Mathematics concept
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] : 101
y
n
(
x
)
=
∑
k
=
0
n
(
n
+
k
)
!
(
n
−
k
)
!
k
!
(
x
2
)
k
.
{\displaystyle 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 [ 2] : 8 [ 3] : 15
θ
n
(
x
)
=
x
n
y
n
(
1
/
x
)
=
∑
k
=
0
n
(
n
+
k
)
!
(
n
−
k
)
!
k
!
x
n
−
k
2
k
.
{\displaystyle \theta _{n}(x)=x^{n}\,y_{n}(1/x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,{\frac {x^{n-k}}{2^{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
)
=
15
x
3
+
15
x
2
+
6
x
+
1
{\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1}
while the third-degree reverse Bessel polynomial is
θ
3
(
x
)
=
x
3
+
6
x
2
+
15
x
+
15.
{\displaystyle \theta _{3}(x)=x^{3}+6x^{2}+15x+15.}
The reverse Bessel polynomial is used in the design of Bessel electronic filters .
Definition in terms of Bessel functions [ edit ] The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.
y
n
(
x
)
=
x
n
θ
n
(
1
/
x
)
{\displaystyle y_{n}(x)=\,x^{n}\theta _{n}(1/x)\,}
y
n
(
x
)
=
2
π
x
e
1
/
x
K
n
+
1
2
(
1
/
x
)
{\displaystyle y_{n}(x)={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{n+{\frac {1}{2}}}(1/x)}
θ
n
(
x
)
=
2
π
x
n
+
1
/
2
e
x
K
n
+
1
2
(
x
)
{\displaystyle \theta _{n}(x)={\sqrt {\frac {2}{\pi }}}\,x^{n+1/2}e^{x}K_{n+{\frac {1}{2}}}(x)}
where K n x ) is a modified Bessel function of the second kind , y n x ) is the ordinary polynomial, and θ n x ) is the reverse polynomial .[ 2] : 7, 34 [ 4]
y
3
(
x
)
=
15
x
3
+
15
x
2
+
6
x
+
1
=
2
π
x
e
1
/
x
K
3
+
1
2
(
1
/
x
)
{\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{3+{\frac {1}{2}}}(1/x)}
Definition as a hypergeometric function [ edit ] The Bessel polynomial may also be defined as a confluent hypergeometric function [ 5] : 8
y
n
(
x
)
=
2
F
0
(
−
n
,
n
+
1
;
;
−
x
/
2
)
=
(
2
x
)
−
n
U
(
−
n
,
−
2
n
,
2
x
)
=
(
2
x
)
n
+
1
U
(
n
+
1
,
2
n
+
2
,
2
x
)
.
{\displaystyle y_{n}(x)=\,_{2}F_{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).}
A similar expression holds true for the generalized Bessel polynomials (see below):[ 2] : 35
y
n
(
x
;
a
,
b
)
=
2
F
0
(
−
n
,
n
+
a
−
1
;
;
−
x
/
b
)
=
(
b
x
)
n
+
a
−
1
U
(
n
+
a
−
1
,
2
n
+
a
,
b
x
)
.
{\displaystyle y_{n}(x;a,b)=\,_{2}F_{0}(-n,n+a-1;;-x/b)=\left({\frac {b}{x}}\right)^{n+a-1}U\left(n+a-1,2n+a,{\frac {b}{x}}\right).}
The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial :
θ
n
(
x
)
=
n
!
(
−
2
)
n
L
n
−
2
n
−
1
(
2
x
)
{\displaystyle \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:
θ
n
(
x
)
=
(
−
2
n
)
n
(
−
2
)
n
1
F
1
(
−
n
;
−
2
n
;
2
x
)
{\displaystyle \theta _{n}(x)={\frac {(-2n)_{n}}{(-2)^{n}}}\,\,_{1}F_{1}(-n;-2n;2x)}
where (−2n )n Pochhammer symbol (rising factorial).
Generating function [ edit ] The Bessel polynomials, with index shifted, have the generating function
∑
n
=
0
∞
2
π
x
n
+
1
2
e
x
K
n
−
1
2
(
x
)
t
n
n
!
=
1
+
x
∑
n
=
1
∞
θ
n
−
1
(
x
)
t
n
n
!
