Bessel polynomials

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In mathematics there are a number of different but closely related definitions of the Bessel polynomial. 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 polynomial (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 order Bessel polynomial is

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

while the third order 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

Contents

[edit] Properties

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

where Kn(x) is a modified Bessel function of the second kind.

[edit] Definition as a hypergeometric function

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

y_n(x)=\,_2F_0(-n,n+1;;-x/2)

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).

[edit] 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)\,

[edit] 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

[edit] Particular values

(See also Sloan's A001498)

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\,

[edit] See also

[edit] References

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