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About Bessel polynomials About Bessel polynomials 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
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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
Article 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
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
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
)
=
1
+
6
x
+
15
x
2
+
15
x
3
{\displaystyle y_{3}(x)=1+6x+15x^{2}+15x^{3}}
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.
Contents 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
Definition as a hypergeometric function The Bessel polynomial may also be defined as a confluent hypergeometric function
y
n
(
x
)
=
2
F
0
(
−
n
,
n
+
1
;
;
−
x
/
2
)
=
(
2
x
)
−
n
U
(
−
n
,
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
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
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
Orthogonality The Bessel polynomials are orthogonal with respect to the weight
e
−
2
/
x
{\displaystyle e^{-2/x}}
integrated over the unit circle of the complex plane. In other words, if
n
≠
m
{\displaystyle n\neq m}
,
∫
0
2
π
y
n
(
e
i
θ
)
y
m
(
e
i
θ
)
i
e
i
θ
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
)
Powers of
x
{\displaystyle x}
are expressed in terms of the generalized Bessel polynomials from the inverse connection formulae which have applications in change of basis to these polynomials.
x
n
=
∑
k
=
0
n
α
(
n
,
k
,
α
,
β
)
y
n
−
k
(
x
;
α
,
β
)
{\displaystyle x^{n}=\sum _{k=0}^{n}\alpha (n,k,\alpha ,\beta )\ y_{n-k}(x;\alpha ,\beta )}
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
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
Zeros If one denotes the zeros of
y
n
(
x
;
α
,
β
)
{\displaystyle y_{n}(x;\alpha ,\beta )}
as
α
k
(
n
)
(
α
,
β
)
{\displaystyle \alpha _{k}^{(n)}(\alpha ,\beta )}
, and that of the
θ
n
(
x
;
α
,
β
)
{\displaystyle \theta _{n}(x;\alpha ,\beta )}
by
Particular values The Bessel polynomials
y
n
(
x
)
{\displaystyle y_{n}(x)}
up to
n
=
5
{\displaystyle n=5}
are
y
0
(
x
)
=
1
y
1
(
x
)
=
x
+
1
y
2
(
x
)
=
3
x
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
3
{\displaystyle y_{n}(x)={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{n+{\frac {1}{2}}}(1/x)}
{\displaystyle \theta _{n}(x)={\sqrt {\frac {2}{\pi }}}\,x^{n+1/2}e^{x}K_{n+{\frac {1}{2}}}(x)}
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:
{\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)}
{\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):
{\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:
{\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:
{\displaystyle \theta _{n}(x)={\frac {(-2n)_{n}}{(-2)^{n}}}\,\,_{1}F_{1}(-n;-2n;2x)}
where (−2n)n is the Pochhammer symbol (rising factorial).
{\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
, yields the generating function for the polynomials
{\displaystyle \{\theta _{n}\}_{n\geq 0}}
{\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
{\displaystyle \sum _{n=0}^{\infty }y_{n-1}(x){\frac {t^{n}}{n!}}=\exp \left({\frac {1-{\sqrt {1-2xt}}}{x}}\right).}
{\displaystyle t=z-xz^{2}/2}
, one has the following representation for the exponential function:
{\displaystyle e^{z}=\sum _{n=0}^{\infty }y_{n-1}(x){\frac {(z-xz^{2}/2)^{n}}{n!}}.}
{\displaystyle y_{n}(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,}
{\displaystyle \theta _{0}(x)=1\,}
{\displaystyle \theta _{1}(x)=x+1\,}
{\displaystyle \theta _{n}(x)=(2n\!-\!1)\theta _{n-1}(x)+x^{2}\theta _{n-2}(x)\,}
{\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}
{\displaystyle x{\frac {d^{2}\theta _{n}(x)}{dx^{2}}}-2(x\!+\!n){\frac {d\theta _{n}(x)}{dx}}+2n\,\theta _{n}(x)=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}
They are also orthogonal with respect to a real weight, provided it is a hyperfunction.
{\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
{\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
{\displaystyle y_{n}(x;\alpha ,\beta )}
{\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}.}
{\displaystyle \theta _{n}(x;\alpha ,\beta )}
polynomials can explicitly be written as follows:
{\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
{\displaystyle \rho (x;\alpha ,\beta ):={}_{1}F_{1}\left(1,\alpha -1,-{\frac {\beta }{x}}\right)}
they are orthogonal, for the relation
{\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).
{\displaystyle \alpha (n,k,\alpha ,\beta )={n \choose k}{\frac {(-1)^{k}\beta ^{n}(2(n-k)+\alpha -1)}{(n-k+\alpha -1)_{n+1}}}}
Similarly, for the reverse generalized Bessel polynomials
{\displaystyle x^{n}=\sum _{k=0}^{n}\alpha _{1}(n,k,\alpha ,\beta )\ \theta _{n-k}(x;\alpha ,\beta )}
{\displaystyle \alpha _{1}(n,k,\alpha ,\beta )=(-1)^{k}{\frac {n \choose k}{\beta ^{k}}}(n+\alpha -1)(n+\alpha -2k)_{k-1}}
{\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 are normalization coefficients.
{\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}
{\displaystyle 0\leq m\leq n}
{\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}}})}
{\displaystyle \beta _{k}^{(n)}(\alpha ,\beta )}
, then the following estimates exist:
{\displaystyle {\frac {2}{n(n+\alpha -1)}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}},}
{\displaystyle {\frac {n+\alpha -1}{2}}\leq \beta _{k}^{(n)}(\alpha ,2)\leq {\frac {n(n+\alpha -1)}{2}},}
{\displaystyle \alpha \geq 2}
. Moreover, all these zeros have negative real part.
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).
One result is the following:
{\displaystyle {\frac {2}{2n+\alpha -{\frac {2}{3}}}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -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.
The reverse Bessel polynomials are obtained by reversing the coefficients.
{\textstyle \theta _{k}(x)=x^{k}y_{k}(1/x)}
This results in the following:
{\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}}}
