Legendre and Related Functions - 14.7 Integer Degree and Order

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.7.E1 𝖯 n 0 ⁑ ( x ) = 𝖯 n ⁑ ( x ) Ferrers-Legendre-P-first-kind 0 𝑛 π‘₯ shorthand-Ferrers-Legendre-P-first-kind 𝑛 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{0}_{n}\left(x\right)=\mathsf{P}_{n}% \left(x\right)}}
\FerrersP[0]{n}@{x} = \FerrersP[]{n}@{x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\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 𝖯 n ⁑ ( x ) = P n 0 ⁑ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝑛 π‘₯ Legendre-P-first-kind 0 𝑛 π‘₯ {\displaystyle{\displaystyle\mathsf{P}_{n}\left(x\right)=P^{0}_{n}\left(x% \right)}}
\FerrersP[]{n}@{x} = \assLegendreP[0]{n}@{x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\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 P n 0 ⁑ ( x ) = P n ⁑ ( x ) Legendre-P-first-kind 0 𝑛 π‘₯ Legendre-spherical-polynomial 𝑛 π‘₯ {\displaystyle{\displaystyle P^{0}_{n}\left(x\right)=P_{n}\left(x\right)}}
\assLegendreP[0]{n}@{x} = \LegendrepolyP{n}@{x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\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 𝖰 n 0 ⁑ ( x ) = 𝖰 n ⁑ ( x ) Ferrers-Legendre-Q-first-kind 0 𝑛 π‘₯ shorthand-Ferrers-Legendre-Q-first-kind 𝑛 π‘₯ {\displaystyle{\displaystyle\mathsf{Q}^{0}_{n}\left(x\right)=\mathsf{Q}_{n}% \left(x\right)}}
\FerrersQ[0]{n}@{x} = \FerrersQ[]{n}@{x}		 β„œ ⁑ ( n + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 0 + 1 ) > 0 , β„œ ⁑ ( n - ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - 0 + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 0 1 0 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 0 1 0 1 2 1 2 π‘₯ 1 {\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 𝖰 n ⁑ ( x ) = 1 2 ⁒ P n ⁑ ( x ) ⁒ ln ⁑ ( 1 + x 1 - x ) - W n - 1 ⁒ ( x ) shorthand-Ferrers-Legendre-Q-first-kind 𝑛 π‘₯ 1 2 Legendre-spherical-polynomial 𝑛 π‘₯ 1 π‘₯ 1 π‘₯ subscript π‘Š 𝑛 1 π‘₯ {\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)		 β„œ ⁑ ( n + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 0 + 1 ) > 0 , β„œ ⁑ ( n - ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - 0 + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 0 1 0 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 0 1 0 1 2 1 2 π‘₯ 1 {\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: .2990381063-3.962388980*I
Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.950961893-8.282078880*I
Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, 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 W n - 1 ⁒ ( x ) = βˆ‘ s = 0 n - 1 ( n + s ) ! ⁒ ( ψ ⁑ ( n + 1 ) - ψ ⁑ ( s + 1 ) ) 2 s ⁒ ( n - s ) ! ⁒ ( s ! ) 2 ⁒ ( x - 1 ) s subscript π‘Š 𝑛 1 π‘₯ superscript subscript 𝑠 0 𝑛 1 𝑛 𝑠 digamma 𝑛 1 digamma 𝑠 1 superscript 2 𝑠 𝑛 𝑠 superscript 𝑠 2 superscript π‘₯ 1 𝑠 {\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+.7500000000*I
Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.950961893+.7500000000*I
Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, 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 W n - 1 ⁒ ( x ) = βˆ‘ k = 1 n 1 k ⁒ P k - 1 ⁑ ( x ) ⁒ P n - k ⁑ ( x ) subscript π‘Š 𝑛 1 π‘₯ superscript subscript π‘˜ 1 𝑛 1 π‘˜ Legendre-spherical-polynomial π‘˜ 1 π‘₯ Legendre-spherical-polynomial 𝑛 π‘˜ π‘₯ {\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+.7500000000*I
Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.950961894+.7500000000*I
Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
 Skipped - Because timed out
14.7#Ex1 W 0 ⁒ ( x ) = 1 subscript π‘Š 0 π‘₯ 1 {\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 W 1 ⁒ ( x ) = 3 2 ⁒ x subscript π‘Š 1 π‘₯ 3 2 π‘₯ {\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 W 2 ⁒ ( x ) = 5 2 ⁒ x 2 - 2 3 subscript π‘Š 2 π‘₯ 5 2 superscript π‘₯ 2 2 3 {\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 Q n 0 ⁑ ( x ) = Q n ⁑ ( x ) Legendre-Q-second-kind 0 𝑛 π‘₯ shorthand-Legendre-Q-second-kind 𝑛 π‘₯ {\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 Q n ⁑ ( x ) = n ! ⁒ 𝑸 n 0 ⁑ ( x ) shorthand-Legendre-Q-second-kind 𝑛 π‘₯ 𝑛 associated-Legendre-black-Q 0 𝑛 π‘₯ {\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)*Pi*I)*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 n ! ⁒ 𝑸 n 0 ⁑ ( x ) = n ! ⁒ 𝑸 n ⁑ ( x ) 𝑛 associated-Legendre-black-Q 0 𝑛 π‘₯ 𝑛 shorthand-associated-Legendre-black-Q 𝑛 π‘₯ {\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)*Pi*I)*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 Q n ⁑ ( x ) = 1 2 ⁒ P n ⁑ ( x ) ⁒ ln ⁑ ( x + 1 x - 1 ) - W n - 1 ⁒ ( x ) shorthand-Legendre-Q-second-kind 𝑛 π‘₯ 1 2 Legendre-spherical-polynomial 𝑛 π‘₯ π‘₯ 1 π‘₯ 1 subscript π‘Š 𝑛 1 π‘₯ {\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+.7500000000*I
Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -5.708333332+1.299038106*I
Test Values: {x = 3/2, W[n-1] = -1/2+1/2*I*3^(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 𝖯 n m ⁑ ( x ) = ( - 1 ) m ⁒ ( 1 - x 2 ) m / 2 ⁒ d m d x m ⁑ 𝖯 n ⁑ ( x ) Ferrers-Legendre-P-first-kind π‘š 𝑛 π‘₯ superscript 1 π‘š superscript 1 superscript π‘₯ 2 π‘š 2 derivative π‘₯ π‘š shorthand-Ferrers-Legendre-P-first-kind 𝑛 π‘₯ {\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}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\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 𝖰 n m ⁑ ( x ) = ( - 1 ) m ⁒ ( 1 - x 2 ) m / 2 ⁒ d m d x m ⁑ 𝖰 n ⁑ ( x ) Ferrers-Legendre-Q-first-kind π‘š 𝑛 π‘₯ superscript 1 π‘š superscript 1 superscript π‘₯ 2 π‘š 2 derivative π‘₯ π‘š shorthand-Ferrers-Legendre-Q-first-kind 𝑛 π‘₯ {\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}		 β„œ ⁑ ( n + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ + m + 1 ) > 0 , β„œ ⁑ ( n - ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - m + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 π‘š 1 0 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 π‘š 1 0 1 2 1 2 π‘₯ 1 {\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 𝖯 n m ⁑ ( x ) = ( - 1 ) m + n ⁒ ( 1 - x 2 ) m / 2 2 n ⁒ n ! ⁒ d m + n d x m + n ⁑ ( 1 - x 2 ) n Ferrers-Legendre-P-first-kind π‘š 𝑛 π‘₯ superscript 1 π‘š 𝑛 superscript 1 superscript π‘₯ 2 π‘š 2 superscript 2 𝑛 𝑛 derivative π‘₯ π‘š 𝑛 superscript 1 superscript π‘₯ 2 𝑛 {\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}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\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 P n m ⁑ ( x ) = ( x 2 - 1 ) m / 2 ⁒ d m d x m ⁑ P n ⁑ ( x ) Legendre-P-first-kind π‘š 𝑛 π‘₯ superscript superscript π‘₯ 2 1 π‘š 2 derivative π‘₯ π‘š Legendre-spherical-polynomial 𝑛 π‘₯ {\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 Q n m ⁑ ( x ) = ( x 2 - 1 ) m / 2 ⁒ d m d x m ⁑ Q n ⁑ ( x ) Legendre-Q-second-kind π‘š 𝑛 π‘₯ superscript superscript π‘₯ 2 1 π‘š 2 derivative π‘₯ π‘š shorthand-Legendre-Q-second-kind 𝑛 π‘₯ {\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 P n ⁑ ( x ) = 1 2 n ⁒ n ! ⁒ d n d x n ⁑ ( x 2 - 1 ) n Legendre-spherical-polynomial 𝑛 π‘₯ 1 superscript 2 𝑛 𝑛 derivative π‘₯ 𝑛 superscript superscript π‘₯ 2 1 𝑛 {\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 P n m ⁑ ( x ) = ( x 2 - 1 ) m / 2 2 n ⁒ n ! ⁒ d m + n d x m + n ⁑ ( x 2 - 1 ) n Legendre-P-first-kind π‘š 𝑛 π‘₯ superscript superscript π‘₯ 2 1 π‘š 2 superscript 2 𝑛 𝑛 derivative π‘₯ π‘š 𝑛 superscript superscript π‘₯ 2 1 𝑛 {\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 P m m ⁑ ( x ) = ( 2 ⁒ m ) ! 2 m ⁒ m ! ⁒ ( x 2 - 1 ) m / 2 Legendre-P-first-kind π‘š π‘š π‘₯ 2 π‘š superscript 2 π‘š π‘š superscript superscript π‘₯ 2 1 π‘š 2 {\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(2*m))/((2)^(m)* factorial(m))*((x)^(2)- 1)^(m/2)

