Legendre and Related Functions - 14.18 Sums

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.18.E1 𝖯 Ξ½ ⁑ ( cos ⁑ ΞΈ 1 ⁒ cos ⁑ ΞΈ 2 + sin ⁑ ΞΈ 1 ⁒ sin ⁑ ΞΈ 2 ⁒ cos ⁑ Ο• ) = 𝖯 Ξ½ ⁑ ( cos ⁑ ΞΈ 1 ) ⁒ 𝖯 Ξ½ ⁑ ( cos ⁑ ΞΈ 2 ) + 2 ⁒ βˆ‘ m = 1 ∞ ( - 1 ) m ⁒ 𝖯 Ξ½ - m ⁑ ( cos ⁑ ΞΈ 1 ) ⁒ 𝖯 Ξ½ m ⁑ ( cos ⁑ ΞΈ 2 ) ⁒ cos ⁑ ( m ⁒ Ο• ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 subscript πœƒ 1 subscript πœƒ 2 subscript πœƒ 1 subscript πœƒ 2 italic-Ο• shorthand-Ferrers-Legendre-P-first-kind 𝜈 subscript πœƒ 1 shorthand-Ferrers-Legendre-P-first-kind 𝜈 subscript πœƒ 2 2 superscript subscript π‘š 1 superscript 1 π‘š Ferrers-Legendre-P-first-kind π‘š 𝜈 subscript πœƒ 1 Ferrers-Legendre-P-first-kind π‘š 𝜈 subscript πœƒ 2 π‘š italic-Ο• {\displaystyle{\displaystyle\mathsf{P}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}% +\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_% {1}\right)\mathsf{P}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)% ^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{\nu}\left(% \cos\theta_{2}\right)\cos\left(m\phi\right)}}
\FerrersP[]{\nu}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \FerrersP[]{\nu}@{\cos@@{\theta_{1}}}\FerrersP[]{\nu}@{\cos@@{\theta_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\FerrersP[-m]{\nu}@{\cos@@{\theta_{1}}}\FerrersP[m]{\nu}@{\cos@@{\theta_{2}}}\cos@{m\phi}		 

LegendreP(nu, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)) = LegendreP(nu, cos(theta[1]))*LegendreP(nu, cos(theta[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cos(theta[1]))*LegendreP(nu, m, cos(theta[2]))*cos(m*phi), m = 1..infinity)

LegendreP[\[Nu], Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], Cos[Subscript[\[Theta], 1]]]*LegendreP[\[Nu], Cos[Subscript[\[Theta], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, Cos[Subscript[\[Theta], 1]]]*LegendreP[\[Nu], m, Cos[Subscript[\[Theta], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]
Aborted Failure Manual Skip! Skipped - Because timed out
14.18.E2 𝖯 n ⁑ ( cos ⁑ ΞΈ 1 ⁒ cos ⁑ ΞΈ 2 + sin ⁑ ΞΈ 1 ⁒ sin ⁑ ΞΈ 2 ⁒ cos ⁑ Ο• ) = βˆ‘ m = - n n ( - 1 ) m ⁒ 𝖯 n - m ⁑ ( cos ⁑ ΞΈ 1 ) ⁒ 𝖯 n m ⁑ ( cos ⁑ ΞΈ 2 ) ⁒ cos ⁑ ( m ⁒ Ο• ) shorthand-Ferrers-Legendre-P-first-kind 𝑛 subscript πœƒ 1 subscript πœƒ 2 subscript πœƒ 1 subscript πœƒ 2 italic-Ο• superscript subscript π‘š 𝑛 𝑛 superscript 1 π‘š Ferrers-Legendre-P-first-kind π‘š 𝑛 subscript πœƒ 1 Ferrers-Legendre-P-first-kind π‘š 𝑛 subscript πœƒ 2 π‘š italic-Ο• {\displaystyle{\displaystyle\mathsf{P}_{n}\left(\cos\theta_{1}\cos\theta_{2}+% \sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\sum_{m=-n}^{n}(-1)^{m}\mathsf{P}^% {-m}_{n}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{n}\left(\cos\theta_{2}% \right)\cos\left(m\phi\right)}}
\FerrersP[]{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \sum_{m=-n}^{n}(-1)^{m}\FerrersP[-m]{n}@{\cos@@{\theta_{1}}}\FerrersP[m]{n}@{\cos@@{\theta_{2}}}\cos@{m\phi}		 

