Bessel polynomials

From Wikipedia, the free encyclopedia

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

1 Properties
2 Particular values
3 See also
4 References

[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 $K n (x)$ 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 $( - 2 n) n$ 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

Eric Weisstein's Math World

[edit] References

Carlitz, L. (1957). "A Note on the Bessel Polynomials". Duke Math. J. 24: 151-162.
Krall, H. L.; Fink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65: 100-115.
Sloane, N. J. A.. The On-Line Encyclopedia of Integer Sequences (HTML). Retrieved on August 16, 2006. (See sequences A001497,A001498, and A104548)
Dita, P.; Grama, N. (May 24 2006). "On Adomian’s Decomposition Method for Solving Differential Equations" (PDF). arXiv:solv-int/9705008 1. Retrieved on 2006-08-16.
Grosswald, E. (1979). 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 (English) (PDF). Retrieved on August 16, 2006.

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Categories: Orthogonal polynomials | Special hypergeometric functions

Bessel polynomials

From Wikipedia, the free encyclopedia

Contents

[edit] Properties

[edit] Definition in terms of Bessel functions

[edit] Definition as a hypergeometric function

[edit] Recursion

[edit] Differential Equation

[edit] Particular values

[edit] See also

[edit] References

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