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| {{More footnotes|date=September 2009}}
| | Hello, my name is Andrew and my wife doesn't like it at all. For a while I've been in Alaska but I will have to transfer in a yr or two. It's not a typical thing but what I like doing is to climb but I don't have the time lately. Distributing production is where my primary income arrives from and it's something I truly enjoy.<br><br>My blog :: [http://www.sirudang.com/siroo_Notice/2110 clairvoyance] |
| In [[mathematics]], the '''Bessel polynomials''' are an [[orthogonal polynomials|orthogonal]] sequence of [[polynomial]]s. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series (Krall & Frink, 1948)
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| :<math>y_n(x)=\sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\,\left(\frac{x}{2}\right)^k</math>
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| Another definition, favored by electrical engineers, is sometimes known as the '''reverse Bessel polynomials''' (See Grosswald 1978, Berg 2000).
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| :<math>\theta_n(x)=x^n\,y_n(1/x)=\sum_{k=0}^n\frac{(2n-k)!}{(n-k)!k!}\,\frac{x^k}{2^{n-k}}</math>
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| The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is
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| :<math>y_3(x)=15x^3+15x^2+6x+1\,</math>
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| while the third-degree reverse Bessel polynomial is | |
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| :<math>\theta_3(x)=x^3+6x^2+15x+15\,</math>
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| The reverse Bessel polynomial is used in the design of [[Bessel filter|Bessel electronic filters]].
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| == Properties ==
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| === Definition in terms of Bessel functions ===
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| The Bessel polynomial may also be defined using [[Bessel function]]s from which the polynomial draws its name.
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| :<math>y_n(x)=\,x^{n}\theta_n(1/x)\,</math>
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| :<math>\theta_n(x)=\sqrt{\frac{2}{\pi}}\,x^{n+1/2}e^{x}K_{n+ \frac 1 2}(x)</math>
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| :<math>y_n(x)=\sqrt{\frac{2}{\pi x}}\,e^{1/x}K_{n+\frac 1 2}(1/x)</math>
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| where ''K''<sub>''n''</sub>(''x'') is a modified Bessel function of the second kind and ''y''<sub>''n''</sub>(''x'') is the reverse polynomial (pag 7 and 34 Grosswald 1978).
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| === Definition as a hypergeometric function ===
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| The Bessel polynomial may also be defined as a [[confluent hypergeometric function]] (Dita, 2006)
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| :<math>y_n(x)=\,_2F_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).</math>
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| The reverse Bessel polynomial may be defined as a generalized [[Laguerre polynomial]]:
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| :<math>\theta_n(x)=\frac{n!}{(-2)^n}\,L_n^{-2n-1}(2x)</math>
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| from which it follows that it may also be defined as a hypergeometric function:
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| :<math>\theta_n(x)=\frac{(-2n)_n}{(-2)^n}\,\,_1F_1(-n;-2n;-2x)</math>
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| where (−2''n'')<sub>''n''</sub> is the [[Pochhammer symbol]] (rising factorial).
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| ===Generating function===
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| The Bessel polynomials have the generating function
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| :<math>\sum_{n=0} \sqrt{\frac 2 \pi} x^{n+\frac 1 2} e^x K_{n-\frac 1 2}(x) \frac {t^n}{n!}= e^{x(1-\sqrt{1-2t})}.</math>
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| === Recursion ===
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| The Bessel polynomial may also be defined by a recursion formula:
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| :<math>y_0(x)=1\,</math>
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| :<math>y_1(x)=x+1\,</math>
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| :<math>y_n(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,</math>
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| and
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| :<math>\theta_0(x)=1\,</math>
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| :<math>\theta_1(x)=x+1\,</math>
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| :<math>\theta_n(x)=(2n\!-\!1)\theta_{n-1}(x)+x^2\theta_{n-2}(x)\,</math>
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| === Differential equation ===
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| The Bessel polynomial obeys the following differential equation:
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| :<math>x^2\frac{d^2y_n(x)}{dx^2}+2(x\!+\!1)\frac{dy_n(x)}{dx}-n(n+1)y_n(x)=0</math>
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| and
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| :<math>x\frac{d^2\theta_n(x)}{dx^2}-2(x\!+\!n)\frac{d\theta_n(x)}{dx}+2n\,\theta_n(x)=0</math>
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| ==Generalization==
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| ===Explicit Form===
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| A generalization of the Bessel polynomials have been suggested in literature (Krall, Fink), as following:
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| :<math>y_n(x;\alpha,\beta):= (-1)^n n! \left(\frac x \beta\right)^n L_n^{(1-2n-\alpha)}\left(\frac \beta x\right),</math>
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| the corresponding reverse polynomials are
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| :<math>\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).</math>
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| For the weighting function
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| :<math>\rho(x;\alpha,\beta):= \, _1F_1\left(1,\alpha-1,-\frac \beta x\right)</math>
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| they are orthogonal, for the relation
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| :<math>0= \oint_c\rho(x;\alpha,\beta)y_n(x;\alpha,\beta) y_m(x;\alpha,\beta)\mathrm d x</math>
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| holds for ''m'' ≠ ''n'' and ''c'' a curve surrounding the 0 point.
