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 (Krall & Frink, 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 polynomials (See Grosswald 1978, Berg 2000).

\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)=15x^{3}+15x^{2}+6x+1\,

while the third-degree 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.

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}}\theta _{n}(1/x)\,

y_{n}(x)={\sqrt {{\frac {2}{\pi x}}}}\,e^{{1/x}}K_{{n+{\frac 12}}}(1/x)

\theta _{n}(x)={\sqrt {{\frac {2}{\pi }}}}\,x^{{n+1/2}}e^{{x}}K_{{n+{\frac 12}}}(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)=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)=\left({\frac 2x}\right)^{{-n}}U\left(-n,-2n,{\frac 2x}\right)=\left({\frac 2x}\right)^{{n+1}}U\left(n+1,2n+2,{\frac 2x}\right).

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}}}\,\,_{1}F_{1}(-n;-2n;-2x)

where (−2n)_n is the Pochhammer symbol (rising factorial).

The inversion for monomials is given by

\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

\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 $t$ , cancelling $x$ , yields the generating function for the polynomials $\{\theta _{n}\}_{n\geq 0}$

\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\,

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)\,

Differential equation

The Bessel polynomial obeys the following differential equation:

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{\frac {d^{2}\theta _{n}(x)}{dx^{2}}}-2(x\!+\!n){\frac {d\theta _{n}(x)}{dx}}+2n\,\theta _{n}(x)=0

Generalization

Explicit Form

A generalization of the Bessel polynomials have been suggested in literature (Krall, Fink), as following:

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

\theta _{n}(x;\alpha ,\beta ):={\frac {n!}{(-\beta )^{n}}}L_{n}^{{(1-2n-\alpha )}}(\beta x)=x^{n}y_{n}\left({\frac 1x};\alpha ,\beta \right).

For the weighting function

\rho (x;\alpha ,\beta ):=\,_{1}F_{1}\left(1,\alpha -1,-{\frac \beta x}\right)

they are orthogonal, for the relation

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).

Rodrigues formula for Bessel polynomials

The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :

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.

Associated Bessel polynomials

According to this generalization we have the following generalized differential equation for associated Bessel polynomials:

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 $0\leq m\leq n$ . The solutions are,

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

{\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}}

none of which factor. Filaseta and Trifonov (Journal for Pure and Applied Mathematics, 550:125-140, 2002) proved that all Bessel polynomials are irreducible.

Resulting in the following reverse bessel polynomials:

{\begin{aligned}\theta _{1}(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}}

References

Carlitz, Leonard (1957). "A Note on the Bessel Polynomials". Duke Math. J. 24 (2): 151–162. MR 0085360. doi:10.1215/S0012-7094-57-02421-3.
Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. JSTOR 1990516. doi:10.2307/1990516.
Sloane, N. J. A. "The On-Line Encyclopedia of Integer Sequences". Missing or empty |url= (help) (See sequences A001497, A001498, and A104548)
Dita, P.; Grama, Grama, N. (May 24, 2006). "On Adomian’s Decomposition Method for Solving Differential Equations". arXiv:solv-int/9705008  [solv-int].
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.

↑ Wolfram Alpha example

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Bessel polynomials", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Bessel Polynomial". MathWorld.

"Sloane's A001498 : Coefficients of Bessel polynomials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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