Difference between revisions of "Bessel polynomial"

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(Properties)
 
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=Properties=
 
=Properties=
{{:Bessel polynomial generalized hypergeometric}}
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[[Bessel polynomial generalized hypergeometric]]<br />
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[[Bessel polynomial in terms of Bessel functions]]<br />
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[[Bessel at n+1/2 in terms of Bessel polynomial]]<br />
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[[Bessel at -n-1/2 in terms of Bessel polynomial]]<br />
  
{{:Bessel polynomial in terms of Bessel functions}}
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{{:Orthogonal polynomials footer}}
 
 
{{:Bessel at n+1/2 in terms of Bessel polynomial}}
 
 
 
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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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{{:Orthogonal polynomials footer}}
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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