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| They specialize to the Bessel polynomials for α = β = 2, in which situation ρ(''x'') = exp(−2 / ''x'').
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| ===Rodrigues formula for Bessel polynomials===
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| The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :
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| :<math>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}})</math>
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| where ''a''{{su|b=''n''|p=(α, β)}} are normalization coefficients. | |
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| ===Associated Bessel polynomials===
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| According to this generalization we have the following generalized associated Bessel polynomials differential equation:
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| :<math>x^2\frac{d^2B_{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</math>
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| where <math>0\leq m\leq n</math>. The solutions are,
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| :<math>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}})</math> | |
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| == Particular values ==
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| :<math> | |
| \begin{align}
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| y_0(x) & = 1 \\
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| y_1(x) & = x + 1 \\
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| y_2(x) & = 3x^2+ 3x + 1 \\
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| y_3(x) & = 15x^3+ 15x^2+ 6x + 1 \\
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| y_4(x) & = 105x^4+105x^3+ 45x^2+ 10x + 1 \\
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| y_5(x) & = 945x^5+945x^4+420x^3+105x^2+15x+1
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| \end{align}
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| </math>
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| == References ==
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| *{{cite journal
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| | last =Carlitz
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| | first = Leonard
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| | authorlink = Leonard Carlitz
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| | coauthors =
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| | year = 1957
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| | month =
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| | title = A Note on the Bessel Polynomials
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| | journal = Duke Math. J.
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| | volume = 24
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| | issue = 2
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| | pages = 151–162
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| | doi = 10.1215/S0012-7094-57-02421-3
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| | mr = 0085360
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| }}
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| *{{cite journal
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| | last = Krall
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| | first = H. L.
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| | coauthors = Frink, O.
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| | year = 1948
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| | month =
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| | title = A New Class of Orthogonal Polynomials: The Bessel Polynomials
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| | journal = Trans. Amer. Math. Soc.
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| | volume = 65
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| | issue = 1
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| | pages = 100–115
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| | doi = 10.2307/1990516
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| | jstor = 1990516
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| | accessdate =
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| | quotes =
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| }}
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| *{{cite web
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| | title = The [[On-Line Encyclopedia of Integer Sequences]]
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| | accessdate = 2006-08-16
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| | author =Sloane, N. J. A.
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| | last =
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| | first =
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| | coauthors =
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| | date =
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| | year =
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| | month =
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| | work =
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| | publisher =
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| | pages =
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| }} (See sequences {{OEIS2C|A001497}}, {{OEIS2C|A001498}}, and {{OEIS2C|A104548}})
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| *{{cite arxiv
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| | last1 = Dita
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| | first1 = P.
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| | last2=Grama
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| | first2= Grama, N.
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| | year = 2006
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| | month = May 24
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| | title = On Adomian’s Decomposition Method for Solving Differential Equations
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| | eprint = solv-int/9705008
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| | quotes =
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| | class = solv-int
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| }}
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| *{{cite journal
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| | last1 = Fakhri
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| | first1 = H.
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| | last2= Chenaghlou
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| | first2 = A.
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| | year = 2006
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| | month =
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| | title = Ladder operators and recursion relations for the associated Bessel polynomials
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| | journal = Physics Letters A
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| | volume = 358
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| | issue = 5–6
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| | pages = 345–353
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| | doi = 10.1016/j.physleta.2006.05.070
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| | bibcode=2006PhLA..358..345F
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| | accessdate =
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| | quotes =
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| }}
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| *{{cite book
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| |last=Grosswald
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| |first=E.
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| |authorlink=Emil Grosswald
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| |coauthors=
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| |title=Bessel Polynomials (Lecture Notes in Mathematics)
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| |year=1978
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| |publisher=Springer
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| |location= New York
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| |isbn=0-387-09104-1
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| }}
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| *{{cite book
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| |last=Roman
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| |first=S.
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| |coauthors=
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| |title=The Umbral Calculus (The Bessel Polynomials §4.1.7)
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| |year= 1984
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| |publisher=Academic Press
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| |location= New York
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| |isbn=0-486-44139-3
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| }}
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| *{{cite web
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| | url = http://www.math.ku.dk/~berg/manus/bessel.pdf
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| | title = Linearization coefficients of Bessel polynomials and properties of Student-t distributions
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| | accessdate = 2006-08-16
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| | author =
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| | last = Berg
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| | first = Christian
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| | coauthors = Vignat, C.
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| | date =
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| | year = 2000
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| | month =
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| | format = PDF
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| | work =
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| | publisher =
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| | pages =
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| }}
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| ==External links==
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| * {{springer|title=Bessel polynomials|id=p/b110410}}
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| *{{MathWorld|title=Bessel Polynomial|urlname=BesselPolynomial}}
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| *{{SloanesRef |sequencenumber=A001498|name=Coefficients of Bessel polynomials }}
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| [[Category:Orthogonal polynomials]]
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| [[Category:Special hypergeometric functions]]
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