- Bessel polynomials
In
mathematics , the Bessel polynomials are an orthogonal sequence ofpolynomial s. 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(x)=sum_{k=0}^nfrac{(n+k)!}{(n-k)!k!},left(frac{x}{2} ight)^k
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials (See Grosswald 1978, Berg 2000).
:heta_n(x)=x^n,y_n(1/x)=sum_{k=0}^nfrac{(2n-k)!}{(n-k)!k!},frac{x^k}{2^{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(x)=15x^3+15x^2+6x+1,
while the third order reverse Bessel polynomial is
:heta_3(x)=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 function s from which the polynomial draws its name.:y_n(x)=,x^{n} heta_n(1/x),:heta_n(x)=sqrt{frac{2}{pi,x^{n+1/2}e^{x}K_{(n+1/2)}(x):y_n(x)=sqrt{frac{2}{pi x,e^{1/x}K_{(n+1/2)}(1/x)where K_n(x) is a modified Bessel function of the second kind and y_n(x) 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(x)=,_2F_0(-n,n+1;;-x/2)
The reverse Bessel polynomial may be defined as a generalized
Laguerre polynomial ::heta_n(x)=frac{n!}{(-2)^n},L_n^{-2n-1}(2x)
from which it follows that it may also be defined as a hypergeometric function:
:heta_n(x)=frac{(-2n)_n}{(-2)^n},,_1F_1(-n;-2n;-2x)
where 2n)_n is the
Pochhammer symbol (rising factorial).Recursion
The Bessel polynomial may also be defined by a recursion formula:
:y_0(x)=1,:y_1(x)=x+1,:y_n(x)=(2n!-!1)x,y_{n-1}(x)+y_{n-2}(x),
and
:heta_0(x)=1,:heta_1(x)=x+1,:heta_n(x)=(2n!-!1) heta_{n-1}(x)+x^2 heta_{n-2}(x),
Differential Equation
The Bessel polynomial obeys the following differential equation:
:x^2frac{d^2y_n(x)}{dx^2}+2(x!+!1)frac{dy_n(x)}{dx}-n(n+1)y_n(x)=0
and
:xfrac{d^2 heta_n(x)}{dx^2}-2(x!+!n)frac{d heta_n(x)}{dx}+2n, heta_n(x)=0
Particular values
(See also [http://www.research.att.com/~njas/sequences/A001498 Sloan's A001498] )
:y_0(x) = 1 ,
:y_1(x) = x + 1 ,
:y_2(x) = 3x^2+ 3x + 1 ,
:y_3(x) = 15x^3+ 15x^2+ 6x + 1 ,
:y_4(x) = 105x^4+105x^3+ 45x^2+ 10x + 1 ,
:y_5(x) = 945x^5+945x^4+420x^3+105x^2+15x+1,
See also
* [http://mathworld.wolfram.com/BesselPolynomial.html Eric Weisstein's Math World]
References
*cite journal
last = Carlitz
first = L.
authorlink =
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.
authorlink =
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 =
authorlink =
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.
authorlink =
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.
authorlink=
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.
authorlink=
coauthors=
title=The Umbral Calculus (The Bessel Polynomials §4.1.7)
year= 1984
publisher=Academic Press
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
authorlink =
coauthors = Vignat, C.
date =
year = 2000
month =
format = PDF
work =
publisher =
pages =
language = English
archiveurl =
archivedate =
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