Confluent Hypergeometric Functions - 14.2 Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.2.E1 ( 1 - x 2 ) ⁢ d 2 w d x 2 - 2 ⁢ x ⁢ d w d x + ν ⁢ ( ν + 1 ) ⁢ w = 0 1 superscript 𝑥 2 derivative 𝑤 𝑥 2 2 𝑥 derivative 𝑤 𝑥 𝜈 𝜈 1 𝑤 0 {\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{% \mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}+\nu(\nu+1)w=0}}
\left(1-x^{2}\right)\deriv[2]{w}{x}-2x\deriv{w}{x}+\nu(\nu+1)w = 0		 

(1 - (x)^(2))*diff(w, [x$(2)])- 2*x*diff(w, x)+ nu*(nu + 1)*w = 0

(1 - (x)^(2))*D[w, {x, 2}]- 2*x*D[w, x]+ \[Nu]*(\[Nu]+ 1)*w == 0
Failure Failure
Failed [300 / 300]
Result: .5000000007+1.866025405*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: .5000000007+1.866025405*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5000000000000004, 1.8660254037844386]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.8660254037844388, -0.5]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.2.E2 ( 1 - x 2 ) ⁢ d 2 w d x 2 - 2 ⁢ x ⁢ d w d x + ( ν ⁢ ( ν + 1 ) - μ 2 1 - x 2 ) ⁢ w = 0 1 superscript 𝑥 2 derivative 𝑤 𝑥 2 2 𝑥 derivative 𝑤 𝑥 𝜈 𝜈 1 superscript 𝜇 2 1 superscript 𝑥 2 𝑤 0 {\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{% \mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}+\left(\nu(\nu+1)-\frac{\mu% ^{2}}{1-x^{2}}\right)w=0}}
\left(1-x^{2}\right)\deriv[2]{w}{x}-2x\deriv{w}{x}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-x^{2}}\right)w = 0		 

(1 - (x)^(2))*diff(w, [x$(2)])- 2*x*diff(w, x)+(nu*(nu + 1)-((mu)^(2))/(1 - (x)^(2)))*w = 0

(1 - (x)^(2))*D[w, {x, 2}]- 2*x*D[w, x]+(\[Nu]*(\[Nu]+ 1)-Divide[\[Mu]^(2),1 - (x)^(2)])*w == 0
Failure Failure
Failed [300 / 300]
Result: .5000000005+2.666025404*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: .4999999998+.5326920710*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5000000000000007, 2.666025403784439]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.8660254037844387, 0.30000000000000043]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.2.E3 𝒲 ⁡ { 𝖯 ν - μ ⁡ ( x ) , 𝖯 ν - μ ⁡ ( - x ) } = 2 Γ ⁡ ( μ - ν ) ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ ( 1 - x 2 ) Wronskian Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 2 Euler-Gamma 𝜇 𝜈 Euler-Gamma 𝜈 𝜇 1 1 superscript 𝑥 2 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x% \right),\mathsf{P}^{-\mu}_{\nu}\left(-x\right)\right\}=\frac{2}{\Gamma\left(% \mu-\nu\right)\Gamma\left(\nu+\mu+1\right)\left(1-x^{2}\right)}}}
\Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersP[-\mu]{\nu}@{-x}} = \frac{2}{\EulerGamma@{\mu-\nu}\EulerGamma@{\nu+\mu+1}\left(1-x^{2}\right)}		 ℜ ⁡ ( μ - ν ) > 0 , ℜ ⁡ ( ν + μ + 1 ) > 0 formulae-sequence 𝜇 𝜈 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0,\Re(\nu+\mu+1)>0}} 
(LegendreP(nu, - mu, x))*diff(LegendreP(nu, - mu, - x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, - mu, - x)) = (2)/(GAMMA(mu - nu)*GAMMA(nu + mu + 1)*(1 - (x)^(2)))

Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreP[\[Nu], - \[Mu], - x]}, x] == Divide[2,Gamma[\[Mu]- \[Nu]]*Gamma[\[Nu]+ \[Mu]+ 1]*(1 - (x)^(2))]
Failure Failure Successful [Tested: 87] Successful [Tested: 96]
14.2.E4 𝒲 ⁡ { 𝖯 ν μ ⁡ ( x ) , 𝖰 ν μ ⁡ ( x ) } = Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 1 ) ⁢ ( 1 - x 2 ) Wronskian Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 1 1 superscript 𝑥 2 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{\mu}_{\nu}\left(x% \right),\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right\}=\frac{\Gamma\left(\nu+\mu% +1\right)}{\Gamma\left(\nu-\mu+1\right)\left(1-x^{2}\right)}}}
\Wronskian@{\FerrersP[\mu]{\nu}@{x},\FerrersQ[\mu]{\nu}@{x}} = \frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\left(1-x^{2}\right)}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}} 
(LegendreP(nu, mu, x))*diff(LegendreQ(nu, mu, x), x)-diff(LegendreP(nu, mu, x), x)*(LegendreQ(nu, mu, x)) = (GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*(1 - (x)^(2)))

