Orthogonal Polynomials - 18.34 Bessel Polynomials

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.34.E1 y n ⁑ ( x ; a ) = F 0 2 ⁑ ( - n , n + a - 1 - ; - x 2 ) Bessel-polynomial-y 𝑛 π‘₯ π‘Ž Gauss-hypergeometric-pFq 2 0 𝑛 𝑛 π‘Ž 1 π‘₯ 2 {\displaystyle{\displaystyle y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-% 1\atop-};-\frac{x}{2}\right)}}
\Besselpolyy{n}@{x}{a} = \genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}}		 

Error

Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x] == HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]]
Missing Macro Error Failure - Error
18.34.E1 F 0 2 ⁑ ( - n , n + a - 1 - ; - x 2 ) = ( n + a - 1 ) n ⁒ ( x 2 ) n ⁒ F 1 1 ⁑ ( - n - 2 ⁒ n - a + 2 ; 2 x ) Gauss-hypergeometric-pFq 2 0 𝑛 𝑛 π‘Ž 1 π‘₯ 2 Pochhammer 𝑛 π‘Ž 1 𝑛 superscript π‘₯ 2 𝑛 Kummer-confluent-hypergeometric-M-as-1F1 𝑛 2 𝑛 π‘Ž 2 2 π‘₯ {\displaystyle{\displaystyle{{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}% \right)={\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left% ({-n\atop-2n-a+2};\frac{2}{x}\right)}}
\genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}} = \Pochhammersym{n+a-1}{n}\left(\frac{x}{2}\right)^{n}\genhyperF{1}{1}@@{-n}{-2n-a+2}{\frac{2}{x}}		 

hypergeom([- n , n + a - 1], [-], -(x)/(2)) = pochhammer(n + a - 1, n)*((x)/(2))^(n)* hypergeom([- n], [- 2*n - a + 2], (2)/(x))

HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]] == Pochhammer[n + a - 1, n]*(Divide[x,2])^(n)* HypergeometricPFQ[{- n}, {- 2*n - a + 2}, Divide[2,x]]
Error Failure - Error
18.34#Ex1 y n ⁒ ( x ) = y n ⁑ ( x ; 2 ) subscript 𝑦 𝑛 π‘₯ Bessel-polynomial-y 𝑛 π‘₯ 2 {\displaystyle{\displaystyle y_{n}(x)=y_{n}\left(x;2\right)}}
y_{n}(x) = \Besselpolyy{n}@{x}{2}		 

Error

Subscript[y, n][x] == Pochhammer[n + 2 - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - 2 + 2, 2/x]
Missing Macro Error Failure -
Failed [89 / 90]
Result: Complex[-1.200961894323342, 0.7499999999999999]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[y, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-10.950961894323342, 0.7499999999999999]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[y, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.34#Ex2 ΞΈ n ⁒ ( x ) = x n ⁒ y n ⁒ ( x - 1 ) subscript πœƒ 𝑛 π‘₯ superscript π‘₯ 𝑛 subscript 𝑦 𝑛 superscript π‘₯ 1 {\displaystyle{\displaystyle\theta_{n}(x)=x^{n}y_{n}(x^{-1})}}
\theta_{n}(x) = x^{n}y_{n}(x^{-1})		 

theta[n](x) = (x)^(n)* y[n]((x)^(- 1))
 
Subscript[\[Theta], n][x] == (x)^(n)* Subscript[y, n][(x)^(- 1)]
 Skipped - no semantic math Skipped - no semantic math - -
18.34#Ex3 y n ⁒ ( x ; a , b ) = y n ⁑ ( 2 ⁒ x / b ; a ) subscript 𝑦 𝑛 π‘₯ π‘Ž 𝑏 Bessel-polynomial-y 𝑛 2 π‘₯ 𝑏 π‘Ž {\displaystyle{\displaystyle y_{n}(x;a,b)=y_{n}\left(2x/b;a\right)}}
y_{n}(x;a,b) = \Besselpolyy{n}@{2x/b}{a}		 

Error

Subscript[y, n][x ; a , b] == Pochhammer[n + a - 1, n] (2*x/b/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/2*x/b]
Translation Error Translation Error - -
18.34#Ex4 ΞΈ n ⁒ ( x ; a , b ) = x n ⁒ y n ⁒ ( x - 1 ; a , b ) subscript πœƒ 𝑛 π‘₯ π‘Ž 𝑏 superscript π‘₯ 𝑛 subscript 𝑦 𝑛 superscript π‘₯ 1 π‘Ž 𝑏 {\displaystyle{\displaystyle\theta_{n}(x;a,b)=x^{n}y_{n}(x^{-1};a,b)}}
\theta_{n}(x;a,b) = x^{n}y_{n}(x^{-1};a,b)		 

theta[n](x ; a , b) = (x)^(n)* y[n]((x)^(- 1); a , b)
 
