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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 & Fink, 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
Definition in terms of Bessel functions

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

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

where $$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+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).
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 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 :

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

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:

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

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}

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. MR0085360.
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". Retrieved 2006-08-16. (See sequences OEIS A001497, OEIS A001498, and OEIS 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). Retrieved 2006-08-16.