LegendreP(n, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)) = sum((- 1)^(m)* LegendreP(n, - m, cos(theta[1]))*LegendreP(n, m, cos(theta[2]))*cos(m*phi), m = - n..n)

LegendreP[n, Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]] == Sum[(- 1)^(m)* LegendreP[n, - m, Cos[Subscript[\[Theta], 1]]]*LegendreP[n, m, Cos[Subscript[\[Theta], 2]]]*Cos[m*\[Phi]], {m, - n, n}, GenerateConditions->None]
Aborted Failure Manual Skip! Skipped - Because timed out
14.18.E3 𝖰 Ξ½ ⁑ ( cos ⁑ ΞΈ 1 ⁒ cos ⁑ ΞΈ 2 + sin ⁑ ΞΈ 1 ⁒ sin ⁑ ΞΈ 2 ⁒ cos ⁑ Ο• ) = 𝖯 Ξ½ ⁑ ( cos ⁑ ΞΈ 1 ) ⁒ 𝖰 Ξ½ ⁑ ( cos ⁑ ΞΈ 2 ) + 2 ⁒ βˆ‘ m = 1 ∞ ( - 1 ) m ⁒ 𝖯 Ξ½ - m ⁑ ( cos ⁑ ΞΈ 1 ) ⁒ 𝖰 Ξ½ m ⁑ ( cos ⁑ ΞΈ 2 ) ⁒ cos ⁑ ( m ⁒ Ο• ) shorthand-Ferrers-Legendre-Q-first-kind 𝜈 subscript πœƒ 1 subscript πœƒ 2 subscript πœƒ 1 subscript πœƒ 2 italic-Ο• shorthand-Ferrers-Legendre-P-first-kind 𝜈 subscript πœƒ 1 shorthand-Ferrers-Legendre-Q-first-kind 𝜈 subscript πœƒ 2 2 superscript subscript π‘š 1 superscript 1 π‘š Ferrers-Legendre-P-first-kind π‘š 𝜈 subscript πœƒ 1 Ferrers-Legendre-Q-first-kind π‘š 𝜈 subscript πœƒ 2 π‘š italic-Ο• {\displaystyle{\displaystyle\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}% +\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_% {1}\right)\mathsf{Q}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)% ^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(% \cos\theta_{2}\right)\cos\left(m\phi\right)}}
\FerrersQ[]{\nu}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \FerrersP[]{\nu}@{\cos@@{\theta_{1}}}\FerrersQ[]{\nu}@{\cos@@{\theta_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\FerrersP[-m]{\nu}@{\cos@@{\theta_{1}}}\FerrersQ[m]{\nu}@{\cos@@{\theta_{2}}}\cos@{m\phi}		 

LegendreQ(nu, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)) = LegendreP(nu, cos(theta[1]))*LegendreQ(nu, cos(theta[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cos(theta[1]))*LegendreQ(nu, m, cos(theta[2]))*cos(m*phi), m = 1..infinity)

LegendreQ[\[Nu], Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], Cos[Subscript[\[Theta], 1]]]*LegendreQ[\[Nu], Cos[Subscript[\[Theta], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, Cos[Subscript[\[Theta], 1]]]*LegendreQ[\[Nu], m, Cos[Subscript[\[Theta], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]
Aborted Failure Manual Skip! Skipped - Because timed out
14.18.E4 P Ξ½ ⁑ ( cosh ⁑ ΞΎ 1 ⁒ cosh ⁑ ΞΎ 2 - sinh ⁑ ΞΎ 1 ⁒ sinh ⁑ ΞΎ 2 ⁒ cos ⁑ Ο• ) = P Ξ½ ⁑ ( cosh ⁑ ΞΎ 1 ) ⁒ P Ξ½ ⁑ ( cosh ⁑ ΞΎ 2 ) + 2 ⁒ βˆ‘ m = 1 ∞ ( - 1 ) m ⁒ P Ξ½ - m ⁑ ( cosh ⁑ ΞΎ 1 ) ⁒ P Ξ½ m ⁑ ( cosh ⁑ ΞΎ 2 ) ⁒ cos ⁑ ( m ⁒ Ο• ) shorthand-Legendre-P-first-kind 𝜈 subscript πœ‰ 1 subscript πœ‰ 2 subscript πœ‰ 1 subscript πœ‰ 2 italic-Ο• shorthand-Legendre-P-first-kind 𝜈 subscript πœ‰ 1 shorthand-Legendre-P-first-kind 𝜈 subscript πœ‰ 2 2 superscript subscript π‘š 1 superscript 1 π‘š Legendre-P-first-kind π‘š 𝜈 subscript πœ‰ 1 Legendre-P-first-kind π‘š 𝜈 subscript πœ‰ 2 π‘š italic-Ο• {\displaystyle{\displaystyle P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1% }\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(% \cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}% \right)P^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right)}}
\assLegendreP[]{\nu}@{\cosh@@{\xi_{1}}\cosh@@{\xi_{2}}-\sinh@@{\xi_{1}}\sinh@@{\xi_{2}}\cos@@{\phi}} = \assLegendreP[]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreP[]{\nu}@{\cosh@@{\xi_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\assLegendreP[-m]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreP[m]{\nu}@{\cosh@@{\xi_{2}}}\cos@{m\phi}		 