Subscript[\[Theta], n][x ; a , b] == (x)^(n)* Subscript[y, n][(x)^(- 1); a , b]
 Skipped - no semantic math Skipped - no semantic math - -
18.34.E4 y n + 1 ⁑ ( x ; a ) = ( A n ⁒ x + B n ) ⁒ y n ⁑ ( x ; a ) - C n ⁒ y n - 1 ⁑ ( x ; a ) Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž subscript 𝐴 𝑛 π‘₯ subscript 𝐡 𝑛 Bessel-polynomial-y 𝑛 π‘₯ π‘Ž subscript 𝐢 𝑛 Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž {\displaystyle{\displaystyle y_{n+1}\left(x;a\right)=(A_{n}x+B_{n})y_{n}\left(% x;a\right)-C_{n}y_{n-1}\left(x;a\right)}}
\Besselpolyy{n+1}@{x}{a} = (A_{n}x+B_{n})\Besselpolyy{n}@{x}{a}-C_{n}\Besselpolyy{n-1}@{x}{a}		 

Error

Pochhammer[n + 1 + a - 1, n + 1] (x/2)^n + 1 Hypergeometric1F1[-n + 1, -2 n + 1 - a + 2, 2/x] == (Subscript[A, n]*x + Subscript[B, n])*Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]-(Divide[- n*(2*n + a),(n + a - 1)*(2*n + a - 2)])*Pochhammer[n - 1 + a - 1, n - 1] (x/2)^n - 1 Hypergeometric1F1[-n - 1, -2 n - 1 - a + 2, 2/x]
Missing Macro Error Aborted -
Failed [300 / 300]
Result: Complex[-1.0464966909469928, 0.15625000000000006]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-13.266992864557183, -0.13671874999999994]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.34.E7 x 2 ⁒ y n β€²β€² ⁑ ( x ; a ) + ( a ⁒ x + 2 ) ⁒ y n β€² ⁑ ( x ; a ) - n ⁒ ( n + a - 1 ) ⁒ y n ⁑ ( x ; a ) = 0 superscript π‘₯ 2 diffop Bessel-polynomial-y 𝑛 2 π‘₯ π‘Ž π‘Ž π‘₯ 2 diffop Bessel-polynomial-y 𝑛 1 π‘₯ π‘Ž 𝑛 𝑛 π‘Ž 1 Bessel-polynomial-y 𝑛 π‘₯ π‘Ž 0 {\displaystyle{\displaystyle x^{2}y_{n}''\left(x;a\right)+(ax+2)y_{n}'\left(x;% a\right)-n(n+a-1)y_{n}\left(x;a\right)=0}}
x^{2}\Besselpolyy{n}''@{x}{a}+(ax+2)\Besselpolyy{n}'@{x}{a}-n(n+a-1)\Besselpolyy{n}@{x}{a} = 0		 

Error

(x)^(2)* D[Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x], {x, 2}]+(a*x + 2)*D[Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x], {x, 1}]- n*(n + a - 1)*Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x] == 0
Missing Macro Error Successful -
Failed [9 / 54]
Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 2], Rule[x, 1.5]}


Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 3], Rule[x, 1.5]}

... skip entries to safe data
18.34.E8 lim Ξ± β†’ ∞ ⁑ P n ( Ξ± , a - Ξ± - 2 ) ⁑ ( 1 + Ξ± ⁒ x ) P n ( Ξ± , a - Ξ± - 2 ) ⁑ ( 1 ) = y n ⁑ ( x ; a ) subscript β†’ 𝛼 Jacobi-polynomial-P 𝛼 π‘Ž 𝛼 2 𝑛 1 𝛼 π‘₯ Jacobi-polynomial-P 𝛼 π‘Ž 𝛼 2 𝑛 1 Bessel-polynomial-y 𝑛 π‘₯ π‘Ž {\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)% }_{n}\left(1+\alpha x\right)}{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}% \left(x;a\right)}}
\lim_{\alpha\to\infty}\frac{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1+\alpha x}}{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1}} = \Besselpolyy{n}@{x}{a}		 

Error

Limit[Divide[JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1 + \[Alpha]*x],JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1]], \[Alpha] -> Infinity, GenerateConditions->None] == Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]
Missing Macro Error Aborted - Skipped - Because timed out
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