Bessel

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Bessel

Hypergeometric representation

y n ⁑ ( x ; a ) = \HyperpFq 20 @ @ - n , n + a + 1 - - x 2 fragments Bessel-polynomial-y 𝑛 π‘₯ π‘Ž \HyperpFq 20 @ @ n , n a 1 π‘₯ 2 {\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a\right)=\HyperpFq{2% }{0}@@{-n,n+a+1}{-}{-\frac{x}{2}}}}} {\displaystyle \BesselPoly{n}@{x}{a}=\HyperpFq{2}{0}@@{-n,n+a+1}{-}{-\frac{x}{2}} }

Constraint(s): n = 0 , 1 , 2 , … , N 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}


y n ⁑ ( x ; a ) = ( n + a + 1 ) n ⁒ ( x 2 ) n ⁒ \HyperpFq ⁒ 11 ⁒ @ ⁒ @ - n - 2 ⁒ n - a ⁒ 2 x Bessel-polynomial-y 𝑛 π‘₯ π‘Ž Pochhammer-symbol 𝑛 π‘Ž 1 𝑛 superscript π‘₯ 2 𝑛 \HyperpFq 11 @ @ 𝑛 2 𝑛 π‘Ž 2 π‘₯ {\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a\right)={\left(n+a+% 1\right)_{n}}\left(\frac{x}{2}\right)^{n}\,\HyperpFq{1}{1}@@{-n}{-2n-a}{\frac{% 2}{x}}}}} {\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)

∫ 0 ∞ x a ⁒ e - 2 x ⁒ y m ⁑ ( x ; a ) ⁒ y n ⁑ ( x ; a ) ⁒ 𝑑 x = - 2 a + 1 2 ⁒ n + a + 1 ⁒ Ξ“ ⁑ ( - n - a ) ⁒ n ! ⁒ Ξ΄ m , n superscript subscript 0 superscript π‘₯ π‘Ž 2 π‘₯ Bessel-polynomial-y π‘š π‘₯ π‘Ž Bessel-polynomial-y 𝑛 π‘₯ π‘Ž differential-d π‘₯ superscript 2 π‘Ž 1 2 𝑛 π‘Ž 1 Euler-Gamma 𝑛 π‘Ž 𝑛 Kronecker-delta π‘š 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}x^{a}{\mathrm{e}^{-% \frac{2}{x}}}y_{m}\!\left(x;a\right)y_{n}\!\left(x;a\right)\,dx=-\frac{2^{a+1}% }{2n+a+1}\Gamma\left(-n-a\right)n!\,\delta_{m,n}}}} {\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): a < - 2 ⁒ N - 1 π‘Ž 2 𝑁 1 {\displaystyle{\displaystyle{\displaystyle a<-2N-1}}}


Recurrence relation

2 ⁒ ( n + a + 1 ) ⁒ ( 2 ⁒ n + a ) ⁒ y n + 1 ⁑ ( x ; a ) = ( 2 ⁒ n + a + 1 ) ⁒ [ 2 ⁒ a + ( 2 ⁒ n + a ) ⁒ ( 2 ⁒ n + a + 2 ) ⁒ x ] ⁒ y n ⁑ ( x ; a ) + 2 ⁒ n ⁒ ( 2 ⁒ n + a + 2 ) ⁒ y n - 1 ⁑ ( x ; a ) 2 𝑛 π‘Ž 1 2 𝑛 π‘Ž Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž 2 𝑛 π‘Ž 1 delimited-[] 2 π‘Ž 2 𝑛 π‘Ž 2 𝑛 π‘Ž 2 π‘₯ Bessel-polynomial-y 𝑛 π‘₯ π‘Ž 2 𝑛 2 𝑛 π‘Ž 2 Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž {\displaystyle{\displaystyle{\displaystyle 2(n+a+1)(2n+a)y_{n+1}\!\left(x;a% \right){}=(2n+a+1)\left[2a+(2n+a)(2n+a+2)x\right]y_{n}\!\left(x;a\right){}+2n(% 2n+a+2)y_{n-1}\!\left(x;a\right)}}} {\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