=
e
x
(
1
−
1
−
2
t
)
.
{\displaystyle \sum _{n=0}^{\infty }{\sqrt {\frac {2}{\pi }}}x^{n+{\frac {1}{2}}}e^{x}K_{n-{\frac {1}{2}}}(x){\frac {t^{n}}{n!}}=1+x\sum _{n=1}^{\infty }\theta _{n-1}(x){\frac {t^{n}}{n!}}=e^{x(1-{\sqrt {1-2t}})}.}
Differentiating with respect to
t
{\displaystyle t}
x
{\displaystyle x}
{
θ
n
}
n
≥
0
{\displaystyle \{\theta _{n}\}_{n\geq 0}}
∑
n
=
0
∞
θ
n
(
x
)
t
n
n
!
=
1
1
−
2
t
e
x
(
1
−
1
−
2
t
)
.
{\displaystyle \sum _{n=0}^{\infty }\theta _{n}(x){\frac {t^{n}}{n!}}={\frac {1}{\sqrt {1-2t}}}e^{x(1-{\sqrt {1-2t}})}.}
Similar generating function exists for the
y
n
{\displaystyle y_{n}}
[ 1] : 106
∑
n
=
0
∞
y
n
−
1
(
x
)
t
n
n
!
=
exp
(
1
−
1
−
2
x
t
x
)
.
{\displaystyle \sum _{n=0}^{\infty }y_{n-1}(x){\frac {t^{n}}{n!}}=\exp \left({\frac {1-{\sqrt {1-2xt}}}{x}}\right).}
Upon setting
t
=
z
−
x
z
2
/
2
{\displaystyle t=z-xz^{2}/2}
exponential function :[ 1] : 107
e
z
=
∑
n
=
0
∞
y
n
−
1
(
x
)
(
z
−
x
z
2
/
2
)
n
n
!
.
{\displaystyle e^{z}=\sum _{n=0}^{\infty }y_{n-1}(x){\frac {(z-xz^{2}/2)^{n}}{n!}}.}
The Bessel polynomial may also be defined by a recursion formula:
y
0
(
x
)
=
1
{\displaystyle y_{0}(x)=1\,}
y
1
(
x
)
=
x
+
1
{\displaystyle y_{1}(x)=x+1\,}
y
n
(
x
)
=
(
2
n
−
1
)
x
y
n
−
1
(
x
)
+
y
n
−
2
(
x
)
{\displaystyle y_{n}(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,}
and
θ
0
(
x
)
=
1
{\displaystyle \theta _{0}(x)=1\,}
θ
1
(
x
)
=
x
+
1
{\displaystyle \theta _{1}(x)=x+1\,}
θ
n
(
x
)
=
(
2
n
−
1
)
θ
n
−
1
(
x
)
+
x
2
θ
n
−
2
(
x
)
{\displaystyle \theta _{n}(x)=(2n\!-\!1)\theta _{n-1}(x)+x^{2}\theta _{n-2}(x)\,}
Differential equation [ edit ] The Bessel polynomial obeys the following differential equation:
x
2
d
2
y
n
(
x
)
d
x
2
+
2
(
x
+
1
)
d
y
n
(
x
)
d
x
−
n
(
n
+
1
)
y
n
(
x
)
=
0
{\displaystyle x^{2}{\frac {d^{2}y_{n}(x)}{dx^{2}}}+2(x\!+\!1){\frac {dy_{n}(x)}{dx}}-n(n+1)y_{n}(x)=0}
and
x
d
2
θ
n
(
x
)
d
x
2
−
2
(
x
+
n
)
d
θ
n
(
x
)
d
x
+
2
n
θ
n
(
x
)
=
0
{\displaystyle x{\frac {d^{2}\theta _{n}(x)}{dx^{2}}}-2(x\!+\!n){\frac {d\theta _{n}(x)}{dx}}+2n\,\theta _{n}(x)=0}
The Bessel polynomials are orthogonal with respect to the weight
e
−
2
/
x
{\displaystyle e^{-2/x}}
[ 1] : 104
n
≠
m
{\displaystyle n\neq m}
∫
0
2
π
y
n
(
e
i
θ
)
y
m
(
e
i
θ
)
i
e
i
θ
d
θ
=
0
{\displaystyle \int _{0}^{2\pi }y_{n}\left(e^{i\theta }\right)y_{m}\left(e^{i\theta }\right)ie^{i\theta }\mathrm {d} \theta =0}
A generalization of the Bessel polynomials have been suggested in literature, as following:
y
n
(
x
;
α
,
β
)
:=
(
−
1
)
n
n
!
