14.7: Difference between revisions

Latest revision as of 11:36, 28 June 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.7.E1	${\displaystyle{\displaystyle\mathsf{P}^{0}_{n}\left(x\right)=\mathsf{P}_{n}% \left(x\right)}}$ `\FerrersP[0]{n}@{x} = \FerrersP[]{n}@{x}`	${\displaystyle{\displaystyle\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreP(n, 0, x) = LegendreP(n, x)	LegendreP[n, 0, x] == LegendreP[n, x]	Successful	Successful	-	Successful [Tested: 3]
14.7.E1	${\displaystyle{\displaystyle\mathsf{P}_{n}\left(x\right)=P^{0}_{n}\left(x% \right)}}$ `\FerrersP[]{n}@{x} = \assLegendreP[0]{n}@{x}`	${\displaystyle{\displaystyle\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreP(n, x) = LegendreP(n, 0, x)	LegendreP[n, x] == LegendreP[n, 0, 3, x]	Successful	Successful	-	Successful [Tested: 3]
14.7.E1	${\displaystyle{\displaystyle P^{0}_{n}\left(x\right)=P_{n}\left(x\right)}}$ `\assLegendreP[0]{n}@{x} = \LegendrepolyP{n}@{x}`	${\displaystyle{\displaystyle\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreP(n, 0, x) = LegendreP(n, x)	LegendreP[n, 0, 3, x] == LegendreP[n, x]	Successful	Successful	-	Successful [Tested: 3]
14.7.E2	${\displaystyle{\displaystyle\mathsf{Q}^{0}_{n}\left(x\right)=\mathsf{Q}_{n}% \left(x\right)}}$ `\FerrersQ[0]{n}@{x} = \FerrersQ[]{n}@{x}`	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+0+1)>0,\Re(n-\mu+1)>0,\Re(% \nu-0+1)>0,\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreQ(n, 0, x) = LegendreQ(n, x)	LegendreQ[n, 0, x] == LegendreQ[n, x]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 9]
14.7.E2	${\displaystyle{\displaystyle\mathsf{Q}_{n}\left(x\right)=\frac{1}{2}P_{n}\left% (x\right)\ln\left(\frac{1+x}{1-x}\right)-W_{n-1}(x)}}$ `\FerrersQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{1+x}{1-x}}-W_{n-1}(x)`	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+0+1)>0,\Re(n-\mu+1)>0,\Re(% \nu-0+1)>0,\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreQ(n, x) = (1)/(2)LegendreP(n, x)ln((1 + x)/(1 - x))- W[n - 1](x)	LegendreQ[n, x] == Divide[1,2]LegendreP[n, x]Log[Divide[1 + x,1 - x]]- Subscript[W, n - 1][x]	Failure	Failure	Failed [88 / 90] Result: .2990381063-3.962388980I Test Values: {x = 3/2, W[n-1] = 1/23^(1/2)+1/2I, n = 1} Result: -.950961893-8.282078880I Test Values: {x = 3/2, W[n-1] = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [88 / 90] Result: Complex[0.299038105676658, -3.9623889803846897] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.9509618943233424, -8.282078879070655] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.7.E3	${\displaystyle{\displaystyle W_{n-1}(x)=\sum_{s=0}^{n-1}\frac{(n+s)!(\psi\left% (n+1\right)-\psi\left(s+1\right))}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}}}$ `W_{n-1}(x) = \sum_{s=0}^{n-1}\frac{(n+s)!(\digamma@{n+1}-\digamma@{s+1})}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}`		W[n - 1](x) = sum((factorial(n + s)(Psi(n + 1)- Psi(s + 1)))/((2)^(s)factorial(n - s)(factorial(s))^(2))(x - 1)^(s), s = 0..n - 1)	Subscript[W, n - 1][x] == Sum[Divide[(n + s)!(PolyGamma[n + 1]- PolyGamma[s + 1]),(2)^(s)(n - s)!((s)!)^(2)](x - 1)^(s), {s, 0, n - 1}, GenerateConditions->None]	Failure	Failure	Failed [85 / 90] Result: .2990381061+.7500000000I Test Values: {x = 3/2, W[n-1] = 1/23^(1/2)+1/2I, n = 1} Result: -.950961893+.7500000000I Test Values: {x = 3/2, W[n-1] = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [88 / 90] Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[0.5, Plus[-0.845568670196934, Times[2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Plus[1, , 1], Plus[2, , 1], Power[Plus[-1, 1.5], 2], []], Times[Plus[-1, Times[-1, ], 1], Plus[2, , 1], Plus[-1, 1.5], Plus[6, Times[11, ], Times[5, Power[, 2]], Times[-1, 1], Times[-1, Power[1, 2]], Times[-1, , 1.5], Times[-1, Power[, 2], 1.5], Times[1, 1.5], Times[Power[1, 2], 1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-22, Times[-37, ], Times[-21, Power[, 2]], Times[-4, Power[, 3]], Times[3, 1], Times[2, , 1], Times[3, Power[1, 2]], Times[2, , Power[1, 2]], Times[6, 1.5], Times[13, , 1.5], Times[9, Power[, 2], 1.5], Times[2, Power[, 3], 1.5], Times[-3, 1, 1.5], Times[-2, , 1, 1.5], Times[-3, Power[1, 2], 1.5], Times[-2, , Power[1, 2], 1.5]], [Plus[2, ]]], Times[4, Plus[1, ], Power[Plus[2, ], 3], [Pl<syntaxhighlight lang=mathematica>Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[0.0625, Plus[-36.91137340393869, Times[16.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Plus[1, , 2], Plus[2, , 2], Power[Plus[-1, 1.5], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[2, , 2], Plus[-1, 1.5], Plus[6, Times[11, ], Times[5, Power[, 2]], Times[-1, 2], Times[-1, Power[2, 2]], Times[-1, , 1.5], Times[-1, Power[, 2], 1.5], Times[2, 1.5], Times[Power[2, 2], 1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-22, Times[-37, ], Times[-21, Power[, 2]], Times[-4, Power[, 3]], Times[3, 2], Times[2, , 2], Times[3, Power[2, 2]], Times[2, , Power[2, 2]], Times[6, 1.5], Times[13, , 1.5], Times[9, Power[, 2], 1.5], Times[2, Power[, 3], 1.5], Times[-3, 2, 1.5], Times[-2, , 2, 1.5], Times[-3, Power[2, 2], 1.5], Times[-2, , Power[2, 2], 1.5]], [Plus[2, ]]], Times[4, Plus[1, ], Power[Plus[2, ], 3], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[-1, EulerGamma]], Equal[[2], Plus[Times[-1, EulerGamma], Times[Rational[1, 2], Plus[1, Times[-1, EulerGamma]], 2, Plus[1, 2], Plus[-1, 1.5]]]]}]][2.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.7.E4	${\displaystyle{\displaystyle W_{n-1}(x)=\sum_{k=1}^{n}\frac{1}{k}P_{k-1}\left(% x\right)P_{n-k}\left(x\right)}}$ `W_{n-1}(x) = \sum_{k=1}^{n}\frac{1}{k}\LegendrepolyP{k-1}@{x}\LegendrepolyP{n-k}@{x}`		W[n - 1](x) = sum((1)/(k)LegendreP(k - 1, x)LegendreP(n - k, x), k = 1..n)	Subscript[W, n - 1][x] == Sum[Divide[1,k]LegendreP[k - 1, x]LegendreP[n - k, x], {k, 1, n}, GenerateConditions->None]	Failure	Failure	Failed [85 / 90] Result: .299038106+.7500000000I Test Values: {x = 3/2, W[n-1] = 1/23^(1/2)+1/2I, n = 1} Result: -.950961894+.7500000000I Test Values: {x = 3/2, W[n-1] = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Skipped - Because timed out
14.7#Ex1	${\displaystyle{\displaystyle W_{0}(x)=1}}$ `W_{0}(x) = 1`		W[0](x) = 1	Subscript[W, 0][x] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
14.7#Ex2	${\displaystyle{\displaystyle W_{1}(x)=\tfrac{3}{2}x}}$ `W_{1}(x) = \tfrac{3}{2}x`		W[1](x) = (3)/(2)*x	Subscript[W, 1][x] == Divide[3,2]*x	Skipped - no semantic math	Skipped - no semantic math	-	-
14.7#Ex3	${\displaystyle{\displaystyle W_{2}(x)=\tfrac{5}{2}x^{2}-\tfrac{2}{3}}}$ `W_{2}(x) = \tfrac{5}{2}x^{2}-\tfrac{2}{3}`		W[2](x) = (5)/(2)*(x)^(2)-(2)/(3)	Subscript[W, 2][x] == Divide[5,2]*(x)^(2)-Divide[2,3]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.7.E6	${\displaystyle{\displaystyle Q^{0}_{n}\left(x\right)=Q_{n}\left(x\right)}}$ `\assLegendreQ[0]{n}@{x} = \assLegendreQ[]{n}@{x}`		LegendreQ(n, 0, x) = LegendreQ(n, x)	LegendreQ[n, 0, 3, x] == LegendreQ[n, 0, 3, x]	Successful	Successful	-	Successful [Tested: 9]
