18.7

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.7.E1	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)}}$ \ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{\Pochhammersym{\lambda+\frac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{\lambda-\frac{1}{2}}{n}@{x}		GegenbauerC(n, lambda, x) = (pochhammer(2*lambda, n))/(pochhammer(lambda +(1)/(2), n))*JacobiP(n, lambda -(1)/(2), lambda -(1)/(2), x)	GegenbauerC[n, \[Lambda], x] == Divide[Pochhammer[2*\[Lambda], n],Pochhammer[\[Lambda]+Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], \[Lambda]-Divide[1,2], x]	Successful	Successful	-	Failed [15 / 90] Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, -1.5]} Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -1.5]} ... skip entries to safe data
18.7.E2	${\displaystyle{\displaystyle P^{(\alpha,\alpha)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{{\left(2\alpha+1\right)_{n}}}C^{(\alpha+\frac{1}{2})}_{n}\left(x\right)}}$ \JacobipolyP{\alpha}{\alpha}{n}@{x} = \frac{\Pochhammersym{\alpha+1}{n}}{\Pochhammersym{2\alpha+1}{n}}\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{x}		JacobiP(n, alpha, alpha, x) = (pochhammer(alpha + 1, n))/(pochhammer(2*alpha + 1, n))*GegenbauerC(n, alpha +(1)/(2), x)	JacobiP[n, \[Alpha], \[Alpha], x] == Divide[Pochhammer[\[Alpha]+ 1, n],Pochhammer[2*\[Alpha]+ 1, n]]*GegenbauerC[n, \[Alpha]+Divide[1,2], x]	Successful	Successful	-	Successful [Tested: 27]
18.7.E3	${\displaystyle{\displaystyle T_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(1\right)}}}$ \ChebyshevpolyT{n}@{x} = \ifrac{\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}}{\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{1}}		ChebyshevT(n, x) = (JacobiP(n, -(1)/(2), -(1)/(2), x))/(JacobiP(n, -(1)/(2), -(1)/(2), 1))	ChebyshevT[n, x] == Divide[JacobiP[n, -Divide[1,2], -Divide[1,2], x],JacobiP[n, -Divide[1,2], -Divide[1,2], 1]]	Successful	Successful	-	Successful [Tested: 9]
18.7.E4	${\displaystyle{\displaystyle U_{n}\left(x\right)=C^{(1)}_{n}\left(x\right)}}$ \ChebyshevpolyU{n}@{x} = \ultrasphpoly{1}{n}@{x}		ChebyshevU(n, x) = GegenbauerC(n, 1, x)	ChebyshevU[n, x] == GegenbauerC[n, 1, x]	Successful	Successful	-	Successful [Tested: 9]
18.7.E4	${\displaystyle{\displaystyle C^{(1)}_{n}\left(x\right)=\ifrac{(n+1)P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(1\right)}}}$ \ultrasphpoly{1}{n}@{x} = \ifrac{(n+1)\JacobipolyP{\frac{1}{2}}{\frac{1}{2}}{n}@{x}}{\JacobipolyP{\frac{1}{2}}{\frac{1}{2}}{n}@{1}}		GegenbauerC(n, 1, x) = ((n + 1)*JacobiP(n, (1)/(2), (1)/(2), x))/(JacobiP(n, (1)/(2), (1)/(2), 1))	GegenbauerC[n, 1, x] == Divide[(n + 1)*JacobiP[n, Divide[1,2], Divide[1,2], x],JacobiP[n, Divide[1,2], Divide[1,2], 1]]	Successful	Successful	-	Successful [Tested: 9]
18.7.E9	${\displaystyle{\displaystyle P_{n}\left(x\right)=C^{(\frac{1}{2})}_{n}\left(x\right)}}$ \LegendrepolyP{n}@{x} = \ultrasphpoly{\frac{1}{2}}{n}@{x}		LegendreP(n, x) = GegenbauerC(n, (1)/(2), x)	LegendreP[n, x] == GegenbauerC[n, Divide[1,2], x]	Successful	Successful	-	Successful [Tested: 9]