LegendreP(nu, cosh(xi[1])*cosh(xi[2])- sinh(xi[1])*sinh(xi[2])*cos(phi)) = LegendreP(nu, cosh(xi[1]))*LegendreP(nu, cosh(xi[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cosh(xi[1]))*LegendreP(nu, m, cosh(xi[2]))*cos(m*phi), m = 1..infinity)

LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]*Cosh[Subscript[\[Xi], 2]]- Sinh[Subscript[\[Xi], 1]]*Sinh[Subscript[\[Xi], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreP[\[Nu], m, 3, Cosh[Subscript[\[Xi], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]
Aborted Failure Manual Skip! Skipped - Because timed out
14.18.E5 Q Ξ½ ⁑ ( cosh ⁑ ΞΎ 1 ⁒ cosh ⁑ ΞΎ 2 - sinh ⁑ ΞΎ 1 ⁒ sinh ⁑ ΞΎ 2 ⁒ cos ⁑ Ο• ) = P Ξ½ ⁑ ( cosh ⁑ ΞΎ 1 ) ⁒ Q Ξ½ ⁑ ( cosh ⁑ ΞΎ 2 ) + 2 ⁒ βˆ‘ m = 1 ∞ ( - 1 ) m ⁒ P Ξ½ - m ⁑ ( cosh ⁑ ΞΎ 1 ) ⁒ Q Ξ½ m ⁑ ( cosh ⁑ ΞΎ 2 ) ⁒ cos ⁑ ( m ⁒ Ο• ) shorthand-Legendre-Q-second-kind 𝜈 subscript πœ‰ 1 subscript πœ‰ 2 subscript πœ‰ 1 subscript πœ‰ 2 italic-Ο• shorthand-Legendre-P-first-kind 𝜈 subscript πœ‰ 1 shorthand-Legendre-Q-second-kind 𝜈 subscript πœ‰ 2 2 superscript subscript π‘š 1 superscript 1 π‘š Legendre-P-first-kind π‘š 𝜈 subscript πœ‰ 1 Legendre-Q-second-kind π‘š 𝜈 subscript πœ‰ 2 π‘š italic-Ο• {\displaystyle{\displaystyle Q_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1% }\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)Q_{\nu}\left(% \cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}% \right)Q^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right)}}
\assLegendreQ[]{\nu}@{\cosh@@{\xi_{1}}\cosh@@{\xi_{2}}-\sinh@@{\xi_{1}}\sinh@@{\xi_{2}}\cos@@{\phi}} = \assLegendreP[]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreQ[]{\nu}@{\cosh@@{\xi_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\assLegendreP[-m]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreQ[m]{\nu}@{\cosh@@{\xi_{2}}}\cos@{m\phi}		 

LegendreQ(nu, cosh(xi[1])*cosh(xi[2])- sinh(xi[1])*sinh(xi[2])*cos(phi)) = LegendreP(nu, cosh(xi[1]))*LegendreQ(nu, cosh(xi[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cosh(xi[1]))*LegendreQ(nu, m, cosh(xi[2]))*cos(m*phi), m = 1..infinity)