14.7.E6	${\displaystyle{\displaystyle Q_{n}\left(x\right)=n!\boldsymbol{Q}^{0}_{n}\left% (x\right)}}$ `\assLegendreQ[]{n}@{x} = n!\assLegendreOlverQ[0]{n}@{x}`		LegendreQ(n, x) = factorial(n)exp(-(0)PiI)LegendreQ(n,0,x)/GAMMA(n+0+1)	LegendreQ[n, 0, 3, x] == (n)!*Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1]	Successful	Successful	-	Successful [Tested: 9]
14.7.E6	${\displaystyle{\displaystyle n!\boldsymbol{Q}^{0}_{n}\left(x\right)=n!% \boldsymbol{Q}_{n}\left(x\right)}}$ `n!\assLegendreOlverQ[0]{n}@{x} = n!\assLegendreOlverQ[]{n}@{x}`		factorial(n)exp(-(0)PiI)LegendreQ(n,0,x)/GAMMA(n+0+1) = factorial(n)*LegendreQ(n,x)/GAMMA(n+1)	(n)!Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1] == (n)!Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3]	Successful	Failure	-	Failed [9 / 9] Result: Complex[0.47374510099224165, -6.531449595452549^-17] Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: Complex[-0.012907674693808963, 1.8730892901368242^-17] Test Values: {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E7	${\displaystyle{\displaystyle Q_{n}\left(x\right)=\frac{1}{2}P_{n}\left(x\right% )\ln\left(\frac{x+1}{x-1}\right)-W_{n-1}(x)}}$ `\assLegendreQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{x+1}{x-1}}-W_{n-1}(x)`		LegendreQ(n, x) = (1)/(2)LegendreP(n, x)ln((x + 1)/(x - 1))- W[n - 1](x)	LegendreQ[n, 0, 3, x] == Divide[1,2]LegendreP[n, x]Log[Divide[x + 1,x - 1]]- Subscript[W, n - 1][x]	Failure	Failure	Failed [30 / 30] Result: -3.659295226+.7500000000I Test Values: {x = 3/2, W[n-1] = 1/23^(1/2)+1/2I, n = 3} Result: -5.708333332+1.299038106I Test Values: {x = 3/2, W[n-1] = -1/2+1/2I3^(1/2), n = 3} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-3.659295227656675, 0.7499999999999999] Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-5.708333333333333, 1.299038105676658] Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.7.E8	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x% ^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{P}_{n}\left% (x\right)}}$ `\FerrersP[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersP[]{n}@{x}`	${\displaystyle{\displaystyle\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreP(n, m, x) = (- 1)^(m)(1 - (x)^(2))^(m/2) diff(LegendreP(n, x), [x$(m)])	LegendreP[n, m, x] == (- 1)^(m)(1 - (x)^(2))^(m/2) D[LegendreP[n, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
14.7.E9	${\displaystyle{\displaystyle\mathsf{Q}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x% ^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{Q}_{n}\left% (x\right)}}$ `\FerrersQ[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersQ[]{n}@{x}`	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+m+1)>0,\Re(n-\mu+1)>0,\Re(% \nu-m+1)>0,\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreQ(n, m, x) = (- 1)^(m)(1 - (x)^(2))^(m/2) diff(LegendreQ(n, x), [x$(m)])	LegendreQ[n, m, x] == (- 1)^(m)(1 - (x)^(2))^(m/2) D[LegendreQ[n, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
14.7.E10	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m+n}\frac{% \left(1-x^{2}\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m% +n}}\left(1-x^{2}\right)^{n}}}$ `\FerrersP[m]{n}@{x} = (-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(1-x^{2}\right)^{n}`	${\displaystyle{\displaystyle\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreP(n, m, x) = (- 1)^(m + n)((1 - (x)^(2))^(m/2))/((2)^(n) factorial(n))*diff((1 - (x)^(2))^(n), [x$(m + n)])	LegendreP[n, m, x] == (- 1)^(m + n)Divide[(1 - (x)^(2))^(m/2),(2)^(n) (n)!]*D[(1 - (x)^(2))^(n), {x, m + n}]	Failure	Failure	Failed [18 / 27] Result: Float(undefined)+Float(undefined)I Test Values: {x = 3/2, m = 1, n = 1} Result: Float(undefined)+Float(undefined)I Test Values: {x = 3/2, m = 1, n = 2} ... skip entries to safe data	Failed [27 / 27] Result: Plus[Complex[0.0, -1.118033988749895], Times[Complex[0.0, -0.5590169943749475], D[-1.25 Test Values: {1.5, 2.0}]]], {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]} Result: Plus[Complex[0.0, -5.031152949374526], Times[Complex[0.0, 0.13975424859373686], D[1.5625 Test Values: {1.5, 3.0}]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E11	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\left(x^{2}-1\right)^{m/2% }\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}P_{n}\left(x\right)}}$ `\assLegendreP[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\LegendrepolyP{n}@{x}`		LegendreP(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreP(n, x), [x$(m)])	LegendreP[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreP[n, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
14.7.E12	${\displaystyle{\displaystyle Q^{m}_{n}\left(x\right)=\left(x^{2}-1\right)^{m/2% }\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}Q_{n}\left(x\right)}}$ `\assLegendreQ[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\assLegendreQ[]{n}@{x}`		LegendreQ(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreQ(n, x), [x$(m)])	LegendreQ[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreQ[n, 0, 3, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Failed [18 / 27] Result: Plus[Complex[-0.4419376420578732, 5.412175187689032^-17], Times[-1.118033988749895, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[Times[-1, ], 1], Plus[1, , 1], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[1, 0, 3, 1.5]], Equal[[1], Times[-1, Plus[1, 1], Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[1.5, LegendreQ[1, 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, 1], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]} Result: Plus[Complex[-0.1998650072605977, 2.447640414032535^-17], Times[-1.118033988749895, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[Times[-1, ], 2], Plus[1, , 2], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[2, 0, 3, 1.5]], Equal[[1], Times[-1, Plus[1, 2], Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[1.5, LegendreQ[2, 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, 2], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E13	${\displaystyle{\displaystyle P_{n}\left(x\right)=\frac{1}{2^{n}n!}\frac{{% \mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{2}-1\right)^{n}}}$ `\LegendrepolyP{n}@{x} = \frac{1}{2^{n}n!