Difference between revisions of "Bessel polynomial"

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=Properties=
 
=Properties=
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[[Bessel polynomial generalized hypergeometric]]<br />
<strong>Theorem:</strong> The following formula holds:
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[[Bessel polynomial in terms of Bessel functions]]<br />
$$y_n(x)={}_2F_0 \left( -n, 1+n;-; -\dfrac{1}{2}x \right),$$
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[[Bessel at n+1/2 in terms of Bessel polynomial]]<br />
where $y_n(x)$ denotes a [[Bessel polynomial]] and ${}_2F_0$ denotes the [[generalized hypergeometric function]].
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[[Bessel at -n-1/2 in terms of Bessel polynomial]]<br />
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<strong>Proof:</strong> █
 
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{{:Orthogonal polynomials footer}}
<strong>Theorem:</strong> The following formula holds:
 
$$y_n\left( \dfrac{1}{ir} \right) = \left(\dfrac{\pi r}{2} \right)^{\frac{1}{2}} e^{ir} \left[ \dfrac{J_{n +\frac{1}{2}}(r)}{i^{n+1}}+i^nJ_{-n-\frac{1}{2}}(r) \right],$$
 
where $y_n$ denotes a [[Bessel polynomial]] and $J_{\nu}$ denotes a [[Bessel function]].
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$J_{n +\frac{1}{2}}(r) = (2\pi r)^{-\frac{1}{2}} \left[\dfrac{e^{ir}}{i^{n+1}} y_n \left( -\dfrac{1}{ir} \right) + i^{n+1}e^{-ir}y_n\left( \dfrac{1}{ir} \right) \right],$$
 
where $J_{n+\frac{1}{2}}$ denotes a [[Bessel function]] and $y_n$ denotes a [[Bessel polynomial]].
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$
 
where $J_{-n-\frac{1}{2}}$ denotes a [[Bessel function]] and $y_n$ denotes a [[Bessel polynomial]].
 
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<strong>Proof:</strong> █
 
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[[Category:SpecialFunction]]

Latest revision as of 20:10, 9 June 2016

The Bessel polynomials are orthogonal polynomials defined by $$y_n(x) = \displaystyle\sum_{k=0}^n \dfrac{(n+k)!}{(n-k)!k!} \left( \dfrac{x}{2} \right)^k.$$

Properties

Bessel polynomial generalized hypergeometric
Bessel polynomial in terms of Bessel functions
Bessel at n+1/2 in terms of Bessel polynomial
Bessel at -n-1/2 in terms of Bessel polynomial

Orthogonal polynomials