18.7.E9	${\displaystyle{\displaystyle C^{(\frac{1}{2})}_{n}\left(x\right)=P^{(0,0)}_{n}\left(x\right)}}$ \ultrasphpoly{\frac{1}{2}}{n}@{x} = \JacobipolyP{0}{0}{n}@{x}		GegenbauerC(n, (1)/(2), x) = JacobiP(n, 0, 0, x)	GegenbauerC[n, Divide[1,2], x] == JacobiP[n, 0, 0, x]	Successful	Successful	-	Successful [Tested: 9]
18.7.E10	${\displaystyle{\displaystyle P^{*}_{n}\left(x\right)=P_{n}\left(2x-1\right)}}$ \shiftLegendrepolyP{n}@{x} = \LegendrepolyP{n}@{2x-1}		LegendreP(n, 2*(x) - 1) = LegendreP(n, 2*x - 1)	Error	Successful	Missing Macro Error	-	-
18.7.E13	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}_{2n}\left(1\right)}=\frac{P^{(\alpha,-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,-\frac{1}{2})}_{n}\left(1\right)}}}$ \frac{\JacobipolyP{\alpha}{\alpha}{2n}@{x}}{\JacobipolyP{\alpha}{\alpha}{2n}@{1}} = \frac{\JacobipolyP{\alpha}{-\frac{1}{2}}{n}@{2x^{2}-1}}{\JacobipolyP{\alpha}{-\frac{1}{2}}{n}@{1}}		(JacobiP(2*n, alpha, alpha, x))/(JacobiP(2*n, alpha, alpha, 1)) = (JacobiP(n, alpha, -(1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, -(1)/(2), 1))	Divide[JacobiP[2*n, \[Alpha], \[Alpha], x],JacobiP[2*n, \[Alpha], \[Alpha], 1]] == Divide[JacobiP[n, \[Alpha], -Divide[1,2], 2*(x)^(2)- 1],JacobiP[n, \[Alpha], -Divide[1,2], 1]]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
18.7.E14	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha)}_{2n+1}\left(1\right)}=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,\frac{1}{2})}_{n}\left(1\right)}}}$ \frac{\JacobipolyP{\alpha}{\alpha}{2n+1}@{x}}{\JacobipolyP{\alpha}{\alpha}{2n+1}@{1}} = \frac{x\JacobipolyP{\alpha}{\frac{1}{2}}{n}@{2x^{2}-1}}{\JacobipolyP{\alpha}{\frac{1}{2}}{n}@{1}}		(JacobiP(2*n + 1, alpha, alpha, x))/(JacobiP(2*n + 1, alpha, alpha, 1)) = (x*JacobiP(n, alpha, (1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, (1)/(2), 1))	Divide[JacobiP[2*n + 1, \[Alpha], \[Alpha], x],JacobiP[2*n + 1, \[Alpha], \[Alpha], 1]] == Divide[x*JacobiP[n, \[Alpha], Divide[1,2], 2*(x)^(2)- 1],JacobiP[n, \[Alpha], Divide[1,2], 1]]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
18.7.E15	${\displaystyle{\displaystyle C^{(\lambda)}_{2n}\left(x\right)=\frac{{\left(\lambda\right)_{n}}}{{\left(\tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}$ \ultrasphpoly{\lambda}{2n}@{x} = \frac{\Pochhammersym{\lambda}{n}}{\Pochhammersym{\tfrac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{-\frac{1}{2}}{n}@{2x^{2}-1}		GegenbauerC(2*n, lambda, x) = (pochhammer(lambda, n))/(pochhammer((1)/(2), n))*JacobiP(n, lambda -(1)/(2), -(1)/(2), 2*(x)^(2)- 1)	GegenbauerC[2*n, \[Lambda], x] == Divide[Pochhammer[\[Lambda], n],Pochhammer[Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], -Divide[1,2], 2*(x)^(2)- 1]	Failure	Failure	Failed [15 / 90] Result: Float(infinity)+Float(infinity)*I Test Values: {lambda = -3/2, x = 3/2, n = 2} Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -3/2, x = 3/2, n = 3} ... skip entries to safe data	Successful [Tested: 90]