}\deriv[n]{}{x}\left(x^{2}-1\right)^{n}`		LegendreP(n, x) = (1)/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(n)])	LegendreP[n, x] == Divide[1,(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, n}]	Failure	Failure	Error	Failed [6 / 9] Result: Plus[1.5, Times[-0.5, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-2, 1]], []], Times[-2, Plus[-1, Times[-1, ], 1], 1.5, [Plus[1, ]]], Times[Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], Power[Plus[-1, Power[1.5, 2]], 1]], Equal[[1], Times[2, 1, 1.5, Power[Plus[-1, Power[1.5, 2]], Plus[-1, 1]]]]}]][1.0]]], {Rule[n, 1], Rule[x, 1.5]} Result: Plus[2.875, Times[-0.25, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-2, 2]], []], Times[-2, Plus[-1, Times[-1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], Power[Plus[-1, Power[1.5, 2]], 2]], Equal[[1], Times[2, 2, 1.5, Power[Plus[-1, Power[1.5, 2]], Plus[-1, 2]]]]}]][2.0]]], {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E14	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right% )^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(x^{2}-1% \right)^{n}}}$ `\assLegendreP[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(x^{2}-1\right)^{n}`		LegendreP(n, m, x) = (((x)^(2)- 1)^(m/2))/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(m + n)])	LegendreP[n, m, 3, x] == Divide[((x)^(2)- 1)^(m/2),(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, m + n}]	Failure	Failure	Failed [18 / 27] Result: Float(undefined)+Float(undefined)I Test Values: {x = 3/2, m = 1, n = 1} Result: Float(undefined)+Float(undefined)I Test Values: {x = 3/2, m = 1, n = 2} ... skip entries to safe data	Failed [27 / 27] Result: Plus[1.118033988749895, Times[-0.5590169943749475, D[1.25 Test Values: {1.5, 2.0}]]], {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]} Result: Plus[5.031152949374526, Times[-0.13975424859373686, D[1.5625 Test Values: {1.5, 3.0}]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E15	${\displaystyle{\displaystyle P^{m}_{m}\left(x\right)=\frac{(2m)!}{2^{m}m!}% \left(x^{2}-1\right)^{m/2}}}$ `\assLegendreP[m]{m}@{x} = \frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}`		LegendreP(m, m, x) = (factorial(2m))/((2)^(m) factorial(m))*((x)^(2)- 1)^(m/2)	LegendreP[m, m, 3, x] == Divide[(2m)!,(2)^(m) (m)!]*((x)^(2)- 1)^(m/2)	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
14.7.E16	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=P^{m}_{n}\left(x% \right)}}$ `\FerrersP[m]{n}@{x} = \assLegendreP[m]{n}@{x}`	${\displaystyle{\displaystyle m>n,\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreP(n, m, x) = LegendreP(n, m, x)	LegendreP[n, m, x] == LegendreP[n, m, 3, x]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 9]
14.7.E16	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=0}}$ `\assLegendreP[m]{n}@{x} = 0`	${\displaystyle{\displaystyle m>n,\|(\tfrac{1}{2}-\tfrac{1}{2}x)\|<1}}$	LegendreP(n, m, x) = 0	LegendreP[n, m, 3, x] == 0	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
14.7.E17	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(-x\right)=(-1)^{n-m}% \mathsf{P}^{m}_{n}\left(x\right)}}$ `\FerrersP[m]{n}@{-x} = (-1)^{n-m}\FerrersP[m]{n}@{x}`	${\displaystyle{\displaystyle\|(\tfrac{1}{2}-\tfrac{1}{2}(-x))\|<1,\|(\tfrac{1}{2}% -\tfrac{1}{2}x)\|<1}}$	LegendreP(n, m, - x) = (- 1)^(n - m)* LegendreP(n, m, x)	LegendreP[n, m, - x] == (- 1)^(n - m)* LegendreP[n, m, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
14.7.E18	${\displaystyle{\displaystyle\mathsf{Q}^{+m}_{n}\left(-x\right)=(-1)^{n-m-1}% \mathsf{Q}^{+m}_{n}\left(x\right)}}$ `\FerrersQ[+ m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[+ m]{n}@{x}`		LegendreQ(n, + m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, + m, x)	LegendreQ[n, + m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, + m, x]	Failure	Failure	Error	Successful [Tested: 9]
14.7.E18	${\displaystyle{\displaystyle\mathsf{Q}^{-m}_{n}\left(-x\right)=(-1)^{n-m-1}% \mathsf{Q}^{-m}_{n}\left(x\right)}}$ `\FerrersQ[- m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[- m]{n}@{x}`	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+(-m)+1)>0,\Re(n-\mu+1)>0,% \Re(\nu-(-m)+1)>0,\|(\tfrac{1}{2}-\tfrac{1}{2}(-x))\|<1,\|(\tfrac{1}{2}-\tfrac{1}% {2}x)\|<1}}$	LegendreQ(n, - m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, - m, x)	LegendreQ[n, - m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, - m, x]	Failure	Failure	Error	Failed [3 / 9] Result: Indeterminate Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 0.5]} Result: Indeterminate Test Values: {Rule[m, 3], Rule[n, 1], Rule[x, 0.5]} ... skip entries to safe data
14.7.E19	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{% n}=\left(1-2xh+h^{2}\right)^{-1/2}}}$ `\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{n} = \left(1-2xh+h^{2}\right)^{-1/2}`		sum(LegendreP(n, x)(h)^(n), n = 0..infinity) = (1 - 2x*h + (h)^(2))^(- 1/2)	Sum[LegendreP[n, x](h)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2x*h + (h)^(2))^(- 1/2)	Failure	Successful	Error	Successful [Tested: 30]
14.7.E20	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{Q}_{n}\left(x\right)h^{% n}=\frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\\ln\left(\frac{x-h+\left(1-2xh+h^% {2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}\right)}}$ `\sum_{n=0}^{\infty}\FerrersQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}}`		sum(LegendreQ(n, x)(h)^(n), n = 0..infinity) = (1)/((1 - 2xh + (h)^(2))^(1/2)) ln((x - h +(1 - 2xh + (h)^(2))^(1/2))/((1 - (x)^(2))^(1/2)))	Sum[LegendreQ[n, x](h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2xh + (h)^(2))^(1/2)] Log[Divide[x - h +(1 - 2xh + (h)^(2))^(1/2),(1 - (x)^(2))^(1/2)]]	Failure	Failure	Manual Skip!	Skipped - Because timed out
14.7.E21	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{% -n-1}=\left(1-2xh+h^{2}\right)^{-1/2}}}$ `\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{-n-1} = \left(1-2xh+h^{2}\right)^{-1/2}`		sum(LegendreP(n, x)(h)^(- n - 1), n = 0..infinity) = (1 - 2x*h + (h)^(2))^(- 1/2)	Sum[LegendreP[n, x](h)^(- n - 1), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2x*h + (h)^(2))^(- 1/2)	Failure	Failure	Error	Failed [20 / 30] Result: Complex[-0.45970084338098294, -1.7156269037800917] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]} Result: Complex[-0.3437237693334403, -1.2827945709214845] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 2]} ... skip entries to safe data
14.7.E22	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}Q_{n}\left(x\right)h^{n}=\frac{% 1}{\left(1-2xh+h^{2}\right)^{1/2}}\\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right% )^{1/2}}{\left(x^{2}-1\right)^{1/2}}\right)}}$ `\sum_{n=0}^{\infty}\assLegendreQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}}`		sum(LegendreQ(n, x)(h)^(n), n = 0..infinity) = (1)/((1 - 2xh + (h)^(2))^(1/2)) ln((x - h +(1 - 2xh + (h)^(2))^(1/2))/(((x)^(2)- 1)^(1/2)))	Sum[LegendreQ[n, 0, 3, x](h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2xh + (h)^(2))^(1/2)] Log[Divide[x - h +(1 - 2xh + (h)^(2))^(1/2),((x)^(2)- 1)^(1/2)]]	Failure	Failure	Successful [Tested: 30]	Skipped - Because timed out

@@ Line 14: / Line 14: @@
 ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
 |-
