Bessel polynomials

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 & Fink, 1948)

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

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

$y_3(x)=15x^3%2B15x^2%2B6x%2B1\,$

while the third-degree reverse Bessel polynomial is

$\theta_3(x)=x^3%2B6x^2%2B15x%2B15\,$

The reverse Bessel polynomial is used in the design of Bessel electronic filters.

1 Properties
2 Generalization
3 Particular values
4 References
5 External links

Properties

Definition in terms of Bessel functions

The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.

$y_n(x)=\,x^{n}\theta_n(1/x)\,$

$\theta_n(x)=\sqrt{\frac{2}{\pi}}\,x^{n%2B1/2}e^{x}K_{n%2B \frac 1 2}(x)$

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

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 confluent hypergeometric function (Dita, 2006)

$y_n(x)=\,_2F_0(-n,n%2B1;;-x/2)= \left(\frac 2 x\right)^{-n} U\left(-n,-2n,\frac 2 x\right)= \left(\frac 2 x\right)^{n%2B1} U\left(n%2B1,2n%2B2,\frac 2 x \right).$

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).

Generating function

The Bessel polynomials have the generating function

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

Recursion

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

$y_0(x)=1\,$

$y_1(x)=x%2B1\,$

$y_n(x)=(2n\!-\!1)x\,y_{n-1}(x)%2By_{n-2}(x)\,$

and

$\theta_0(x)=1\,$

$\theta_1(x)=x%2B1\,$

$\theta_n(x)=(2n\!-\!1)\theta_{n-1}(x)%2Bx^2\theta_{n-2}(x)\,$

Differential equation

The Bessel polynomial obeys the following differential equation:

$x^2\frac{d^2y_n(x)}{dx^2}%2B2(x\!%2B\!1)\frac{dy_n(x)}{dx}-n(n%2B1)y_n(x)=0$

and

$x\frac{d^2\theta_n(x)}{dx^2}-2(x\!%2B\!n)\frac{d\theta_n(x)}{dx}%2B2n\,\theta_n(x)=0$

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 $m \neq n$ and $c$ a curve surrounding the 0 point.

They specialize to the Bessel polynomials for $\alpha=\beta=2$ , in which situation $\rho(x)=e^{-2/x}$ .

Rodrigues formula for Bessel polynomials

The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :

$B_n^{(\alpha,\beta)}(x)=\frac{a_n^{(\alpha,\beta)}}{x^{\alpha} e^{\frac{(-\beta)}{x}}} \left(\frac{d}{dx}\right)^n (x^{\alpha%2B2n} e^{\frac{(-\beta)}{x}})$

where $a_n^{(\alpha,\beta)}$ are normalization coefficients.

Associated Bessel polynomials

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

$x^2\frac{d^2B_{n,m}^{(\alpha,\beta)}(x)}{dx^2} %2B [(\alpha%2B2)x%2B\beta]\frac{dB_{n,m}^{(\alpha,\beta)}(x)}{dx} - \left[ n(\alpha%2Bn%2B1) %2B \frac{m \beta}{x} \right] B_{n,m}^{(\alpha,\beta)}(x)=0$

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

$B_{n,m}^{(\alpha,\beta)}(x)=\frac{a_{n,m}^{(\alpha,\beta)}}{x^{\alpha%2Bm} e^{\frac{(-\beta)}{x}}} \left(\frac{d}{dx}\right)^{n-m} (x^{\alpha%2B2n} e^{\frac{(-\beta)}{x}})$

Particular values

$\begin{align} y_0(x) & = 1 \\ y_1(x) & = x %2B 1 \\ y_2(x) & = 3x^2%2B 3x %2B 1 \\ y_3(x) & = 15x^3%2B 15x^2%2B 6x %2B 1 \\ y_4(x) & = 105x^4%2B105x^3%2B 45x^2%2B 10x %2B 1 \\ y_5(x) & = 945x^5%2B945x^4%2B420x^3%2B105x^2%2B15x%2B1 \end{align}$

References

Carlitz, Leonard (1957). "A Note on the Bessel Polynomials". Duke Math. J. 24 (2): 151–162. doi:10.1215/S0012-7094-57-02421-3. MR 0085360.
Krall, H. L.; Fink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516. JSTOR 1990516.
"The On-Line Encyclopedia of Integer Sequences". http://www.research.att.com/~njas/sequences. Retrieved 2006-08-16. (See sequences A001497, A001498, and A104548)
Dita, P.; Grama, Grama, N. (May 24 2006). "On Adomian’s Decomposition Method for Solving Differential Equations". arXiv:solv-int/9705008 [solv-int].
Fakhri, H.; Chenaghlou, A. (2006). "Ladder operators and recursion relations for the associated Bessel polynomials". Physics Letters A 358 (5–6): 345–353. Bibcode 2006PhLA..358..345F. doi:10.1016/j.physleta.2006.05.070.
Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 0-387-09104-1.
Roman, S. (1984). The Umbral Calculus (The Bessel Polynomials §4.1.7). New York: Academic Press. ISBN 0-486-44139-3.
Berg, Christian; Vignat, C. (2000). "Linearization coefficients of Bessel polynomials and properties of Student-t distributions" (PDF). http://www.math.ku.dk/~berg/manus/bessel.pdf. Retrieved 2006-08-16.

External links

Weisstein, Eric W., "Bessel Polynomial" from MathWorld.
Sloane's A001498 : Coefficients of Bessel polynomials. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.