18.7.E16	${\displaystyle{\displaystyle C^{(\lambda)}_{2n+1}\left(x\right)=\frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n+1}}}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}$ \ultrasphpoly{\lambda}{2n+1}@{x} = \frac{\Pochhammersym{\lambda}{n+1}}{\Pochhammersym{\frac{1}{2}}{n+1}}x\JacobipolyP{\lambda-\frac{1}{2}}{\frac{1}{2}}{n}@{2x^{2}-1}		GegenbauerC(2*n + 1, lambda, x) = (pochhammer(lambda, n + 1))/(pochhammer((1)/(2), n + 1))*x*JacobiP(n, lambda -(1)/(2), (1)/(2), 2*(x)^(2)- 1)	GegenbauerC[2*n + 1, \[Lambda], x] == Divide[Pochhammer[\[Lambda], n + 1],Pochhammer[Divide[1,2], n + 1]]*x*JacobiP[n, \[Lambda]-Divide[1,2], Divide[1,2], 2*(x)^(2)- 1]	Failure	Failure	Failed [15 / 90] Result: Float(infinity)+Float(infinity)*I Test Values: {lambda = -3/2, x = 3/2, n = 2} Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -3/2, x = 3/2, n = 3} ... skip entries to safe data	Successful [Tested: 90]
18.7.E19	${\displaystyle{\displaystyle H_{2n}\left(x\right)=(-1)^{n}2^{2n}n!L^{(-\frac{1}{2})}_{n}\left(x^{2}\right)}}$ \HermitepolyH{2n}@{x} = (-1)^{n}2^{2n}n!\LaguerrepolyL[-\frac{1}{2}]{n}@{x^{2}}		HermiteH(2*n, x) = (- 1)^(n)* (2)^(2*n)* factorial(n)*LaguerreL(n, -(1)/(2), (x)^(2))	HermiteH[2*n, x] == (- 1)^(n)* (2)^(2*n)* (n)!*LaguerreL[n, -Divide[1,2], (x)^(2)]	Missing Macro Error	Failure	-	Successful [Tested: 9]
18.7.E20	${\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}2^{2n+1}n!\,xL^{(\frac{1}{2})}_{n}\left(x^{2}\right)}}$ \HermitepolyH{2n+1}@{x} = (-1)^{n}2^{2n+1}n!\,x\LaguerrepolyL[\frac{1}{2}]{n}@{x^{2}}		HermiteH(2*n + 1, x) = (- 1)^(n)* (2)^(2*n + 1)* factorial(n)*x*LaguerreL(n, (1)/(2), (x)^(2))	HermiteH[2*n + 1, x] == (- 1)^(n)* (2)^(2*n + 1)* (n)!*x*LaguerreL[n, Divide[1,2], (x)^(2)]	Missing Macro Error	Failure	-	Successful [Tested: 9]
18.7.E21	${\displaystyle{\displaystyle\lim_{\beta\to\infty}P^{(\alpha,\beta)}_{n}\left(1-(\ifrac{2x}{\beta})\right)=L^{(\alpha)}_{n}\left(x\right)}}$ \lim_{\beta\to\infty}\JacobipolyP{\alpha}{\beta}{n}@{1-(\ifrac{2x}{\beta})} = \LaguerrepolyL[\alpha]{n}@{x}		limit(JacobiP(n, alpha, beta, 1 -((2*x)/(beta))), beta = infinity) = LaguerreL(n, alpha, x)	Limit[JacobiP[n, \[Alpha], \[Beta], 1 -(Divide[2*x,\[Beta]])], \[Beta] -> Infinity, GenerateConditions->None] == LaguerreL[n, \[Alpha], x]	Missing Macro Error	Failure	-	Failed [27 / 27] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[α, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5]} ... skip entries to safe data