-| [https://dlmf.nist.gov/14.7.E1 14.7.E1] || [[Item:Q4751|<math>\FerrersP[0]{n}@{x} = \FerrersP[]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[0]{n}@{x} = \FerrersP[]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, 0, x) = LegendreP(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, 0, x] == LegendreP[n, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/14.7.E1 14.7.E1] || <math qid="Q4751">\FerrersP[0]{n}@{x} = \FerrersP[]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[0]{n}@{x} = \FerrersP[]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, 0, x) = LegendreP(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, 0, x] == LegendreP[n, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/14.7.E1 14.7.E1] || [[Item:Q4751|<math>\FerrersP[]{n}@{x} = \assLegendreP[0]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[]{n}@{x} = \assLegendreP[0]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, x) = LegendreP(n, 0, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, x] == LegendreP[n, 0, 3, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/14.7.E1 14.7.E1] || <math qid="Q4751">\FerrersP[]{n}@{x} = \assLegendreP[0]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[]{n}@{x} = \assLegendreP[0]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, x) = LegendreP(n, 0, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, x] == LegendreP[n, 0, 3, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/14.7.E1 14.7.E1] || [[Item:Q4751|<math>\assLegendreP[0]{n}@{x} = \LegendrepolyP{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[0]{n}@{x} = \LegendrepolyP{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, 0, x) = LegendreP(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, 0, 3, x] == LegendreP[n, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/14.7.E1 14.7.E1] || <math qid="Q4751">\assLegendreP[0]{n}@{x} = \LegendrepolyP{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[0]{n}@{x} = \LegendrepolyP{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, 0, x) = LegendreP(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, 0, 3, x] == LegendreP[n, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/14.7.E2 14.7.E2] || [[Item:Q4752|<math>\FerrersQ[0]{n}@{x} = \FerrersQ[]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[0]{n}@{x} = \FerrersQ[]{n}@{x}</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+0+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-0+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, 0, x) = LegendreQ(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, x] == LegendreQ[n, x]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E2 14.7.E2] || <math qid="Q4752">\FerrersQ[0]{n}@{x} = \FerrersQ[]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[0]{n}@{x} = \FerrersQ[]{n}@{x}</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+0+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-0+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, 0, x) = LegendreQ(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, x] == LegendreQ[n, x]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E2 14.7.E2] || [[Item:Q4752|<math>\FerrersQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{1+x}{1-x}}-W_{n-1}(x)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{1+x}{1-x}}-W_{n-1}(x)</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+0+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-0+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, x) = (1)/(2)*LegendreP(n, x)*ln((1 + x)/(1 - x))- W[n - 1](x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, x] == Divide[1,2]*LegendreP[n, x]*Log[Divide[1 + x,1 - x]]- Subscript[W, n - 1][x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [88 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2990381063-3.962388980*I
+| [https://dlmf.nist.gov/14.7.E2 14.7.E2] || <math qid="Q4752">\FerrersQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{1+x}{1-x}}-W_{n-1}(x)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{1+x}{1-x}}-W_{n-1}(x)</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+0+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-0+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, x) = (1)/(2)*LegendreP(n, x)*ln((1 + x)/(1 - x))- W[n - 1](x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, x] == Divide[1,2]*LegendreP[n, x]*Log[Divide[1 + x,1 - x]]- Subscript[W, n - 1][x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [88 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2990381063-3.962388980*I
 Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.950961893-8.282078880*I
 Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [88 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.299038105676658, -3.9623889803846897]
@@ Line 28: / Line 28: @@
 Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E3 14.7.E3] || [[Item:Q4753|<math>W_{n-1}(x) = \sum_{s=0}^{n-1}\frac{(n+s)!(\digamma@{n+1}-\digamma@{s+1})}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>W_{n-1}(x) = \sum_{s=0}^{n-1}\frac{(n+s)!(\digamma@{n+1}-\digamma@{s+1})}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>W[n - 1](x) = sum((factorial(n + s)*(Psi(n + 1)- Psi(s + 1)))/((2)^(s)*factorial(n - s)*(factorial(s))^(2))*(x - 1)^(s), s = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[W, n - 1][x] == Sum[Divide[(n + s)!*(PolyGamma[n + 1]- PolyGamma[s + 1]),(2)^(s)*(n - s)!*((s)!)^(2)]*(x - 1)^(s), {s, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [85 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2990381061+.7500000000*I
+| [https://dlmf.nist.gov/14.7.E3 14.7.E3] || <math qid="Q4753">W_{n-1}(x) = \sum_{s=0}^{n-1}\frac{(n+s)!(\digamma@{n+1}-\digamma@{s+1})}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>W_{n-1}(x) = \sum_{s=0}^{n-1}\frac{(n+s)!(\digamma@{n+1}-\digamma@{s+1})}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>W[n - 1](x) = sum((factorial(n + s)*(Psi(n + 1)- Psi(s + 1)))/((2)^(s)*factorial(n - s)*(factorial(s))^(2))*(x - 1)^(s), s = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[W, n - 1][x] == Sum[Divide[(n + s)!*(PolyGamma[n + 1]- PolyGamma[s + 1]),(2)^(s)*(n - s)!*((s)!)^(2)]*(x - 1)^(s), {s, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [85 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2990381061+.7500000000*I
 Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.950961893+.7500000000*I
 Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [88 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[0.5, Plus[-0.845568670196934, Times[2.0, DifferenceRoot[Function[{, }
@@ Line 34: / Line 34: @@
 Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Plus[1, , 2], Plus[2, , 2], Power[Plus[-1, 1.5], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[2, , 2], Plus[-1, 1.5], Plus[6, Times[11, ], Times[5, Power[, 2]], Times[-1, 2], Times[-1, Power[2, 2]], Times[-1, , 1.5], Times[-1, Power[, 2], 1.5], Times[2, 1.5], Times[Power[2, 2], 1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-22, Times[-37, ], Times[-21, Power[, 2]], Times[-4, Power[, 3]], Times[3, 2], Times[2, , 2], Times[3, Power[2, 2]], Times[2, , Power[2, 2]], Times[6, 1.5], Times[13, , 1.5], Times[9, Power[, 2], 1.5], Times[2, Power[, 3], 1.5], Times[-3, 2, 1.5], Times[-2, , 2, 1.5], Times[-3, Power[2, 2], 1.5], Times[-2, , Power[2, 2], 1.5]], [Plus[2, ]]], Times[4, Plus[1, ], Power[Plus[2, ], 3], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[-1, EulerGamma]], Equal[[2], Plus[Times[-1, EulerGamma], Times[Rational[1, 2], Plus[1, Times[-1, EulerGamma]], 2, Plus[1, 2], Plus[-1, 1.5]]]]}]][2.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E4 14.7.E4] || [[Item:Q4754|<math>W_{n-1}(x) = \sum_{k=1}^{n}\frac{1}{k}\LegendrepolyP{k-1}@{x}\LegendrepolyP{n-k}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>W_{n-1}(x) = \sum_{k=1}^{n}\frac{1}{k}\LegendrepolyP{k-1}@{x}\LegendrepolyP{n-k}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>W[n - 1](x) = sum((1)/(k)*LegendreP(k - 1, x)*LegendreP(n - k, x), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[W, n - 1][x] == Sum[Divide[1,k]*LegendreP[k - 1, x]*LegendreP[n - k, x], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [85 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .299038106+.7500000000*I
