Bessel

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Bessel

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Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \BesselPoly{n}@{x}{a}=\HyperpFq{2}{0}@@{-n,n+a+1}{-}{-\frac{x}{2}} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n=0,1,2,\ldots,N}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \BesselPoly{n}@{x}{a}=\pochhammer{n+a+1}{n}\left(\frac{x}{2}\right)^n\,\HyperpFq{1}{1}@@{-n}{-2n-a}{\frac{2}{x}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_0^{\infty}x^a\expe^{-\frac{2}{x}}\BesselPoly{m}@{x}{a}\BesselPoly{n}@{x}{a}\,dx =-\frac{2^{a+1}}{2n+a+1}\EulerGamma@{-n-a}n!\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a<-2N-1}}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2(n+a+1)(2n+a)\BesselPoly{n+1}@{x}{a} {}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]\BesselPoly{n}@{x}{a} {}+2n(2n+a+2)\BesselPoly{n-1}@{x}{a} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicBesselPoly{n}@@{x}{a}=\monicBesselPoly{n+1}@@{x}{a}-\frac{2a}{(2n+a)(2n+a+2)}\monicBesselPoly{n}@@{x}{a} {}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^2(2n+a+1)}\monicBesselPoly{n-1}@@{x}{a} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \BesselPoly{n}@{x}{a}=\frac{\pochhammer{n+a+1}{n}}{2^n}\monicBesselPoly{n}@@{x}{a} }}

Differential equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\BesselPoly{n}@{x}{a}}}

Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\BesselPoly{n}@{x}{a}=\frac{n(n+a+1)}{2}\BesselPoly{n-1}@{x}{a+2} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x^2\frac{d}{dx}\BesselPoly{n}@{x}{a}+(ax+2)\BesselPoly{n}@{x}{a}=2\BesselPoly{n+1}@{x}{a-2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\left[x^a\expe^{-\frac{2}{x}}\BesselPoly{n}@{x}{a}\right]=2x^{a-2}\expe^{-\frac{2}{x}}\BesselPoly{n+1}@{x}{a-2} }}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \BesselPoly{n}@{x}{a}=2^{-n}x^{-a}\expe^{\frac{2}{x}}D^n\left(x^{2n+a}\expe^{-\frac{2}{x}}\right) }}

Generating function

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left(1-2xt\right)^{-\frac{1}{2}}\left(\frac{2}{1+\sqrt{1-2xt}}\right)^a \exp@{\frac{2t}{1+\sqrt{1-2xt}}}=\sum_{n=0}^{\infty}\BesselPoly{n}@{x}{a}\frac{t^n}{n!} }}

Limit relation

Jacobi polynomial to Bessel polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow -\infty} \frac{\Jacobi{\alpha}{a-\alpha}{n}@{1+\alpha x}}{\Jacobi{\alpha}{a-\alpha}{n}@{1}}=\BesselPoly{n}@{x}{a} }}

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \BesselPoly{n}@{x}{a,b}=\BesselPoly{n}@{2b^{-1}x}{a}\quad\textrm{and}\quad\theta_n(x;a,b)=x^n\BesselPoly{n}@{x^{-1}}{a,b} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{(-x)^n}{n!}\BesselPoly{n}@{2x^{-1}}{-2n-\alpha-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{\nu\rightarrow\infty}\frac{\pseudoJacobi{n}@{\nu x}{\nu}{N}}{\nu^n} =\frac{2^n}{\pochhammer{n-2N-1}{n}}\BesselPoly{n}@{x}{-2N-2} }}

<< Laguerre

Bessel

Charlier >>