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