+| [https://dlmf.nist.gov/14.7.E4 14.7.E4] || <math qid="Q4754">W_{n-1}(x) = \sum_{k=1}^{n}\frac{1}{k}\LegendrepolyP{k-1}@{x}\LegendrepolyP{n-k}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>W_{n-1}(x) = \sum_{k=1}^{n}\frac{1}{k}\LegendrepolyP{k-1}@{x}\LegendrepolyP{n-k}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>W[n - 1](x) = sum((1)/(k)*LegendreP(k - 1, x)*LegendreP(n - k, x), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[W, n - 1][x] == Sum[Divide[1,k]*LegendreP[k - 1, x]*LegendreP[n - k, x], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [85 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .299038106+.7500000000*I
 Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.950961894+.7500000000*I
 Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
 |- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/14.7#Ex1 14.7#Ex1] || [[Item:Q4755|<math>W_{0}(x) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>W_{0}(x) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">W[0](x) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[W, 0][x] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+| [https://dlmf.nist.gov/14.7#Ex1 14.7#Ex1] || <math qid="Q4755">W_{0}(x) = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>W_{0}(x) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">W[0](x) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[W, 0][x] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
 |- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/14.7#Ex2 14.7#Ex2] || [[Item:Q4756|<math>W_{1}(x) = \tfrac{3}{2}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>W_{1}(x) = \tfrac{3}{2}x</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">W[1](x) = (3)/(2)*x</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[W, 1][x] == Divide[3,2]*x</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+| [https://dlmf.nist.gov/14.7#Ex2 14.7#Ex2] || <math qid="Q4756">W_{1}(x) = \tfrac{3}{2}x</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>W_{1}(x) = \tfrac{3}{2}x</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">W[1](x) = (3)/(2)*x</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[W, 1][x] == Divide[3,2]*x</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
 |- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/14.7#Ex3 14.7#Ex3] || [[Item:Q4757|<math>W_{2}(x) = \tfrac{5}{2}x^{2}-\tfrac{2}{3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>W_{2}(x) = \tfrac{5}{2}x^{2}-\tfrac{2}{3}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">W[2](x) = (5)/(2)*(x)^(2)-(2)/(3)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[W, 2][x] == Divide[5,2]*(x)^(2)-Divide[2,3]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+| [https://dlmf.nist.gov/14.7#Ex3 14.7#Ex3] || <math qid="Q4757">W_{2}(x) = \tfrac{5}{2}x^{2}-\tfrac{2}{3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>W_{2}(x) = \tfrac{5}{2}x^{2}-\tfrac{2}{3}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">W[2](x) = (5)/(2)*(x)^(2)-(2)/(3)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[W, 2][x] == Divide[5,2]*(x)^(2)-Divide[2,3]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
 |-
-| [https://dlmf.nist.gov/14.7.E6 14.7.E6] || [[Item:Q4758|<math>\assLegendreQ[0]{n}@{x} = \assLegendreQ[]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[0]{n}@{x} = \assLegendreQ[]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, 0, x) = LegendreQ(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, 3, x] == LegendreQ[n, 0, 3, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E6 14.7.E6] || <math qid="Q4758">\assLegendreQ[0]{n}@{x} = \assLegendreQ[]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[0]{n}@{x} = \assLegendreQ[]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, 0, x) = LegendreQ(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, 3, x] == LegendreQ[n, 0, 3, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E6 14.7.E6] || [[Item:Q4758|<math>\assLegendreQ[]{n}@{x} = n!\assLegendreOlverQ[0]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[]{n}@{x} = n!\assLegendreOlverQ[0]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, x) = factorial(n)*exp(-(0)*Pi*I)*LegendreQ(n,0,x)/GAMMA(n+0+1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, 3, x] == (n)!*Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E6 14.7.E6] || <math qid="Q4758">\assLegendreQ[]{n}@{x} = n!\assLegendreOlverQ[0]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[]{n}@{x} = n!\assLegendreOlverQ[0]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, x) = factorial(n)*exp(-(0)*Pi*I)*LegendreQ(n,0,x)/GAMMA(n+0+1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, 3, x] == (n)!*Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E6 14.7.E6] || [[Item:Q4758|<math>n!\assLegendreOlverQ[0]{n}@{x} = n!\assLegendreOlverQ[]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>n!\assLegendreOlverQ[0]{n}@{x} = n!\assLegendreOlverQ[]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>factorial(n)*exp(-(0)*Pi*I)*LegendreQ(n,0,x)/GAMMA(n+0+1) = factorial(n)*LegendreQ(n,x)/GAMMA(n+1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n)!*Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1] == (n)!*Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.47374510099224165, -6.531449595452549*^-17]
+| [https://dlmf.nist.gov/14.7.E6 14.7.E6] || <math qid="Q4758">n!\assLegendreOlverQ[0]{n}@{x} = n!\assLegendreOlverQ[]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>n!\assLegendreOlverQ[0]{n}@{x} = n!\assLegendreOlverQ[]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>factorial(n)*exp(-(0)*Pi*I)*LegendreQ(n,0,x)/GAMMA(n+0+1) = factorial(n)*LegendreQ(n,x)/GAMMA(n+1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n)!*Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1] == (n)!*Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.47374510099224165, -6.531449595452549*^-17]
 Test Values: {Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.012907674693808963, 1.8730892901368242*^-17]
 Test Values: {Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E7 14.7.E7] || [[Item:Q4759|<math>\assLegendreQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{x+1}{x-1}}-W_{n-1}(x)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{x+1}{x-1}}-W_{n-1}(x)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, x) = (1)/(2)*LegendreP(n, x)*ln((x + 1)/(x - 1))- W[n - 1](x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, 3, x] == Divide[1,2]*LegendreP[n, x]*Log[Divide[x + 1,x - 1]]- Subscript[W, n - 1][x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.659295226+.7500000000*I