(
x
β
)
n
L
n
(
−
1
−
2
n
−
α
)
(
β
x
)
,
{\displaystyle 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
θ
n
(
x
;
α
,
β
)
:=
n
!
(
−
β
)
n
L
n
(
−
1
−
2
n
−
α
)
(
β
x
)
=
x
n
y
n
(
1
x
;
α
,
β
)
.
{\displaystyle \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).}
The explicit coefficients of the
y
n
(
x
;
α
,
β
)
{\displaystyle y_{n}(x;\alpha ,\beta )}
[ 1] : 108
y
n
(
x
;
α
,
β
)
=
∑
k
=
0
n
(
n
k
)
(
n
+
k
+
α
−
2
)
k
_
(
x
β
)
k
.
{\displaystyle y_{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(n+k+\alpha -2)^{\underline {k}}\left({\frac {x}{\beta }}\right)^{k}.}
Consequently, the
θ
n
(
x
;
α
,
β
)
{\displaystyle \theta _{n}(x;\alpha ,\beta )}
θ
n
(
x
;
α
,
β
)
=
∑
k
=
0
n
(
n
k
)
(
2
n
−
k
+
α
−
2
)
n
−
k
_
x
k
β
n
−
k
.
{\displaystyle \theta _{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(2n-k+\alpha -2)^{\underline {n-k}}{\frac {x^{k}}{\beta ^{n-k}}}.}
For the weighting function
ρ
(
x
;
α
,
β
)
:=
1
F
1
(
1
,
α
−
1
,
−
β
x
)
{\displaystyle \rho (x;\alpha ,\beta ):={}_{1}F_{1}\left(1,\alpha -1,-{\frac {\beta }{x}}\right)}
they are orthogonal, for the relation
0
=
∮
c
ρ
(
x
;
α
,
β
)
y
n
(
x
;
α
,
β
)
y
m
(
x
;
α
,
β
)
d
x
{\displaystyle 0=\oint _{c}\rho (x;\alpha ,\beta )y_{n}(x;\alpha ,\beta )y_{m}(x;\alpha ,\beta )\,\mathrm {d} x}
holds for m ≠ n and c a curve surrounding the 0 point.
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 :
B
n
(
α
,
β
)
(
x
)
=
a
n
(
α
,
β
)
x
α
e
−
β
x
(
d
d
x
)
n
(
x
α
+
2
n
e
−
β
x
)
{\displaystyle 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 (α, β) n
Associated Bessel polynomials [ edit ] According to this generalization we have the following generalized differential equation for associated Bessel polynomials:
x
2
d
2
B
n
,
m
(
α
,
β
)
(
x
)
d
x
2
+
[
(
α
+
2
)
x
+
β
]
d
B
n
,
m
(
α
,
β
)
(
x
)
d
x
−
[
n
(
α
+
n
+
1
)
+
m
β
x
]
B
n
,
m
(
α
,
β
)
(
x
)
=
0
{\displaystyle x^{2}{\frac {d^{2}B_{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
≤
m
≤
n
{\displaystyle 0\leq m\leq n}
B
n
,
m
(
α
,
β
)
(
x
)
=
a
n
,
m
(
α
,
β
)
x
α
+
m
e
−
β
x
(
d
d
x
)
n
−
m
(
x
α
+
2
n
e
−
β
x
)
{\displaystyle 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}}})}
If one denotes the zeros of
y
n
(
x
;
α
,
β
)
{\displaystyle y_{n}(x;\alpha ,\beta )}
α
k
(
n
)
(
α
,
β
)
{\displaystyle \alpha _{k}^{(n)}(\alpha ,\beta )}
θ
n
(
x
;
α
,
β
)
{\displaystyle \theta _{n}(x;\alpha ,\beta )}
β
k
(
n
)
(
α
,
β
)
{\displaystyle \beta _{k}^{(n)}(\alpha ,\beta )}
[ 2] : 82
2
n
(
n
+
α
−
1
)
≤
α
k
(
n
)
(
α
,
2
)
≤
2
n
+
α
−
1
,
{\displaystyle {\frac {2}{n(n+\alpha -1)}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}},}
and
n
+
α
−
1
2
≤
β
k
(
n
)
(
α
,
2
)
≤
n
(
n
+
α
−
1
)
2
,
{\displaystyle {\frac {n+\alpha -1}{2}}\leq \beta _{k}^{(n)}(\alpha ,2)\leq {\frac {n(n+\alpha -1)}{2}},}
for all
α
≥
2
{\displaystyle \alpha \geq 2}
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).[ 2] : 88 [ 6] [ 7]
2
2
n
+
α
−
2
3
≤
α
k
(
n
)
(
α
,
2
)
≤
2
n
+
α
−
1
.