LegendreQ[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]*Cosh[Subscript[\[Xi], 2]]- Sinh[Subscript[\[Xi], 1]]*Sinh[Subscript[\[Xi], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreQ[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreQ[\[Nu], m, 3, Cosh[Subscript[\[Xi], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]
Aborted Failure Manual Skip! Skipped - Because timed out
14.18.E6 ( x - y ) ⁒ βˆ‘ k = 0 n ( 2 ⁒ k + 1 ) ⁒ P k ⁑ ( x ) ⁒ P k ⁑ ( y ) = ( n + 1 ) ⁒ ( P n + 1 ⁑ ( x ) ⁒ P n ⁑ ( y ) - P n ⁑ ( x ) ⁒ P n + 1 ⁑ ( y ) ) π‘₯ 𝑦 superscript subscript π‘˜ 0 𝑛 2 π‘˜ 1 shorthand-Legendre-P-first-kind π‘˜ π‘₯ shorthand-Legendre-P-first-kind π‘˜ 𝑦 𝑛 1 shorthand-Legendre-P-first-kind 𝑛 1 π‘₯ shorthand-Legendre-P-first-kind 𝑛 𝑦 shorthand-Legendre-P-first-kind 𝑛 π‘₯ shorthand-Legendre-P-first-kind 𝑛 1 𝑦 {\displaystyle{\displaystyle(x-y)\sum_{k=0}^{n}(2k+1)P_{k}\left(x\right)P_{k}% \left(y\right)=(n+1)\left(P_{n+1}\left(x\right)P_{n}\left(y\right)-P_{n}\left(% x\right)P_{n+1}\left(y\right)\right)}}
(x-y)\sum_{k=0}^{n}(2k+1)\assLegendreP[]{k}@{x}\assLegendreP[]{k}@{y} = (n+1)\left(\assLegendreP[]{n+1}@{x}\assLegendreP[]{n}@{y}-\assLegendreP[]{n}@{x}\assLegendreP[]{n+1}@{y}\right)		 

(x - y)*sum((2*k + 1)*LegendreP(k, x)*LegendreP(k, y), k = 0..n) = (n + 1)*(LegendreP(n + 1, x)*LegendreP(n, y)- LegendreP(n, x)*LegendreP(n + 1, y))