+| [https://dlmf.nist.gov/14.7.E7 14.7.E7] || <math qid="Q4759">\assLegendreQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{x+1}{x-1}}-W_{n-1}(x)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{x+1}{x-1}}-W_{n-1}(x)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, x) = (1)/(2)*LegendreP(n, x)*ln((x + 1)/(x - 1))- W[n - 1](x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, 0, 3, x] == Divide[1,2]*LegendreP[n, x]*Log[Divide[x + 1,x - 1]]- Subscript[W, n - 1][x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.659295226+.7500000000*I
 Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.708333332+1.299038106*I
 Test Values: {x = 3/2, W[n-1] = -1/2+1/2*I*3^(1/2), n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-3.659295227656675, 0.7499999999999999]
@@ Line 58: / Line 58: @@
 Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E8 14.7.E8] || [[Item:Q4760|<math>\FerrersP[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersP[]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersP[]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreP(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreP[n, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || Successful [Tested: 27]
+| [https://dlmf.nist.gov/14.7.E8 14.7.E8] || <math qid="Q4760">\FerrersP[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersP[]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersP[]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreP(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreP[n, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || Successful [Tested: 27]
 |-
-| [https://dlmf.nist.gov/14.7.E9 14.7.E9] || [[Item:Q4761|<math>\FerrersQ[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersQ[]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersQ[]{n}@{x}</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+m+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-m+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreQ(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreQ[n, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || Successful [Tested: 27]
+| [https://dlmf.nist.gov/14.7.E9 14.7.E9] || <math qid="Q4761">\FerrersQ[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersQ[]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersQ[]{n}@{x}</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+m+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-m+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreQ(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreQ[n, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || Successful [Tested: 27]
 |-
-| [https://dlmf.nist.gov/14.7.E10 14.7.E10] || [[Item:Q4762|<math>\FerrersP[m]{n}@{x} = (-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(1-x^{2}\right)^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{x} = (-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(1-x^{2}\right)^{n}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = (- 1)^(m + n)*((1 - (x)^(2))^(m/2))/((2)^(n)* factorial(n))*diff((1 - (x)^(2))^(n), [x$(m + n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, x] == (- 1)^(m + n)*Divide[(1 - (x)^(2))^(m/2),(2)^(n)* (n)!]*D[(1 - (x)^(2))^(n), {x, m + n}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+| [https://dlmf.nist.gov/14.7.E10 14.7.E10] || <math qid="Q4762">\FerrersP[m]{n}@{x} = (-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(1-x^{2}\right)^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{x} = (-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(1-x^{2}\right)^{n}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = (- 1)^(m + n)*((1 - (x)^(2))^(m/2))/((2)^(n)* factorial(n))*diff((1 - (x)^(2))^(n), [x$(m + n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, x] == (- 1)^(m + n)*Divide[(1 - (x)^(2))^(m/2),(2)^(n)* (n)!]*D[(1 - (x)^(2))^(n), {x, m + n}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
 Test Values: {x = 3/2, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
 Test Values: {x = 3/2, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [27 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, -1.118033988749895], Times[Complex[0.0, -0.5590169943749475], D[-1.25
@@ Line 68: / Line 68: @@
 Test Values: {1.5, 3.0}]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E11 14.7.E11] || [[Item:Q4763|<math>\assLegendreP[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\LegendrepolyP{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\LegendrepolyP{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreP(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreP[n, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || Successful [Tested: 27]
+| [https://dlmf.nist.gov/14.7.E11 14.7.E11] || <math qid="Q4763">\assLegendreP[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\LegendrepolyP{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\LegendrepolyP{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreP(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreP[n, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || Successful [Tested: 27]
 |-
-| [https://dlmf.nist.gov/14.7.E12 14.7.E12] || [[Item:Q4764|<math>\assLegendreQ[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\assLegendreQ[]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\assLegendreQ[]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreQ(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreQ[n, 0, 3, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.4419376420578732, 5.412175187689032*^-17], Times[-1.118033988749895, DifferenceRoot[Function[{, }
+| [https://dlmf.nist.gov/14.7.E12 14.7.E12] || <math qid="Q4764">\assLegendreQ[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\assLegendreQ[]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\assLegendreQ[]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreQ(n, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreQ[n, 0, 3, x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.4419376420578732, 5.412175187689032*^-17], Times[-1.118033988749895, DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[-1, Plus[Times[-1, ], 1], Plus[1, , 1], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[1, 0, 3, 1.5]], Equal[[1], Times[-1, Plus[1, 1], Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[1.5, LegendreQ[1, 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, 1], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.1998650072605977, 2.447640414032535*^-17], Times[-1.118033988749895, DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[-1, Plus[Times[-1, ], 2], Plus[1, , 2], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[2, 0, 3, 1.5]], Equal[[1], Times[-1, Plus[1, 2], Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[1.5, LegendreQ[2, 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, 2], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E13 14.7.E13] || [[Item:Q4765|<math>\LegendrepolyP{n}@{x} = \frac{1}{2^{n}n!}\deriv[n]{}{x}\left(x^{2}-1\right)^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LegendrepolyP{n}@{x} = \frac{1}{2^{n}n!}\deriv[n]{}{x}\left(x^{2}-1\right)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(n, x) = (1)/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, x] == Divide[1,(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, n}]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.5, Times[-0.5, DifferenceRoot[Function[{, }