{\displaystyle {\frac {2}{2n+\alpha -{\frac {2}{3}}}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}}.}
The Bessel polynomials
y
n
(
x
)
{\displaystyle y_{n}(x)}
n
=
5
{\displaystyle n=5}
[ 8]
y
0
(
x
)
=
1
y
1
(
x
)
=
x
+
1
y
2
(
x
)
=
3
x
2
+
3
x
+
1
y
3
(
x
)
=
15
x
3
+
15
x
2
+
6
x
+
1
y
4
(
x
)
=
105
x
4
+
105
x
3
+
45
x
2
+
10
x
+
1
y
5
(
x
)
=
945
x
5
+
945
x
4
+
420
x
3
+
105
x
2
+
15
x
+
1
{\displaystyle {\begin{aligned}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{aligned}}}
No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.[ 9]
θ
k
(
x
)
=
x
k
y
k
(
1
/
x
)
{\textstyle \theta _{k}(x)=x^{k}y_{k}(1/x)}
θ
0
(
x
)
=
1
θ
1
(
x
)
=
x
+
1
θ
2
(
x
)
=
x
2
+
3
x
+
3
θ
3
(
x
)
=
x
3
+
6
x
2
+
15
x
+
15
θ
4
(
x
)
=
x
4
+
10
x
3
+
45
x
2
+
105
x
+
105
θ
5
(
x
)
=
x
5
+
15
x
4
+
105
x
3
+
420
x
2
+
945
x
+
945
{\displaystyle {\begin{aligned}\theta _{0}(x)&=1\\\theta _{1}(x)&=x+1\\\theta _{2}(x)&=x^{2}+3x+3\\\theta _{3}(x)&=x^{3}+6x^{2}+15x+15\\\theta _{4}(x)&=x^{4}+10x^{3}+45x^{2}+105x+105\\\theta _{5}(x)&=x^{5}+15x^{4}+105x^{3}+420x^{2}+945x+945\\\end{aligned}}}
^ a b c d e Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials" . Trans. Amer. Math. Soc . 65 (1): 100–115. doi :10.2307/1990516 ^ a b c d e Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics) . New York: Springer. ISBN 978-0-387-09104-4 ^ Berg, Christian; Vignat, Christophe (2008). "Linearization coefficients of Bessel polynomials and properties of Student-t distributions" (PDF) . Constructive Approximation . 27 : 15–32. doi :10.1007/s00365-006-0643-6 . Retrieved 2006-08-16 . ^ Wolfram Alpha example ^ Dita, Petre; Grama, Nicolae (May 14, 1997). "On Adomian's Decomposition Method for Solving Differential Equations". arXiv :solv-int/9705008 ^ Saff, E. B.; Varga, R. S. (1976). "Zero-free parabolic regions for sequences of polynomials". SIAM J. Math. Anal . 7 (3): 344–357. doi :10.1137/0507028 . ^ de Bruin, M. G.; Saff, E. B.; Varga, R. S. (1981). "On the zeros of generalized Bessel polynomials. I". Indag. Math . 84 (1): 1–13. ^ *Sloane, N. J. A. (ed.). "Sequence A001498 (Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ Filaseta, Michael; Trifinov, Ognian (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik . 2002 (550): 125–140. CiteSeerX 10.1.1.6.9538 doi :10.1515/crll.2002.069 .
![{\displaystyle x^{2}{\frac {d^{2}B_{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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4919329568ec11fc84335d890e2e7e1b1cc8cf46)
