Milne model

2013-10-26T01:02:03Z

99.29.152.248: Astronomer's don't measure spatial curvature. That's performed by experimental cosmologists through detection of CMB, BAO, SNe, etc.

{{More footnotes|date=September 2009}}
In [[mathematics]], the '''Bessel polynomials''' are an [[orthogonal polynomials|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 & Frink, 1948)

:<math>y_n(x)=\sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\,\left(\frac{x}{2}\right)^k</math>

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

:<math>\theta_n(x)=x^n\,y_n(1/x)=\sum_{k=0}^n\frac{(2n-k)!}{(n-k)!k!}\,\frac{x^k}{2^{n-k}}</math>

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

:<math>y_3(x)=15x^3+15x^2+6x+1\,</math>

while the third-degree reverse Bessel polynomial is

:<math>\theta_3(x)=x^3+6x^2+15x+15\,</math>

The reverse Bessel polynomial is used in the design of [[Bessel filter|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.
:<math>y_n(x)=\,x^{n}\theta_n(1/x)\,</math>
:<math>\theta_n(x)=\sqrt{\frac{2}{\pi}}\,x^{n+1/2}e^{x}K_{n+ \frac 1 2}(x)</math>
:<math>y_n(x)=\sqrt{\frac{2}{\pi x}}\,e^{1/x}K_{n+\frac 1 2}(1/x)</math>

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 [[confluent hypergeometric function]] (Dita, 2006)

:<math>y_n(x)=\,_2F_0(-n,n+1;;-x/2)= \left(\frac 2 x\right)^{-n} U\left(-n,-2n,\frac 2 x\right)= \left(\frac 2 x\right)^{n+1} U\left(n+1,2n+2,\frac 2 x \right).</math>

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

:<math>\theta_n(x)=\frac{n!}{(-2)^n}\,L_n^{-2n-1}(2x)</math>

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

:<math>\theta_n(x)=\frac{(-2n)_n}{(-2)^n}\,\,_1F_1(-n;-2n;-2x)</math>

where (−2''n'')''n'' is the [[Pochhammer symbol]] (rising factorial).

===Generating function===
The Bessel polynomials have the generating function
:<math>\sum_{n=0} \sqrt{\frac 2 \pi} x^{n+\frac 1 2} e^x K_{n-\frac 1 2}(x) \frac {t^n}{n!}= e^{x(1-\sqrt{1-2t})}.</math>

=== Recursion ===

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

:<math>y_0(x)=1\,</math>
:<math>y_1(x)=x+1\,</math>
:<math>y_n(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,</math>

and

:<math>\theta_0(x)=1\,</math>
:<math>\theta_1(x)=x+1\,</math>
:<math>\theta_n(x)=(2n\!-\!1)\theta_{n-1}(x)+x^2\theta_{n-2}(x)\,</math>

=== Differential equation ===

The Bessel polynomial obeys the following differential equation:

:<math>x^2\frac{d^2y_n(x)}{dx^2}+2(x\!+\!1)\frac{dy_n(x)}{dx}-n(n+1)y_n(x)=0</math>

and

:<math>x\frac{d^2\theta_n(x)}{dx^2}-2(x\!+\!n)\frac{d\theta_n(x)}{dx}+2n\,\theta_n(x)=0</math>

==Generalization==
===Explicit Form===
A generalization of the Bessel polynomials have been suggested in literature (Krall, Fink), as following:

:<math>y_n(x;\alpha,\beta):= (-1)^n n! \left(\frac x \beta\right)^n L_n^{(1-2n-\alpha)}\left(\frac \beta x\right),</math>
the corresponding reverse polynomials are
:<math>\theta_n(x;\alpha, \beta):= \frac{n!}{(-\beta)^n}L_n^{(1-2n-\alpha)}(\beta x)=x^n y_n\left(\frac 1 x;\alpha,\beta\right).</math>

For the weighting function
:<math>\rho(x;\alpha,\beta):= \, _1F_1\left(1,\alpha-1,-\frac \beta x\right)</math>
they are orthogonal, for the relation

:<math>0= \oint_c\rho(x;\alpha,\beta)y_n(x;\alpha,\beta) y_m(x;\alpha,\beta)\mathrm d x</math>
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 :

:<math>B_n^{(\alpha,\beta)}(x)=\frac{a_n^{(\alpha,\beta)}}{x^{\alpha} e^{\frac{(-\beta)}{x}}} \left(\frac{d}{dx}\right)^n (x^{\alpha+2n} e^{\frac{(-\beta)}{x}})</math>

where ''a''{{su|b=''n''|p=(α, β)}} are normalization coefficients.