18.7.E22	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}P^{(\alpha,\beta)}_{n}\left((2x/\alpha)-1\right)=(-1)^{n}L^{(\beta)}_{n}\left(x\right)}}$ \lim_{\alpha\to\infty}\JacobipolyP{\alpha}{\beta}{n}@{(2x/\alpha)-1} = (-1)^{n}\LaguerrepolyL[\beta]{n}@{x}		limit(JacobiP(n, alpha, beta, (2*x/alpha)- 1), alpha = infinity) = (- 1)^(n)* LaguerreL(n, beta, x)	Limit[JacobiP[n, \[Alpha], \[Beta], (2*x/\[Alpha])- 1], \[Alpha] -> Infinity, GenerateConditions->None] == (- 1)^(n)* LaguerreL[n, \[Beta], x]	Missing Macro Error	Failure	-	Failed [27 / 27] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[β, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[β, 1.5]} ... skip entries to safe data
18.7.E23	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}}}$ \lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}\JacobipolyP{\alpha}{\alpha}{n}@{\alpha^{-\frac{1}{2}}x} = \frac{\HermitepolyH{n}@{x}}{2^{n}n!}		limit((alpha)^(-(1)/(2)*n)* JacobiP(n, alpha, alpha, (alpha)^(-(1)/(2))* x), alpha = infinity) = (HermiteH(n, x))/((2)^(n)* factorial(n))	Limit[\[Alpha]^(-Divide[1,2]*n)* JacobiP[n, \[Alpha], \[Alpha], \[Alpha]^(-Divide[1,2])* x], \[Alpha] -> Infinity, GenerateConditions->None] == Divide[HermiteH[n, x],(2)^(n)* (n)!]	Failure	Aborted	Error	Successful [Tested: 9]
18.7.E24	${\displaystyle{\displaystyle\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}C^{(\lambda)}_{n}\left(\lambda^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}}}$ \lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}\ultrasphpoly{\lambda}{n}@{\lambda^{-\frac{1}{2}}x} = \frac{\HermitepolyH{n}@{x}}{n!}		limit((lambda)^(-(1)/(2)*n)* GegenbauerC(n, lambda, (lambda)^(-(1)/(2))* x), lambda = infinity) = (HermiteH(n, x))/(factorial(n))	Limit[\[Lambda]^(-Divide[1,2]*n)* GegenbauerC[n, \[Lambda], \[Lambda]^(-Divide[1,2])* x], \[Lambda] -> Infinity, GenerateConditions->None] == Divide[HermiteH[n, x],(n)!]	Failure	Aborted	Successful [Tested: 9]	Successful [Tested: 9]
18.7.E25	${\displaystyle{\displaystyle\lim_{\lambda\to 0}\frac{1}{\lambda}C^{(\lambda)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)}}$ \lim_{\lambda\to 0}\frac{1}{\lambda}\ultrasphpoly{\lambda}{n}@{x} = \frac{2}{n}\ChebyshevpolyT{n}@{x}	${\displaystyle{\displaystyle n\geq 1}}$	limit((1)/(lambda)*GegenbauerC(n, lambda, x), lambda = 0) = (2)/(n)*ChebyshevT(n, x)	Limit[Divide[1,\[Lambda]]*GegenbauerC[n, \[Lambda], x], \[Lambda] -> 0, GenerateConditions->None] == Divide[2,n]*ChebyshevT[n, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.7.E26	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{(\alpha)}_{n}\left((2\alpha)^{\frac{1}{2}}x+\alpha\right)=\frac{(-1)^{n}}{n!}H_{n}\left(x\right)}}$ \lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}\LaguerrepolyL[\alpha]{n}@{(2\alpha)^{\frac{1}{2}}x+\alpha} = \frac{(-1)^{n}}{n!}\HermitepolyH{n}@{x}		limit(((2)/(alpha))^((1)/(2)*n)* LaguerreL(n, alpha, (2*alpha)^((1)/(2))* x + alpha), alpha = infinity) = ((- 1)^(n))/(factorial(n))*HermiteH(n, x)	Limit[(Divide[2,\[Alpha]])^(Divide[1,2]*n)* LaguerreL[n, \[Alpha], (2*\[Alpha])^(Divide[1,2])* x + \[Alpha]], \[Alpha] -> Infinity, GenerateConditions->None] == Divide[(- 1)^(n),(n)!]*HermiteH[n, x]	Missing Macro Error	Aborted	-	Successful [Tested: 9]