x ⁒ y ^ n ⁑ ( x ) = y ^ n + 1 ⁑ ( x ) - 2 ⁒ a ( 2 ⁒ n + a ) ⁒ ( 2 ⁒ n + a + 2 ) ⁒ y ^ n ⁑ ( x ) - 4 ⁒ n ⁒ ( n + a ) ( 2 ⁒ n + a - 1 ) ⁒ ( 2 ⁒ n + a ) 2 ⁒ ( 2 ⁒ n + a + 1 ) ⁒ y ^ n - 1 ⁑ ( x ) π‘₯ Bessel-polynomial-monic-p 𝑛 π‘₯ π‘Ž Bessel-polynomial-monic-p 𝑛 1 π‘₯ π‘Ž 2 π‘Ž 2 𝑛 π‘Ž 2 𝑛 π‘Ž 2 Bessel-polynomial-monic-p 𝑛 π‘₯ π‘Ž 4 𝑛 𝑛 π‘Ž 2 𝑛 π‘Ž 1 superscript 2 𝑛 π‘Ž 2 2 𝑛 π‘Ž 1 Bessel-polynomial-monic-p 𝑛 1 π‘₯ π‘Ž {\displaystyle{\displaystyle{\displaystyle x{\widehat{y}}_{n}\!\left(x\right)=% {\widehat{y}}_{n+1}\!\left(x\right)-\frac{2a}{(2n+a)(2n+a+2)}{\widehat{y}}_{n}% \!\left(x\right){}-\frac{4n(n+a)}{(2n+a-1)(2n+a)^{2}(2n+a+1)}{\widehat{y}}_{n-% 1}\!\left(x\right)}}} {\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} }
y n ⁑ ( x ; a ) = ( n + a + 1 ) n 2 n ⁒ y ^ n ⁑ ( x ) Bessel-polynomial-y 𝑛 π‘₯ π‘Ž Pochhammer-symbol 𝑛 π‘Ž 1 𝑛 superscript 2 𝑛 Bessel-polynomial-monic-p 𝑛 π‘₯ π‘Ž {\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a\right)=\frac{{% \left(n+a+1\right)_{n}}}{2^{n}}{\widehat{y}}_{n}\!\left(x\right)}}} {\displaystyle \BesselPoly{n}@{x}{a}=\frac{\pochhammer{n+a+1}{n}}{2^n}\monicBesselPoly{n}@@{x}{a} }

Differential equation

x 2 ⁒ y β€²β€² ⁒ ( x ) + [ ( a + 2 ) ⁒ x + 2 ] ⁒ y β€² ⁒ ( x ) - n ⁒ ( n + a + 1 ) ⁒ y ⁒ ( x ) = 0 superscript π‘₯ 2 superscript 𝑦 β€²β€² π‘₯ delimited-[] π‘Ž 2 π‘₯ 2 superscript 𝑦 β€² π‘₯ 𝑛 𝑛 π‘Ž 1 𝑦 π‘₯ 0 {\displaystyle{\displaystyle{\displaystyle x^{2}y^{\prime\prime}(x)+\left[(a+2% )x+2\right]y^{\prime}(x)-n(n+a+1)y(x)=0}}} {\displaystyle x^2y''(x)+\left[(a+2)x+2\right]y'(x)-n(n+a+1)y(x)=0 }

Substitution(s): y ⁒ ( x ) = y n ⁑ ( x ; a ) 𝑦 π‘₯ Bessel-polynomial-y 𝑛 π‘₯ π‘Ž {\displaystyle{\displaystyle{\displaystyle y(x)=y_{n}\!\left(x;a\right)}}}