LegendreP[m, m, 3, x] == Divide[(2*m)!,(2)^(m)* (m)!]*((x)^(2)- 1)^(m/2)
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
14.7.E16 𝖯 n m ⁑ ( x ) = P n m ⁑ ( x ) Ferrers-Legendre-P-first-kind π‘š 𝑛 π‘₯ Legendre-P-first-kind π‘š 𝑛 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=P^{m}_{n}\left(x% \right)}}
\FerrersP[m]{n}@{x} = \assLegendreP[m]{n}@{x}		 m > n , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence π‘š 𝑛 1 2 1 2 π‘₯ 1 {\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 P n m ⁑ ( x ) = 0 Legendre-P-first-kind π‘š 𝑛 π‘₯ 0 {\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=0}}
\assLegendreP[m]{n}@{x} = 0		 m > n , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence π‘š 𝑛 1 2 1 2 π‘₯ 1 {\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 𝖯 n m ⁑ ( - x ) = ( - 1 ) n - m ⁒ 𝖯 n m ⁑ ( x ) Ferrers-Legendre-P-first-kind π‘š 𝑛 π‘₯ superscript 1 𝑛 π‘š Ferrers-Legendre-P-first-kind π‘š 𝑛 π‘₯ {\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}		 | ( 1 2 - 1 2 ⁒ ( - x ) ) | < 1 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 1 2 1 2 π‘₯ 1 1 2 1 2 π‘₯ 1 {\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 𝖰 n + m ⁑ ( - x ) = ( - 1 ) n - m - 1 ⁒ 𝖰 n + m ⁑ ( x ) Ferrers-Legendre-Q-first-kind π‘š 𝑛 π‘₯ superscript 1 𝑛 π‘š 1 Ferrers-Legendre-Q-first-kind π‘š 𝑛 π‘₯ {\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 𝖰 n - m ⁑ ( - x ) = ( - 1 ) n - m - 1 ⁒ 𝖰 n - m ⁑ ( x ) Ferrers-Legendre-Q-first-kind π‘š 𝑛 π‘₯ superscript 1 𝑛 π‘š 1 Ferrers-Legendre-Q-first-kind π‘š 𝑛 π‘₯ {\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}		 β„œ ⁑ ( n + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ + ( - m ) + 1 ) > 0 , β„œ ⁑ ( n - ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ( - m ) + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ ( - x ) ) | < 1 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 π‘š 1 0 formulae-sequence 𝑛 πœ‡ 1 0 formulae-sequence 𝜈 π‘š 1 0 formulae-sequence 1 2 1 2 π‘₯ 1 1 2 1 2 π‘₯ 1 {\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 βˆ‘ n = 0 ∞ 𝖯 n ⁑ ( x ) ⁒ h n = ( 1 - 2 ⁒ x ⁒ h + h 2 ) - 1 / 2 superscript subscript 𝑛 0 shorthand-Ferrers-Legendre-P-first-kind 𝑛 π‘₯ superscript β„Ž 𝑛 superscript 1 2 π‘₯ β„Ž superscript β„Ž 2 1 2 {\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 - 2*x*h + (h)^(2))^(- 1/2)
 
