heta_n\left(x\right)=x^n,y_n\left(1/x\right)=sum_\left\{k=0\right\}^nfrac\left\{\left(2n-k\right)!\right\}\left\{\left(n-k\right)!k!\right\},frac\left\{x^k\right\}\left\{2^\left\{n-k$

The coefficients of the second definition are the same as the first but in reverse order. For example, the third order Bessel polynomial is

:$y_3\left(x\right)=15x^3+15x^2+6x+1,$

while the third order reverse Bessel polynomial is

:$heta_3\left(x\right)=x^3+6x^2+15x+15,$

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.:$y_n\left(x\right)=,x^\left\{n\right\} heta_n\left(1/x\right),$:$heta_n\left(x\right)=sqrt\left\{frac\left\{2\right\}\left\{pi,x^\left\{n+1/2\right\}e^\left\{x\right\}K_\left\{\left(n+1/2\right)\right\}\left(x\right)$:$y_n\left(x\right)=sqrt\left\{frac\left\{2\right\}\left\{pi x,e^\left\{1/x\right\}K_\left\{\left(n+1/2\right)\right\}\left(1/x\right)$

where $K_n\left(x\right)$ is a modified Bessel function of the second kind and $y_n\left(x\right)$ is the reverse polynomial (pag 7 and 34 Grosswald 1978).

Definition as a hypergeometric function

The Bessel polynomial may also be defined as a hypergeometric function (Dita, 2006)

:$y_n\left(x\right)=,_2F_0\left(-n,n+1;;-x/2\right)$

The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:

:$heta_n\left(x\right)=frac\left\{n!\right\}\left\{\left(-2\right)^n\right\},L_n^\left\{-2n-1\right\}\left(2x\right)$

from which it follows that it may also be defined as a hypergeometric function:

:$heta_n\left(x\right)=frac\left\{\left(-2n\right)_n\right\}\left\{\left(-2\right)^n\right\},,_1F_1\left(-n;-2n;-2x\right)$

where $\left(-2n\right)_n$ is the Pochhammer symbol (rising factorial).

Recursion

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

:$y_0\left(x\right)=1,$:$y_1\left(x\right)=x+1,$:$y_n\left(x\right)=\left(2n!-!1\right)x,y_\left\{n-1\right\}\left(x\right)+y_\left\{n-2\right\}\left(x\right),$

and

:$heta_0\left(x\right)=1,$:$heta_1\left(x\right)=x+1,$:$heta_n\left(x\right)=\left(2n!-!1\right) heta_\left\{n-1\right\}\left(x\right)+x^2 heta_\left\{n-2\right\}\left(x\right),$

Differential Equation

The Bessel polynomial obeys the following differential equation:

:$x^2frac\left\{d^2y_n\left(x\right)\right\}\left\{dx^2\right\}+2\left(x!+!1\right)frac\left\{dy_n\left(x\right)\right\}\left\{dx\right\}-n\left(n+1\right)y_n\left(x\right)=0$

and

:$xfrac\left\{d^2 heta_n\left(x\right)\right\}\left\{dx^2\right\}-2\left(x!+!n\right)frac\left\{d heta_n\left(x\right)\right\}\left\{dx\right\}+2n, heta_n\left(x\right)=0$

Particular values

:$y_0\left(x\right) = 1 ,$

:$y_1\left(x\right) = x + 1 ,$

:$y_2\left(x\right) = 3x^2+ 3x + 1 ,$

:$y_3\left(x\right) = 15x^3+ 15x^2+ 6x + 1 ,$

:$y_4\left(x\right) = 105x^4+105x^3+ 45x^2+ 10x + 1 ,$

:$y_5\left(x\right) = 945x^5+945x^4+420x^3+105x^2+15x+1,$

* [http://mathworld.wolfram.com/BesselPolynomial.html Eric Weisstein's Math World]

References

*cite journal
last = Carlitz
first = L.
coauthors =
year = 1957
month =
title = A Note on the Bessel Polynomials
journal = Duke Math. J.
volume = 24
issue =
pages = 151–162
doi =
id =
url =
format =
accessdate =
quotes =

*cite journal
last = Krall
first = H. L.
coauthors = Fink, O.
year = 1948
month =
title = A New Class of Orthogonal Polynomials: The Bessel Polynomials
journal = Trans. Amer. Math. Soc.
volume = 65
issue =
pages = 100–115
doi =
id =
url =
format =
accessdate =
quotes =

*cite web
url = http://www.research.att.com/~njas/sequences
title = The On-Line Encyclopedia of Integer Sequences
accessdate = 2006-08-16
accessmonthday =
accessyear =
author =Sloane, N. J. A.
last =
first =
coauthors =
date =
year =
month =
format = HTML
work =
publisher =
pages =
language =
archiveurl =
archivedate =
(See sequences [http://www.research.att.com/~njas/sequences/A001497 A001497] , [http://www.research.att.com/~njas/sequences/A001498 A001498] , and [http://www.research.att.com/~njas/sequences/A104548 A104548] )
*cite journal
last = Dita
first = P.
coauthors = Grama, N.
year = 2006
month = May 24
title = On Adomian’s Decomposition Method for Solving Differential Equations
journal = arXiv:solv-int/9705008
volume = 1
issue =
pages =
doi =
id =
url = http://arxiv.org/pdf/solv-int/9705008
format = PDF
accessdate = 2006-08-16
quotes =

*cite book
last=Grosswald
first=E.
coauthors=
title=Bessel Polynomials (Lecture Notes in Mathematics)
year=1978
publisher=Springer
location= New York
id=ISBN 0-387-09104-1

*cite book
last=Roman
first=S.
coauthors=
title=The Umbral Calculus (The Bessel Polynomials &sect;4.1.7)
year= 1984
location= New York
id=ISBN 0-486-44139-3

*cite web
url = http://www.math.ku.dk/~berg/manus/bessel.pdf
title = Linearization coefficients of Bessel polynomials and properties of Student-t distributions
accessdate = 2006-08-16
accessmonthday =
accessyear =
author =
last = Berg
first = Christian
coauthors = Vignat, C.
date =
year = 2000
month =
format = PDF
work =
publisher =
pages =
language = English
archiveurl =
archivedate =

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