Forward shift operator

d d ⁒ x ⁒ y n ⁑ ( x ; a ) = n ⁒ ( n + a + 1 ) 2 ⁒ y n - 1 ⁑ ( x ; a + 2 ) 𝑑 𝑑 π‘₯ Bessel-polynomial-y 𝑛 π‘₯ π‘Ž 𝑛 𝑛 π‘Ž 1 2 Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž 2 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}y_{n}\!\left(x;a\right)=% \frac{n(n+a+1)}{2}y_{n-1}\!\left(x;a+2\right)}}} {\displaystyle \frac{d}{dx}\BesselPoly{n}@{x}{a}=\frac{n(n+a+1)}{2}\BesselPoly{n-1}@{x}{a+2} }

Backward shift operator

x 2 ⁒ d d ⁒ x ⁒ y n ⁑ ( x ; a ) + ( a ⁒ x + 2 ) ⁒ y n ⁑ ( x ; a ) = 2 ⁒ y n + 1 ⁑ ( x ; a - 2 ) superscript π‘₯ 2 𝑑 𝑑 π‘₯ Bessel-polynomial-y 𝑛 π‘₯ π‘Ž π‘Ž π‘₯ 2 Bessel-polynomial-y 𝑛 π‘₯ π‘Ž 2 Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž 2 {\displaystyle{\displaystyle{\displaystyle x^{2}\frac{d}{dx}y_{n}\!\left(x;a% \right)+(ax+2)y_{n}\!\left(x;a\right)=2y_{n+1}\!\left(x;a-2\right)}}} {\displaystyle x^2\frac{d}{dx}\BesselPoly{n}@{x}{a}+(ax+2)\BesselPoly{n}@{x}{a}=2\BesselPoly{n+1}@{x}{a-2} }
d d ⁒ x ⁒ [ x a ⁒ e - 2 x ⁒ y n ⁑ ( x ; a ) ] = 2 ⁒ x a - 2 ⁒ e - 2 x ⁒ y n + 1 ⁑ ( x ; a - 2 ) 𝑑 𝑑 π‘₯ delimited-[] superscript π‘₯ π‘Ž 2 π‘₯ Bessel-polynomial-y 𝑛 π‘₯ π‘Ž 2 superscript π‘₯ π‘Ž 2 2 π‘₯ Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž 2 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[x^{a}{\mathrm{e}^{% -\frac{2}{x}}}y_{n}\!\left(x;a\right)\right]=2x^{a-2}{\mathrm{e}^{-\frac{2}{x}% }}y_{n+1}\!\left(x;a-2\right)}}} {\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

y n ⁑ ( x ; a ) = 2 - n ⁒ x - a ⁒ e 2 x ⁒ D n ⁒ ( x 2 ⁒ n + a ⁒ e - 2 x ) Bessel-polynomial-y 𝑛 π‘₯ π‘Ž superscript 2 𝑛 superscript π‘₯ π‘Ž 2 π‘₯ superscript 𝐷 𝑛 superscript π‘₯ 2 𝑛 π‘Ž 2 π‘₯ {\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a\right)=2^{-n}x^{-a% }{\mathrm{e}^{\frac{2}{x}}}D^{n}\left(x^{2n+a}{\mathrm{e}^{-\frac{2}{x}}}% \right)}}} {\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

( 1 - 2 ⁒ x ⁒ t ) - 1 2 ⁒ ( 2 1 + 1 - 2 ⁒ x ⁒ t ) a ⁒ exp ⁑ ( 2 ⁒ t 1 + 1 - 2 ⁒ x ⁒ t ) = βˆ‘ n = 0 ∞ y n ⁑ ( x ; a ) ⁒ t n n ! superscript 1 2 π‘₯ 𝑑 1 2 superscript 2 1 1 2 π‘₯ 𝑑 π‘Ž 2 𝑑 1 1 2 π‘₯ 𝑑 superscript subscript 𝑛 0 Bessel-polynomial-y 𝑛 π‘₯ π‘Ž superscript 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(1-2xt\right)^{-\frac{1}{2}}% \left(\frac{2}{1+\sqrt{1-2xt}}\right)^{a}\exp\left(\frac{2t}{1+\sqrt{1-2xt}}% \right)=\sum_{n=0}^{\infty}y_{n}\!\left(x;a\right)\frac{t^{n}}{n!}}}} {\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

