18.34

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.34.E1	${\displaystyle{\displaystyle y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)}}$ \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	${\displaystyle{\displaystyle{{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)={\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n\atop-2n-a+2};\frac{2}{x}\right)}}$ \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	${\displaystyle{\displaystyle y_{n}(x)=y_{n}\left(x;2\right)}}$ 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	${\displaystyle{\displaystyle\theta_{n}(x)=x^{n}y_{n}(x^{-1})}}$ \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	${\displaystyle{\displaystyle y_{n}(x;a,b)=y_{n}\left(2x/b;a\right)}}$ 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	${\displaystyle{\displaystyle\theta_{n}(x;a,b)=x^{n}y_{n}(x^{-1};a,b)}}$ \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	${\displaystyle{\displaystyle y_{n+1}\left(x;a\right)=(A_{n}x+B_{n})y_{n}\left(x;a\right)-C_{n}y_{n-1}\left(x;a\right)}}$ \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	${\displaystyle{\displaystyle x^{2}y_{n}''\left(x;a\right)+(ax+2)y_{n}'\left(x;a\right)-n(n+a-1)y_{n}\left(x;a\right)=0}}$ 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	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)}_{n}\left(1+\alpha x\right)}{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}\left(x;a\right)}}$ \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