+| [https://dlmf.nist.gov/14.7.E13 14.7.E13] || <math qid="Q4765">\LegendrepolyP{n}@{x} = \frac{1}{2^{n}n!}\deriv[n]{}{x}\left(x^{2}-1\right)^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LegendrepolyP{n}@{x} = \frac{1}{2^{n}n!}\deriv[n]{}{x}\left(x^{2}-1\right)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(n, x) = (1)/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, x] == Divide[1,(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, n}]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.5, Times[-0.5, DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[Plus[, Times[-2, 1]], []], Times[-2, Plus[-1, Times[-1, ], 1], 1.5, [Plus[1, ]]], Times[Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], Power[Plus[-1, Power[1.5, 2]], 1]], Equal[[1], Times[2, 1, 1.5, Power[Plus[-1, Power[1.5, 2]], Plus[-1, 1]]]]}]][1.0]]], {Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[2.875, Times[-0.25, DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[Plus[, Times[-2, 2]], []], Times[-2, Plus[-1, Times[-1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], Power[Plus[-1, Power[1.5, 2]], 2]], Equal[[1], Times[2, 2, 1.5, Power[Plus[-1, Power[1.5, 2]], Plus[-1, 2]]]]}]][2.0]]], {Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E14 14.7.E14] || [[Item:Q4766|<math>\assLegendreP[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(x^{2}-1\right)^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(x^{2}-1\right)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = (((x)^(2)- 1)^(m/2))/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(m + n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, 3, x] == Divide[((x)^(2)- 1)^(m/2),(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, m + n}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+| [https://dlmf.nist.gov/14.7.E14 14.7.E14] || <math qid="Q4766">\assLegendreP[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(x^{2}-1\right)^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(x^{2}-1\right)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = (((x)^(2)- 1)^(m/2))/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(m + n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, 3, x] == Divide[((x)^(2)- 1)^(m/2),(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, m + n}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
 Test Values: {x = 3/2, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
 Test Values: {x = 3/2, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [27 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.118033988749895, Times[-0.5590169943749475, D[1.25
@@ Line 84: / Line 84: @@
 Test Values: {1.5, 3.0}]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E15 14.7.E15] || [[Item:Q4767|<math>\assLegendreP[m]{m}@{x} = \frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{m}@{x} = \frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(m, m, x) = (factorial(2*m))/((2)^(m)* factorial(m))*((x)^(2)- 1)^(m/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[m, m, 3, x] == Divide[(2*m)!,(2)^(m)* (m)!]*((x)^(2)- 1)^(m/2)</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E15 14.7.E15] || <math qid="Q4767">\assLegendreP[m]{m}@{x} = \frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{m}@{x} = \frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(m, m, x) = (factorial(2*m))/((2)^(m)* factorial(m))*((x)^(2)- 1)^(m/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[m, m, 3, x] == Divide[(2*m)!,(2)^(m)* (m)!]*((x)^(2)- 1)^(m/2)</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E16 14.7.E16] || [[Item:Q4768|<math>\FerrersP[m]{n}@{x} = \assLegendreP[m]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{x} = \assLegendreP[m]{n}@{x}</syntaxhighlight> || <math>m > n, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = LegendreP(n, m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, x] == LegendreP[n, m, 3, x]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E16 14.7.E16] || <math qid="Q4768">\FerrersP[m]{n}@{x} = \assLegendreP[m]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{x} = \assLegendreP[m]{n}@{x}</syntaxhighlight> || <math>m > n, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = LegendreP(n, m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, x] == LegendreP[n, m, 3, x]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E16 14.7.E16] || [[Item:Q4768|<math>\assLegendreP[m]{n}@{x} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{n}@{x} = 0</syntaxhighlight> || <math>m > n, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, 3, x] == 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E16 14.7.E16] || <math qid="Q4768">\assLegendreP[m]{n}@{x} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{n}@{x} = 0</syntaxhighlight> || <math>m > n, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, x) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, 3, x] == 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E17 14.7.E17] || [[Item:Q4769|<math>\FerrersP[m]{n}@{-x} = (-1)^{n-m}\FerrersP[m]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{-x} = (-1)^{n-m}\FerrersP[m]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}(-x))| < 1, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, - x) = (- 1)^(n - m)* LegendreP(n, m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, - x] == (- 1)^(n - m)* LegendreP[n, m, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E17 14.7.E17] || <math qid="Q4769">\FerrersP[m]{n}@{-x} = (-1)^{n-m}\FerrersP[m]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{n}@{-x} = (-1)^{n-m}\FerrersP[m]{n}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}(-x))| < 1, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(n, m, - x) = (- 1)^(n - m)* LegendreP(n, m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, m, - x] == (- 1)^(n - m)* LegendreP[n, m, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E18 14.7.E18] || [[Item:Q4770|<math>\FerrersQ[+ m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[+ m]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[+ m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[+ m]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, + m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, + m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, + m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, + m, x]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 9]