Sum[LegendreP[n, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*h + (h)^(2))^(- 1/2)
 Failure Successful Error Successful [Tested: 30]
14.7.E20 βˆ‘ n = 0 ∞ 𝖰 n ⁑ ( x ) ⁒ h n = 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 ) superscript subscript 𝑛 0 shorthand-Ferrers-Legendre-Q-first-kind 𝑛 π‘₯ superscript β„Ž 𝑛 1 superscript 1 2 π‘₯ β„Ž superscript β„Ž 2 1 2 π‘₯ β„Ž superscript 1 2 π‘₯ β„Ž superscript β„Ž 2 1 2 superscript 1 superscript π‘₯ 2 1 2 {\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 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/2))/((1 - (x)^(2))^(1/2)))
 
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)]]
 Failure Failure Manual Skip! Skipped - Because timed out
14.7.E21 βˆ‘ n = 0 ∞ 𝖯 n ⁑ ( x ) ⁒ h - n - 1 = ( 1 - 2 ⁒ x ⁒ h + h 2 ) - 1 / 2 superscript subscript 𝑛 0 shorthand-Ferrers-Legendre-P-first-kind 𝑛 π‘₯ superscript β„Ž 𝑛 1 superscript 1 2 π‘₯ β„Ž superscript β„Ž 2 1 2 {\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 - 2*x*h + (h)^(2))^(- 1/2)
 
Sum[LegendreP[n, x]*(h)^(- n - 1), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*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 βˆ‘ n = 0 ∞ Q n ⁑ ( x ) ⁒ h n = 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 ) superscript subscript 𝑛 0 shorthand-Legendre-Q-second-kind 𝑛 π‘₯ superscript β„Ž 𝑛 1 superscript 1 2 π‘₯ β„Ž superscript β„Ž 2 1 2 π‘₯ β„Ž superscript 1 2 π‘₯ β„Ž superscript β„Ž 2 1 2 superscript superscript π‘₯ 2 1 1 2 {\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 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (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 - 2*x*h + (h)^(2))^(1/2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/2),((x)^(2)- 1)^(1/2)]]
 Failure Failure Successful [Tested: 30] Skipped - Because timed out
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