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 & Frink, 1948)
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 (See Grosswald 1978, Berg 2000).
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 .
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
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 (pg 7 and 34 Grosswald 1978). For example:[ 1]
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
The Bessel polynomial may also be defined as a confluent hypergeometric function (Dita, 2006)
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).}
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).
The inversion for monomials is given by
(
2
x
)
n
n
!
=
(
−
1
)
n
∑
j
=
0
n
n
+
1
j
+
1
(
j
+
1
n
−
j
)
L
j
−
2
j
−
1
(
2
x
)
=
2
n
n
!
∑
i
=
0
n
i
!
(
2
i
+
1
)
(
2
n
+
1
n
−
i
)
x
i
L
i
(
−
2
i
−
1
)
(
1
x
)
.
{\displaystyle {\frac {(2x)^{n}}{n!}}=(-1)^{n}\sum _{j=0}^{n}{\frac {n+1}{j+1}}{j+1 \choose n-j}L_{j}^{-2j-1}(2x)={\frac {2^{n}}{n!}}\sum _{i=0}^{n}i!(2i+1){2n+1 \choose n-i}x^{i}L_{i}^{(-2i-1)}\left({\frac {1}{x}}\right).}
Generating function
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}})}.}
Recursion
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
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}
Generalization
A generalization of the Bessel polynomials have been suggested in literature (Krall, Fink), 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).}
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
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}}})}
Particular values
The first five Bessel Polynomials are expressed as:
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-ordered polynomials with strictly rational coefficients.[ 2]
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}}}
See also
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 . MR 0085360 .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 . JSTOR 1990516 . Sloane, N. J. A. "The On-Line Encyclopedia of Integer Sequences ". OEIS : A001497 OEIS : A001498 OEIS : A104548 Dita, P.; Grama, Grama, N. (May 24, 2006). "On Adomian's Decomposition Method for Solving Differential Equations". arXiv :solv-int/9705008 CS1 maint: multiple names: authors list (link ) 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 .
External links
![{\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)
