# Bessel polynomials

(Redirected from Bessel polynomial)

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)

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

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

$\theta_n(x)=x^n\,y_n(1/x)=\sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\,\frac{x^{n-k}}{2^{k}}$

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

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

while the third-degree reverse Bessel polynomial is

$\theta_3(x)=x^3+6x^2+15x+15\,$

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.

$y_n(x)=\,x^{n}\theta_n(1/x)\,$
$y_n(x)=\sqrt{\frac{2}{\pi x}}\,e^{1/x}K_{n+\frac 1 2}(1/x)$
$\theta_n(x)=\sqrt{\frac{2}{\pi}}\,x^{n+1/2}e^{x}K_{n+ \frac 1 2}(x)$

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]

$y_3(x)=15x^3+15x^2+6x+1 = \sqrt{\frac{2}{\pi x}}\,e^{1/x}K_{3+\frac 1 2}(1/x)$

### Definition as a hypergeometric function

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

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

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

$\theta_n(x)=\frac{n!}{(-2)^n}\,L_n^{-2n-1}(2x)$

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

$\theta_n(x)=\frac{(-2n)_n}{(-2)^n}\,\,_1F_1(-n;-2n;-2x)$

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

The inversion for monomials is given by

$\frac{(2x)^n}{n!}=(-1)^n \sum_{j=0}^n \frac{n+1}{j+1}{j+1\choose n-j}L_j^{-2j-1}(2x)= \frac{2^n}{n!}\sum_{i=0}^n i!(2i+1){2n+1\choose n-i}x^i L_i^{(-2i-1)}\left(\frac 1 x\right).$

### Generating function

The Bessel polynomials have the generating function

$\sum_{n=0} \sqrt{\frac 2 \pi} x^{n+\frac 1 2} e^x K_{n-\frac 1 2}(x) \frac {t^n}{n!}= e^{x(1-\sqrt{1-2t})}.$

### 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

$\theta_0(x)=1\,$
$\theta_1(x)=x+1\,$
$\theta_n(x)=(2n\!-\!1)\theta_{n-1}(x)+x^2\theta_{n-2}(x)\,$

### Differential equation

The Bessel polynomial obeys the following differential equation:

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

and

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

## Generalization

### Explicit Form

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

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

the corresponding reverse polynomials are

$\theta_n(x;\alpha, \beta):= \frac{n!}{(-\beta)^n}L_n^{(1-2n-\alpha)}(\beta x)=x^n y_n\left(\frac 1 x;\alpha,\beta\right).$

For the weighting function

$\rho(x;\alpha,\beta):= \, _1F_1\left(1,\alpha-1,-\frac \beta x\right)$

they are orthogonal, for the relation

$0= \oint_c\rho(x;\alpha,\beta)y_n(x;\alpha,\beta) y_m(x;\alpha,\beta)\mathrm d x$

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 :

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

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:

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

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

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

## Particular values

\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}

none of which factor. Filaseta and Trifonov (Journal for Pure and Applied Mathematics, 550:125-140, 2002) proved that all Bessel polynomials are irreducible.