(x - y)*Sum[(2*k + 1)*LegendreP[k, 0, 3, x]*LegendreP[k, 0, 3, y], {k, 0, n}, GenerateConditions->None] == (n + 1)*(LegendreP[n + 1, 0, 3, x]*LegendreP[n, 0, 3, y]- LegendreP[n, 0, 3, x]*LegendreP[n + 1, 0, 3, y])
Aborted Aborted Manual Skip!
Failed [42 / 54]
Result: Plus[17.25, Times[0.75, Plus[-28.0625, Times[8.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[, 2], Power[-1.5,<syntaxhighlight lang=mathematica>Result: Plus[-106.73437499999997, Times[0.75, Plus[-28.0625, Times[8.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[, 2], Power[-1.5, 2]], Times[68, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[2, ]]], Times[, Plus[1, ], Plus[-165, Times[-271, ], Times[-162, Power[, 2]], Times[-42, Power[, 3]], Times[-4, Power[, 4]], Times[315, Power[1.5, 2]], Times[531, , Power[1.5, 2]], Times[322, Power[, 2], Power[1.5, 2]], Times[84, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[294, 1.5, -1.5], Times[511, , 1.5, -1.5], Times[318, Power[, 2], 1.5, -1.5], Times[84, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[315, Power[-1.5, 2]], Times[531, , Power[-1.5, 2]], Times[322, Power[, 2], Power[-1.5, 2]], Times[84, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[3, Times[2, ]], Plus[12, Times[7, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[, Plus[1, ], Plus[2, ], Plus[3, ], Plus[4, ], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[1], 0], Equal[[2], Times[1.5, -1.5]], Equal[[3], Plus[Times[1.5, -1.5], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]], Equal[[4], Plus[Times[1.5, -1.5], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]], Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[Times[-2, -1.5], Times[Rational[5, 2], -1.5, Plus[-1, Times[3, Power[-1.5, 2]]]]]]]], Equal[[5], Plus[Times[1.5, -1.5], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]], Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[Times[-2, -1.5], Times[Rational[5, 2], -1.5, Plus[-1, Times[3, Power[-1.5, 2]]]]]], Times[Rational[1, 24], Plus[1, Times[-3, Power[1.5, 2]], Times[-8, Plus[-1, Times[3, Power[1.5, 2]]]], Times[7, 1.5, Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]]]], Plus[1, Times[-3, Power[-1.5, 2]], Times[Rational[1, 2], Plus[1, Times[-3, Power[-1.5, 2]]]], Times[Rational[7, 3], -1.5, Plus[Times[-2, -1.5], Times[Rational[5, 2], -1.5, Plus[-1, Times[3, Power[-1.5, 2]]]]]]]]]]}]][3.0]], Times[4.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Power[Plus[1, ], 2], Plus[7, Times[2, ]], []], Times[Plus[7, Times[2, ]], Plus[1, Times[2, ], Power[, 2], Times[9, 1.5, -1.5], Times[12, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[1, ]]], Times[Plus[55, Times[72, ], Times[30, Power[, 2]], Times[4, Power[, 3]], Times[-105, Power[1.5, 2]], Times[-142, , Power[1.5, 2]], Times[-60, Power[, 2], Power[1.5, 2]], Times[-8, Power[, 3], Power[1.5, 2]], Times[-63, 1.5, -1.5], Times[-102, , 1.5, -1.5], Times[-52, Power[, 2], 1.5, -1.5], Times[-8, Power[, 3], 1.5, -1.5], Times[-105, Power[-1.5, 2]], Times[-142, , Power[-1.5, 2]], Times[-60, Power[, 2], Power[-1.5, 2]], Times[-8, Power[, 3], Power[-1.5, 2]]], [Plus[2, ]]], Times[Plus[-55, Times[-72, ], Times[-30, Power[, 2]], Times[-4, Power[, 3]], Times[105, Power[1.5, 2]], Times[142, , Power[1.5, 2]], Times[60, Power[, 2], Power[1.5, 2]], Times[8, Power[, 3], Power[1.5, 2]], Times[147, 1.5, -1.5], Times[182, , 1.5, -1.5], Times[68, Power[, 2], 1.5, -1.5], Times[8, Power[, 3], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[142, , Power[-1.5, 2]], Times[60, Power[, 2], Power[-1.5, 2]], Times[8, Power[, 3], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, Plus[3, Times[2, ]], Plus[16, Times[8, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[Power[Plus[4, ], 2], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[-3], 0], Equal[[-2], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]], Equal[[-1], Plus[Times[1.5, -1.5], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]], Equal[[0], Plus[1, Times[1.5, -1.5], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]], Equal[[1], Plus[2, Times[1.5, -1.5], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]]}]][3.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]}

... skip entries to safe data
14.18.E7 ( x - y ) ⁒ βˆ‘ k = 0 n ( 2 ⁒ k + 1 ) ⁒ P k ⁑ ( x ) ⁒ Q k ⁑ ( y ) = ( n + 1 ) ⁒ ( P n + 1 ⁑ ( x ) ⁒ Q n ⁑ ( y ) - P n ⁑ ( x ) ⁒ Q n + 1 ⁑ ( y ) ) - 1 π‘₯ 𝑦 superscript subscript π‘˜ 0 𝑛 2 π‘˜ 1 shorthand-Legendre-P-first-kind π‘˜ π‘₯ shorthand-Legendre-Q-second-kind π‘˜ 𝑦 𝑛 1 shorthand-Legendre-P-first-kind 𝑛 1 π‘₯ shorthand-Legendre-Q-second-kind 𝑛 𝑦 shorthand-Legendre-P-first-kind 𝑛 π‘₯ shorthand-Legendre-Q-second-kind 𝑛 1 𝑦 1 {\displaystyle{\displaystyle(x-y)\sum_{k=0}^{n}(2k+1)P_{k}\left(x\right)Q_{k}% \left(y\right)=(n+1)\left(P_{n+1}\left(x\right)Q_{n}\left(y\right)-P_{n}\left(% x\right)Q_{n+1}\left(y\right)\right)-1}}
(x-y)\sum_{k=0}^{n}(2k+1)\assLegendreP[]{k}@{x}\assLegendreQ[]{k}@{y} = (n+1)\left(\assLegendreP[]{n+1}@{x}\assLegendreQ[]{n}@{y}-\assLegendreP[]{n}@{x}\assLegendreQ[]{n+1}@{y}\right)-1		 