===Associated Bessel polynomials===

According to this generalization we have the following generalized associated Bessel polynomials differential equation:

:<math>x^2\frac{d^2B_{n,m}^{(\alpha,\beta)}(x)}{dx^2} + [(\alpha+2)x+\beta]\frac{dB_{n,m}^{(\alpha,\beta)}(x)}{dx} - \left[ n(\alpha+n+1) + \frac{m \beta}{x} \right] B_{n,m}^{(\alpha,\beta)}(x)=0</math>

where <math>0\leq m\leq n</math>. The solutions are,

:<math>B_{n,m}^{(\alpha,\beta)}(x)=\frac{a_{n,m}^{(\alpha,\beta)}}{x^{\alpha+m} e^{\frac{(-\beta)}{x}}} \left(\frac{d}{dx}\right)^{n-m} (x^{\alpha+2n} e^{\frac{(-\beta)}{x}})</math>

== Particular values ==

:<math>
\begin{align}
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
\end{align}
</math>

== References ==
*{{cite journal
| last =Carlitz
| first = Leonard
| authorlink = Leonard Carlitz
| coauthors =
| year = 1957
| month =
| title = A Note on the Bessel Polynomials
| journal = Duke Math. J.
| volume = 24
| issue = 2
| pages = 151–162
| doi = 10.1215/S0012-7094-57-02421-3
| mr = 0085360
}}
*{{cite journal
| last = Krall
| first = H. L.
| coauthors = Frink, O.
| year = 1948
| month =
| title = A New Class of Orthogonal Polynomials: The Bessel Polynomials
| journal = Trans. Amer. Math. Soc.
| volume = 65
| issue = 1
| pages = 100–115
| doi = 10.2307/1990516
| jstor = 1990516
| accessdate =
| quotes =
}}
*{{cite web
| title = The [[On-Line Encyclopedia of Integer Sequences]]
| accessdate = 2006-08-16
| author =Sloane, N. J. A.
| last =
| first =
| coauthors =
| date =
| year =
| month =
| work =
| publisher =
| pages =
}} (See sequences {{OEIS2C|A001497}}, {{OEIS2C|A001498}}, and {{OEIS2C|A104548}})
*{{cite arxiv
| last1 = Dita
| first1 = P.
| last2=Grama
| first2= Grama, N.
| year = 2006
| month = May 24
| title = On Adomian’s Decomposition Method for Solving Differential Equations
| eprint = solv-int/9705008
| quotes =
| class = solv-int
}}
*{{cite journal
| last1 = Fakhri
| first1 = H.
| last2= Chenaghlou
| first2 = A.
| year = 2006
| month =
| title = Ladder operators and recursion relations for the associated Bessel polynomials
| journal = Physics Letters A
| volume = 358
| issue = 5–6
| pages = 345–353
| doi = 10.1016/j.physleta.2006.05.070
| bibcode=2006PhLA..358..345F
| accessdate =
| quotes =
}}
*{{cite book
|last=Grosswald
|first=E.
|authorlink=Emil Grosswald
|coauthors=
|title=Bessel Polynomials (Lecture Notes in Mathematics)
|year=1978
|publisher=Springer
|location= New York
|isbn=0-387-09104-1
}}
*{{cite book
|last=Roman
|first=S.
|coauthors=
|title=The Umbral Calculus (The Bessel Polynomials §4.1.7)
|year= 1984
|publisher=Academic Press
|location= New York
|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
| author =
| last = Berg
| first = Christian
| coauthors = Vignat, C.
| date =
| year = 2000
| month =
| format = PDF
| work =
| publisher =
| pages =
}}

==External links==
* {{springer|title=Bessel polynomials|id=p/b110410}}
*{{MathWorld|title=Bessel Polynomial|urlname=BesselPolynomial}}
*{{SloanesRef |sequencenumber=A001498|name=Coefficients of Bessel polynomials }}

[[Category:Orthogonal polynomials]]
[[Category:Special hypergeometric functions]]

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Milne model