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[1]
yn(x)=\sum
| |||||
\left( | |||||
k=0 |
x | |
2 |
\right)k.
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials[2] [3]
ny | |
\theta | |
n(1/x)=\sum |
| ||||
k=0 |
xn-k | |
2k |
.
The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is
3+15x | |
y | |
3(x)=15x |
2+6x+1
while the third-degree reverse Bessel polynomial is
3+6x | |
\theta | |
3(x)=x |
2+15x+15.
The reverse Bessel polynomial is used in the design of Bessel electronic filters.
The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.
n | |
y | |
n(x)=x |
\thetan(1/x)
y | ||||
|
\theta | ||||
|
where Kn(x) is a modified Bessel function of the second kind, yn(x) is the ordinary polynomial, and θn(x) is the reverse polynomial . For example:[4]
3+15x | |
y | |
3(x)=15x |
2+6x+1=\sqrt{
2 | |
\pix |
The Bessel polynomial may also be defined as a confluent hypergeometric function[5]
yn(x)=2F0(-n,n+1;;-x/2)=\left(
2 | |
x\right) |
-nU\left(-n,-2n,
2 | \left( | |
x\right)= |
2 | |
x\right) |
n+1U\left(n+1,2n+2,
2 | |
x |
\right).
A similar expression holds true for the generalized Bessel polynomials (see below):
yn(x;a,b)=2F0(-n,n+a-1;;-x/b)=\left(
b | |
x\right) |
n+a-1U\left(n+a-1,2n+a,
b | |
x |
\right).
The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:
\theta | ||||
|
-2n-1 | |
L | |
n |
(2x)
from which it follows that it may also be defined as a hypergeometric function:
\theta | ||||
|
1F1(-n;-2n;2x)
where (-2n)n is the Pochhammer symbol (rising factorial).
The Bessel polynomials, with index shifted, have the generating function
infty | ||
\sum | \sqrt{ | |
n=0 |
2 | |
\pi} |
| ||||
x |
ex
K | ||||
|
(x)
tn | |
n! |
infty | |
=1+x\sum | |
n=1 |
\thetan-1(x)
tn | |
n! |
=ex(1-\sqrt{1-2t)}.
t
x
\{\thetan\}n\ge0
infty | |
\sum | |
n=0 |
\thetan(x)
tn | = | |
n! |
1 | |
\sqrt{1-2t |
Similar generating function exists for the
yn
infty | |
\sum | |
n=0 |
yn-1(x)
tn | =\exp\left( | |
n! |
1-\sqrt{1-2xt | |
Upon setting
t=z-xz2/2
infty | |
e | |
n=0 |
yn-1(x)
(z-xz2/2)n | |
n! |
.
The Bessel polynomial may also be defined by a recursion formula:
y0(x)=1
y1(x)=x+1
yn(x)=(2n-1)xyn-1(x)+yn-2(x)
and
\theta0(x)=1
\theta1(x)=x+1
\thetan(x)=(2n-1)\thetan-1
2\theta | |
(x)+x | |
n-2 |
(x)
The Bessel polynomial obeys the following differential equation:
| |||||||||||
x | +2(x+1) |
dyn(x) | |
dx |
-n(n+1)yn(x)=0
and
x |
| -2(x+n) | ||||||
dx2 |
d\thetan(x) | |
dx |
+2n\thetan(x)=0
The Bessel polynomials are orthogonal with respect to the weight
e-2/x
n ≠ m
2\pi | |
\int | |
0 |
i\theta | |
y | |
n\left(e |
\right)
i\theta | |
y | |
m\left(e |
\right)iei\thetad\theta=0
A generalization of the Bessel polynomials have been suggested in literature, as following:
yn(x;\alpha,\beta):=(-1)nn!\left(
x | |
\beta\right) |
n
(-1-2n-\alpha) | ||
L | \left( | |
n |
\beta | |
x\right), |
\thetan(x;\alpha,\beta):=
n! | |
(-\beta)n |
(-1-2n-\alpha) | |
L | |
n |
(\betax)=xn
y | ||||
|
The explicit coefficients of the
yn(x;\alpha,\beta)
yn(x;\alpha,\beta)=
n\binom{n}{k}(n+k+\alpha-2) | |
\sum | |
k=0 |
\underline{k
\thetan(x;\alpha,\beta)
\thetan(x;\alpha,
n\binom{n}{k}(2n-k+\alpha-2) | |
\beta)=\sum | |
k=0 |
\underline{n-k
For the weighting function
\rho(x;\alpha,\beta):={}1F
|
0=\ointc\rho(x;\alpha,\beta)yn(x;\alpha,\beta)ym(x;\alpha,\beta)dx
They specialize to the Bessel polynomials for α = β = 2, in which situation ρ(x) = exp(−2/x).
The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :
(\alpha,\beta) | ||
B | (x)= | |
n |
| \left( | ||||||||||||
|
d | |
dx |
\right)n(x\alpha+2n
| ||||
e |
)
where a are normalization coefficients.
According to this generalization we have the following generalized differential equation for associated Bessel polynomials:
| |||||||||||||
x |
+[(\alpha+2)x+\beta]
| ||||||||||
dx |
-\left[n(\alpha+n+1)+
m\beta | |
x |
\right]
(\alpha,\beta) | |
B | |
n,m |
(x)=0
where
0\leqm\leqn
(\alpha,\beta) | ||
B | (x)= | |
n,m |
| \left( | ||||||||||||
|
d | |
dx |
\right)n-m(x\alpha+2n
| ||||
e |
)
If one denotes the zeros of
yn(x;\alpha,\beta)
(n) | |
\alpha | |
k |
(\alpha,\beta)
\thetan(x;\alpha,\beta)
(n) | |
\beta | |
k |
(\alpha,\beta)
2 | |
n(n+\alpha-1) |
(n) | ||
\le\alpha | (\alpha,2)\le | |
k |
2 | |
n+\alpha-1 |
,
n+\alpha-1 | |
2 |
(n) | ||
\le\beta | (\alpha,2)\le | |
k |
n(n+\alpha-1) | |
2 |
,
\alpha\ge2
Sharper results can be said if one resorts to more powerful theorems regarding the estimates of zeros of polynomials (more concretely, the Parabola Theorem of Saff and Varga, or differential equations techniques).[6] One result is the following:[7]
2 | |||
|
(n) | ||
\le\alpha | (\alpha,2)\le | |
k |
2 | |
n+\alpha-1 |
.
The Bessel polynomials
yn(x)
n=5
\begin{align} y0(x)&=1\\ y1(x)&=x+1\\ y2(x)&=3x2+3x+1\\ y3(x)&=15x3+15x2+6x+1\\ y4(x)&=105x4+105x3+45x2+10x+1\\ y5(x)&=945x5+945x4+420x3+105x2+15x+1 \end{align}
No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.[9] The reverse Bessel polynomials are obtained by reversing the coefficients.Equivalently, .This results in the following:
\begin{align} \theta0(x)&=1\\ \theta1(x)&=x+1\\ \theta2(x)&=x2+3x+3\\ \theta3(x)&=x3+6x2+15x+15\\ \theta4(x)&=x4+10x3+45x2+105x+105\\ \theta5(x)&=x5+15x4+105x3+420x2+945x+945\\ \end{align}
. Emil Grosswald . Bessel Polynomials (Lecture Notes in Mathematics) . 1978 . Springer . New York . 978-0-387-09104-4 . Grosswald .