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)

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{(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-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[edit]

Definition in terms of Bessel functions[edit]

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

where Kn(x) is a modified Bessel function of the second kind and yn(x) is the reverse polynomial (pag 7 and 34 Grosswald 1978).

Definition as a hypergeometric function[edit]

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 (rising 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).

Generating function[edit]

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[edit]

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[edit]

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[edit]

Explicit Form[edit]

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[edit]

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[edit]

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

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[edit]


\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.

References[edit]

External links[edit]