(x - y)*sum((2*k + 1)*LegendreP(k, x)*LegendreQ(k, y), k = 0..n) = (n + 1)*(LegendreP(n + 1, x)*LegendreQ(n, y)- LegendreP(n, x)*LegendreQ(n + 1, y))- 1

(x - y)*Sum[(2*k + 1)*LegendreP[k, 0, 3, x]*LegendreQ[k, 0, 3, y], {k, 0, n}, GenerateConditions->None] == (n + 1)*(LegendreP[n + 1, 0, 3, x]*LegendreQ[n, 0, 3, y]- LegendreP[n, 0, 3, x]*LegendreQ[n + 1, 0, 3, y])- 1
Aborted Aborted Manual Skip!
Failed [42 / 54]
Result: Plus[Complex[-0.38140199474411474, 0.0], Times[3.0, Plus[Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[<syntaxhighlight lang=mathematica>Result: Plus[Complex[2.3599248424792147, 0.0], Times[3.0, Plus[Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[, 2], Power[-1.5, 2]], Times[68, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[2, ]]], Times[, Plus[1, ], Plus[-165, Times[-271, ], Times[-162, Power[, 2]], Times[-42, Power[, 3]], Times[-4, Power[, 4]], Times[315, Power[1.5, 2]], Times[531, , Power[1.5, 2]], Times[322, Power[, 2], Power[1.5, 2]], Times[84, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[294, 1.5, -1.5], Times[511, , 1.5, -1.5], Times[318, Power[, 2], 1.5, -1.5], Times[84, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[315, Power[-1.5, 2]], Times[531, , Power[-1.5, 2]], Times[322, Power[, 2], Power[-1.5, 2]], Times[84, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[3, Times[2, ]], Plus[12, Times[7, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[, Plus[1, ], Plus[2, ], Plus[3, ], Plus[4, ], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[1], 0], Equal[[2], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]]], Equal[[3], Plus[Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]]]], Equal[[4], Plus[Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]]]], Equal[[5], Plus[Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 24], Plus[1, Times[-3, Power[1.5, 2]], Times[-8, Plus[-1, Times[3, Power[1.5, 2]]]], Times[7, 1.5, Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]]]], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[7, 3], -1.5, Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[-3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[-3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]]}]][3.0]], DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Power[Plus[1, ], 2], Plus[7, Times[2, ]], []], Times[Plus[7, Times[2, ]], Plus[1, Times[2, ], Power[, 2], Times[9, 1.5, -1.5], Times[12, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[1, ]]], Times[Plus[55, Times[72, ], Times[30, Power[, 2]], Times[4, Power[, 3]], Times[-105, Power[1.5, 2]], Times[-142, , Power[1.5, 2]], Times[-60, Power[, 2], Power[1.5, 2]], Times[-8, Power[, 3], Power[1.5, 2]], Times[-63, 1.5, -1.5], Times[-102, , 1.5, -1.5], Times[-52, Power[, 2], 1.5, -1.5], Times[-8, Power[, 3], 1.5, -1.5], Times[-105, Power[-1.5, 2]], Times[-142, , Power[-1.5, 2]], Times[-60, Power[, 2], Power[-1.5, 2]], Times[-8, Power[, 3], Power[-1.5, 2]]], [Plus[2, ]]], Times[Plus[-55, Times[-72, ], Times[-30, Power[, 2]], Times[-4, Power[, 3]], Times[105, Power[1.5, 2]], Times[142, , Power[1.5, 2]], Times[60, Power[, 2], Power[1.5, 2]], Times[8, Power[, 3], Power[1.5, 2]], Times[147, 1.5, -1.5], Times[182, , 1.5, -1.5], Times[68, Power[, 2], 1.5, -1.5], Times[8, Power[, 3], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[142, , Power[-1.5, 2]], Times[60, Power[, 2], Power[-1.5, 2]], Times[8, Power[, 3], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, Plus[3, Times[2, ]], Plus[16, Times[8, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[Power[Plus[4, ], 2], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Equal[[2], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Equal[[3], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Equal[[4], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 9], Plus[Times[-2, 1.5], Times[Rational[5, 2], 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]}]][3.0]]]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]}