lim Ξ± β†’ - ∞ ⁑ P n ( Ξ± , a - Ξ± ) ⁑ ( 1 + Ξ± ⁒ x ) P n ( Ξ± , a - Ξ± ) ⁑ ( 1 ) = y n ⁑ ( x ; a ) subscript β†’ 𝛼 Jacobi-polynomial-P 𝛼 π‘Ž 𝛼 𝑛 1 𝛼 π‘₯ Jacobi-polynomial-P 𝛼 π‘Ž 𝛼 𝑛 1 Bessel-polynomial-y 𝑛 π‘₯ π‘Ž {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow-\infty}\frac% {P^{(\alpha,a-\alpha)}_{n}\left(1+\alpha x\right)}{P^{(\alpha,a-\alpha)}_{n}% \left(1\right)}=y_{n}\!\left(x;a\right)}}} {\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

y n ⁑ ( x ; a , b ) = y n ⁑ ( 2 ⁒ b - 1 ⁒ x ; a )   and   ΞΈ n ⁒ ( x ; a , b ) = x n ⁒ y n ⁑ ( x - 1 ; a , b ) formulae-sequence Bessel-polynomial-y 𝑛 π‘₯ π‘Ž 𝑏 Bessel-polynomial-y 𝑛 2 superscript 𝑏 1 π‘₯ π‘Ž and subscript πœƒ 𝑛 π‘₯ π‘Ž 𝑏 superscript π‘₯ 𝑛 Bessel-polynomial-y 𝑛 superscript π‘₯ 1 π‘Ž 𝑏 {\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a,b\right)=y_{n}\!% \left(2b^{-1}x;a\right)\quad\textrm{and}\quad\theta_{n}(x;a,b)=x^{n}y_{n}\!% \left(x^{-1};a,b\right)}}} {\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} }
L n Ξ± ⁑ ( x ) = ( - x ) n n ! ⁒ y n ⁑ ( 2 ⁒ x - 1 ; - 2 ⁒ n - Ξ± - 1 ) generalized-Laguerre-polynomial-L 𝛼 𝑛 π‘₯ superscript π‘₯ 𝑛 𝑛 Bessel-polynomial-y 𝑛 2 superscript π‘₯ 1 2 𝑛 𝛼 1 {\displaystyle{\displaystyle{\displaystyle L^{\alpha}_{n}\left(x\right)=\frac{% (-x)^{n}}{n!}y_{n}\!\left(2x^{-1};-2n-\alpha-1\right)}}} {\displaystyle \Laguerre[\alpha]{n}@{x}=\frac{(-x)^n}{n!}\BesselPoly{n}@{2x^{-1}}{-2n-\alpha-1} }
lim Ξ½ β†’ ∞ ⁑ P n ⁑ ( Ξ½ ⁒ x ; Ξ½ , N ) Ξ½ n = 2 n ( n - 2 ⁒ N - 1 ) n ⁒ y n ⁑ ( x ; - 2 ⁒ N - 2 ) subscript β†’ 𝜈 pseudo-Jacobi-polynomial 𝑛 𝜈 π‘₯ 𝜈 𝑁 superscript 𝜈 𝑛 superscript 2 𝑛 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 Bessel-polynomial-y 𝑛 π‘₯ 2 𝑁 2 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\nu\rightarrow\infty}% \frac{P_{n}\!\left(\nu x;\nu,N\right)}{\nu^{n}}=\frac{2^{n}}{{\left(n-2N-1% \right)_{n}}}y_{n}\!\left(x;-2N-2\right)}}} {\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} }

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