Bessel polynomials

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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)

${\displaystyle y_{n}(x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,\left({\frac {x}{2}}\right)^{k}}$

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

${\displaystyle \theta _{n}(x)=x^{n}\,y_{n}(1/x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,{\frac {x^{n-k}}{2^{k}}}}$

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

${\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1\,}$

while the third-degree reverse Bessel polynomial is

${\displaystyle \theta _{3}(x)=x^{3}+6x^{2}+15x+15\,}$

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 functions from which the polynomial draws its name.

${\displaystyle y_{n}(x)=\,x^{n}\theta _{n}(1/x)\,}$
${\displaystyle y_{n}(x)={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{n+{\frac {1}{2}}}(1/x)}$
${\displaystyle \theta _{n}(x)={\sqrt {\frac {2}{\pi }}}\,x^{n+1/2}e^{x}K_{n+{\frac {1}{2}}}(x)}$

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]

${\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{3+{\frac {1}{2}}}(1/x)}$

Definition as a hypergeometric function

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

${\displaystyle y_{n}(x)=\,_{2}F_{0}(-n,n+1;;-x/2)=\left({\frac {2}{x}}\right)^{-n}U\left(-n,-2n,{\frac {2}{x}}\right)=\left({\frac {2}{x}}\right)^{n+1}U\left(n+1,2n+2,{\frac {2}{x}}\right).}$

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

${\displaystyle \theta _{n}(x)={\frac {n!}{(-2)^{n}}}\,L_{n}^{-2n-1}(2x)}$

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

${\displaystyle \theta _{n}(x)={\frac {(-2n)_{n}}{(-2)^{n}}}\,\,_{1}F_{1}(-n;-2n;-2x)}$

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

The inversion for monomials is given by

${\displaystyle {\frac {(2x)^{n}}{n!}}=(-1)^{n}\sum _{j=0}^{n}{\frac {n+1}{j+1}}{j+1 \choose n-j}L_{j}^{-2j-1}(2x)={\frac {2^{n}}{n!}}\sum _{i=0}^{n}i!(2i+1){2n+1 \choose n-i}x^{i}L_{i}^{(-2i-1)}\left({\frac {1}{x}}\right).}$

Generating function

The Bessel polynomials, with index shifted, have the generating function

${\displaystyle \sum _{n=0}^{\infty }{\sqrt {\frac {2}{\pi }}}x^{n+{\frac {1}{2}}}e^{x}K_{n-{\frac {1}{2}}}(x){\frac {t^{n}}{n!}}=1+x\sum _{n=1}^{\infty }\theta _{n-1}(x){\frac {t^{n}}{n!}}=e^{x(1-{\sqrt {1-2t}})}.}$

Differentiating with respect to ${\displaystyle t}$, cancelling ${\displaystyle x}$, yields the generating function for the polynomials ${\displaystyle \{\theta _{n}\}_{n\geq 0}}$

${\displaystyle \sum _{n=0}^{\infty }\theta _{n}(x){\frac {t^{n}}{n!}}={\frac {1}{\sqrt {1-2t}}}e^{x(1-{\sqrt {1-2t}})}.}$

Recursion

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

${\displaystyle y_{0}(x)=1\,}$
${\displaystyle y_{1}(x)=x+1\,}$
${\displaystyle y_{n}(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,}$

and

${\displaystyle \theta _{0}(x)=1\,}$
${\displaystyle \theta _{1}(x)=x+1\,}$
${\displaystyle \theta _{n}(x)=(2n\!-\!1)\theta _{n-1}(x)+x^{2}\theta _{n-2}(x)\,}$

Differential equation

The Bessel polynomial obeys the following differential equation:

${\displaystyle x^{2}{\frac {d^{2}y_{n}(x)}{dx^{2}}}+2(x\!+\!1){\frac {dy_{n}(x)}{dx}}-n(n+1)y_{n}(x)=0}$

and

${\displaystyle x{\frac {d^{2}\theta _{n}(x)}{dx^{2}}}-2(x\!+\!n){\frac {d\theta _{n}(x)}{dx}}+2n\,\theta _{n}(x)=0}$

Generalization

Explicit Form

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

${\displaystyle y_{n}(x;\alpha ,\beta ):=(-1)^{n}n!\left({\frac {x}{\beta }}\right)^{n}L_{n}^{(1-2n-\alpha )}\left({\frac {\beta }{x}}\right),}$

the corresponding reverse polynomials are

${\displaystyle \theta _{n}(x;\alpha ,\beta ):={\frac {n!}{(-\beta )^{n}}}L_{n}^{(1-2n-\alpha )}(\beta x)=x^{n}y_{n}\left({\frac {1}{x}};\alpha ,\beta \right).}$

For the weighting function

${\displaystyle \rho (x;\alpha ,\beta ):=\,_{1}F_{1}\left(1,\alpha -1,-{\frac {\beta }{x}}\right)}$

they are orthogonal, for the relation

${\displaystyle 0=\oint _{c}\rho (x;\alpha ,\beta )y_{n}(x;\alpha ,\beta )y_{m}(x;\alpha ,\beta )\mathrm {d} x}$

holds for mn 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 :

${\displaystyle B_{n}^{(\alpha ,\beta )}(x)={\frac {a_{n}^{(\alpha ,\beta )}}{x^{\alpha }e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})}$

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:

${\displaystyle x^{2}{\frac {d^{2}B_{n,m}^{(\alpha ,\beta )}(x)}{dx^{2}}}+[(\alpha +2)x+\beta ]{\frac {dB_{n,m}^{(\alpha ,\beta )}(x)}{dx}}-\left[n(\alpha +n+1)+{\frac {m\beta }{x}}\right]B_{n,m}^{(\alpha ,\beta )}(x)=0}$

where ${\displaystyle 0\leq m\leq n}$. The solutions are,

${\displaystyle B_{n,m}^{(\alpha ,\beta )}(x)={\frac {a_{n,m}^{(\alpha ,\beta )}}{x^{\alpha +m}e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n-m}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})}$

Particular values

The first five Bessel Polynomials are expressed as:

{\displaystyle {\begin{aligned}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{aligned}}}

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, ${\textstyle \theta _{k}(x)=x^{k}y_{k}(1/x)}$. This results in the following:

{\displaystyle {\begin{aligned}\theta _{0}(x)&=1\\\theta _{1}(x)&=x+1\\\theta _{2}(x)&=x^{2}+3x+3\\\theta _{3}(x)&=x^{3}+6x^{2}+15x+15\\\theta _{4}(x)&=x^{4}+10x^{3}+45x^{2}+105x+105\\\theta _{5}(x)&=x^{5}+15x^{4}+105x^{3}+420x^{2}+945x+945\\\end{aligned}}}

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. doi:10.1515/crll.2002.069.