+| [https://dlmf.nist.gov/14.7.E18 14.7.E18] || <math qid="Q4770">\FerrersQ[+ m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[+ m]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[+ m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[+ m]{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(n, + m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, + m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, + m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, + m, x]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/14.7.E18 14.7.E18] || [[Item:Q4770|<math>\FerrersQ[- m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[- m]{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[- m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[- m]{n}@{x}</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+(- m)+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-(- m)+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}(-x))| < 1, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, - m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, - m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, - m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, - m, x]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+| [https://dlmf.nist.gov/14.7.E18 14.7.E18] || <math qid="Q4770">\FerrersQ[- m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[- m]{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[- m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[- m]{n}@{x}</syntaxhighlight> || <math>\realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+(- m)+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-(- m)+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}(-x))| < 1, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(n, - m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, - m, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[n, - m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, - m, x]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
 Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 0.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
 Test Values: {Rule[m, 3], Rule[n, 1], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E19 14.7.E19] || [[Item:Q4771|<math>\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{n} = \left(1-2xh+h^{2}\right)^{-1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{n} = \left(1-2xh+h^{2}\right)^{-1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreP(n, x)*(h)^(n), n = 0..infinity) = (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreP[n, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || Failure || Successful || Error || Successful [Tested: 30]
+| [https://dlmf.nist.gov/14.7.E19 14.7.E19] || <math qid="Q4771">\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{n} = \left(1-2xh+h^{2}\right)^{-1/2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{n} = \left(1-2xh+h^{2}\right)^{-1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreP(n, x)*(h)^(n), n = 0..infinity) = (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreP[n, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || Failure || Successful || Error || Successful [Tested: 30]
 |-
-| [https://dlmf.nist.gov/14.7.E20 14.7.E20] || [[Item:Q4772|<math>\sum_{n=0}^{\infty}\FerrersQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\FerrersQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity) = (1)/((1 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/2))/((1 - (x)^(2))^(1/2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreQ[n, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2*x*h + (h)^(2))^(1/2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/2),(1 - (x)^(2))^(1/2)]]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
+| [https://dlmf.nist.gov/14.7.E20 14.7.E20] || <math qid="Q4772">\sum_{n=0}^{\infty}\FerrersQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\FerrersQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity) = (1)/((1 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/2))/((1 - (x)^(2))^(1/2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreQ[n, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2*x*h + (h)^(2))^(1/2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/2),(1 - (x)^(2))^(1/2)]]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/14.7.E21 14.7.E21] || [[Item:Q4773|<math>\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{-n-1} = \left(1-2xh+h^{2}\right)^{-1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{-n-1} = \left(1-2xh+h^{2}\right)^{-1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreP(n, x)*(h)^(- n - 1), n = 0..infinity) = (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreP[n, x]*(h)^(- n - 1), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.45970084338098294, -1.7156269037800917]
+| [https://dlmf.nist.gov/14.7.E21 14.7.E21] || <math qid="Q4773">\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{-n-1} = \left(1-2xh+h^{2}\right)^{-1/2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{-n-1} = \left(1-2xh+h^{2}\right)^{-1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreP(n, x)*(h)^(- n - 1), n = 0..infinity) = (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreP[n, x]*(h)^(- n - 1), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*h + (h)^(2))^(- 1/2)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.45970084338098294, -1.7156269037800917]
 Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.3437237693334403, -1.2827945709214845]
 Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/14.7.E22 14.7.E22] || [[Item:Q4774|<math>\sum_{n=0}^{\infty}\assLegendreQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\assLegendreQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity) = (1)/((1 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/2))/(((x)^(2)- 1)^(1/2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreQ[n, 0, 3, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2*x*h + (h)^(2))^(1/2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/2),((x)^(2)- 1)^(1/2)]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Skipped - Because timed out
+| [https://dlmf.nist.gov/14.7.E22 14.7.E22] || <math qid="Q4774">\sum_{n=0}^{\infty}\assLegendreQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\assLegendreQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity) = (1)/((1 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/2))/(((x)^(2)- 1)^(1/2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[LegendreQ[n, 0, 3, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2*x*h + (h)^(2))^(1/2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/2),((x)^(2)- 1)^(1/2)]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Skipped - Because timed out
 |}
 </div>

14.7: Difference between revisions

Latest revision as of 11:36, 28 June 2021

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