... skip entries to safe data
14.18.E8 𝖯 Ξ½ ⁑ ( - x ) = sin ⁑ ( Ξ½ ⁒ Ο€ ) Ο€ ⁒ βˆ‘ n = 0 ∞ 2 ⁒ n + 1 ( Ξ½ - n ) ⁒ ( Ξ½ + n + 1 ) ⁒ 𝖯 n ⁑ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 π‘₯ 𝜈 πœ‹ πœ‹ superscript subscript 𝑛 0 2 𝑛 1 𝜈 𝑛 𝜈 𝑛 1 shorthand-Ferrers-Legendre-P-first-kind 𝑛 π‘₯ {\displaystyle{\displaystyle\mathsf{P}_{\nu}\left(-x\right)=\frac{\sin\left(% \nu\pi\right)}{\pi}\sum_{n=0}^{\infty}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}_% {n}\left(x\right)}}
\FerrersP[]{\nu}@{-x} = \frac{\sin@{\nu\pi}}{\pi}\sum_{n=0}^{\infty}\frac{2n+1}{(\nu-n)(\nu+n+1)}\FerrersP[]{n}@{x}		 

LegendreP(nu, - x) = (sin(nu*Pi))/(Pi)*sum((2*n + 1)/((nu - n)*(nu + n + 1))*LegendreP(n, x), n = 0..infinity)

LegendreP[\[Nu], - x] == Divide[Sin[\[Nu]*Pi],Pi]*Sum[Divide[2*n + 1,(\[Nu]- n)*(\[Nu]+ n + 1)]*LegendreP[n, x], {n, 0, Infinity}, GenerateConditions->None]
Aborted Failure Manual Skip!
Failed [3 / 3]
Result: Plus[Complex[0.07218102573226806, -2.034342748581157], Times[0.3183098861837907, NSum[Times[Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, 1.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[Ξ½, Rational[3, 2]]}


Result: Plus[-0.5703494499205765, Times[0.3183098861837907, NSum[Times[Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, 0.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5], Rule[Ξ½, Rational[3, 2]]}

... skip entries to safe data
14.18.E9 𝖯 Ξ½ - ΞΌ ⁑ ( x ) = sin ⁑ ( Ξ½ ⁒ Ο€ ) Ο€ ⁒ βˆ‘ n = 0 ∞ ( - 1 ) n ⁒ 2 ⁒ n + 1 ( Ξ½ - n ) ⁒ ( Ξ½ + n + 1 ) ⁒ 𝖯 n - ΞΌ ⁑ ( x ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 𝜈 πœ‹ πœ‹ superscript subscript 𝑛 0 superscript 1 𝑛 2 𝑛 1 𝜈 𝑛 𝜈 𝑛 1 Ferrers-Legendre-P-first-kind πœ‡ 𝑛 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\sin% \left(\nu\pi\right)}{\pi}\sum_{n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+% 1)}\mathsf{P}^{-\mu}_{n}\left(x\right)}}
\FerrersP[-\mu]{\nu}@{x} = \frac{\sin@{\nu\pi}}{\pi}\sum_{n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+1)}\FerrersP[-\mu]{n}@{x}		 - 1 < x , x ≀ 1 , ΞΌ β‰₯ 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 1 π‘₯ formulae-sequence π‘₯ 1 formulae-sequence πœ‡ 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle-1<x,x\leq 1,\mu\geq 0,|(\tfrac{1}{2}-\tfrac{1}{2}% x)|<1}} 
LegendreP(nu, - mu, x) = (sin(nu*Pi))/(Pi)*sum((- 1)^(n)*(2*n + 1)/((nu - n)*(nu + n + 1))*LegendreP(n, - mu, x), n = 0..infinity)

LegendreP[\[Nu], - \[Mu], x] == Divide[Sin[\[Nu]*Pi],Pi]*Sum[(- 1)^(n)*Divide[2*n + 1,(\[Nu]- n)*(\[Nu]+ n + 1)]*LegendreP[n, - \[Mu], x], {n, 0, Infinity}, GenerateConditions->None]
Aborted Failure Manual Skip!
Failed [3 / 3]
Result: Plus[0.21434568952624797, Times[0.3183098861837907, NSum[Times[Power[-1, n], Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, -1.5, 0.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5], Rule[ΞΌ, 1.5], Rule[Ξ½, Rational[3, 2]]}


Result: Plus[0.37125762464284556, Times[0.3183098861837907, NSum[Times[Power[-1, n], Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, -0.5, 0.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5], Rule[ΞΌ, 0.5], Rule[Ξ½, Rational[3, 2]]}

... skip entries to safe data
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