Wronskian[{LegendreP[\[Nu], \[Mu], x], LegendreQ[\[Nu], \[Mu], x]}, x] == Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*(1 - (x)^(2))]
Failure Failure Successful [Tested: 120] Successful [Tested: 135]
14.2.E5 𝖯 ν + 1 μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) - 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν + 1 μ ⁡ ( x ) = Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 2 ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 1 𝑥 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 2 {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{% \mu}_{\nu}\left(x\right)-\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_% {\nu+1}\left(x\right)=\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+% 2\right)}}}
\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x}-\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x} = \frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+2}}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 2 ) > 0 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 2 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+2)>0}} 
LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x)- LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x) = (GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 2))

LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu], \[Mu], x]- LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x] == Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 2]]
Failure Failure Successful [Tested: 162] Successful [Tested: 174]
14.2.E6 𝒲 ⁡ { 𝖯 ν - μ ⁡ ( x ) , 𝖰 ν μ ⁡ ( x ) } = cos ⁡ ( μ ⁢ π ) 1 - x 2 Wronskian Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 1 superscript 𝑥 2 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x% \right),\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right\}=\frac{\cos\left(\mu\pi% \right)}{1-x^{2}}}}
\Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersQ[\mu]{\nu}@{x}} = \frac{\cos@{\mu\pi}}{1-x^{2}}		 

(LegendreP(nu, - mu, x))*diff(LegendreQ(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreQ(nu, mu, x)) = (cos(mu*Pi))/(1 - (x)^(2))

Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreQ[\[Nu], \[Mu], x]}, x] == Divide[Cos[\[Mu]*Pi],1 - (x)^(2)]
Failure Failure Error
Failed [21 / 300]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -1.5]}


Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -0.5]}

... skip entries to safe data
14.2.E7 𝒲 ⁡ { P ν - μ ⁡ ( x ) , P ν μ ⁡ ( x ) } = 𝒲 ⁡ { 𝖯 ν - μ ⁡ ( x ) , 𝖯 ν μ ⁡ ( x ) } Wronskian Legendre-P-first-kind 𝜇 𝜈 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 Wronskian Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),P^{% \mu}_{\nu}\left(x\right)\right\}=\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}% \left(x\right),\mathsf{P}^{\mu}_{\nu}\left(x\right)\right\}}}
\Wronskian@{\assLegendreP[-\mu]{\nu}@{x},\assLegendreP[\mu]{\nu}@{x}} = \Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersP[\mu]{\nu}@{x}}		 

(LegendreP(nu, - mu, x))*diff(LegendreP(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, mu, x)) = (LegendreP(nu, - mu, x))*diff(LegendreP(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, mu, x))

Wronskian[{LegendreP[\[Nu], - \[Mu], 3, x], LegendreP[\[Nu], \[Mu], 3, x]}, x] == Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreP[\[Nu], \[Mu], x]}, x]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 300]
14.2.E7 𝒲 ⁡ { 𝖯 ν - μ ⁡ ( x ) , 𝖯 ν μ ⁡ ( x ) } = 2 ⁢ sin ⁡ ( μ ⁢ π ) π ⁢ ( 1 - x 2 ) Wronskian Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 2 𝜇 𝜋 𝜋 1 superscript 𝑥 2 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x% \right),\mathsf{P}^{\mu}_{\nu}\left(x\right)\right\}=\frac{2\sin\left(\mu\pi% \right)}{\pi\left(1-x^{2}\right)}}}
\Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersP[\mu]{\nu}@{x}} = \frac{2\sin@{\mu\pi}}{\pi\left(1-x^{2}\right)}		 

(LegendreP(nu, - mu, x))*diff(LegendreP(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, mu, x)) = (2*sin(mu*Pi))/(Pi*(1 - (x)^(2)))

Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreP[\[Nu], \[Mu], x]}, x] == Divide[2*Sin[\[Mu]*Pi],Pi*(1 - (x)^(2))]
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
14.2.E8 𝒲 ⁡ { P ν - μ ⁡ ( x ) , 𝑸 ν μ ⁡ ( x ) } = - 1 Γ ⁡ ( ν + μ + 1 ) ⁢ ( x 2 - 1 ) Wronskian Legendre-P-first-kind 𝜇 𝜈 𝑥 associated-Legendre-black-Q 𝜇 𝜈 𝑥 1 Euler-Gamma 𝜈 𝜇 1 superscript 𝑥 2 1 {\displaystyle{\displaystyle\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),% \boldsymbol{Q}^{\mu}_{\nu}\left(x\right)\right\}=-\frac{1}{\Gamma\left(\nu+\mu% +1\right)\left(x^{2}-1\right)}}}
\Wronskian@{\assLegendreP[-\mu]{\nu}@{x},\assLegendreOlverQ[\mu]{\nu}@{x}} = -\frac{1}{\EulerGamma@{\nu+\mu+1}\left(x^{2}-1\right)}		 ℜ ⁡ ( ν + μ + 1 ) > 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0}} 
(LegendreP(nu, - mu, x))*diff(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1), x)-diff(LegendreP(nu, - mu, x), x)*(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1)) = -(1)/(GAMMA(nu + mu + 1)*((x)^(2)- 1))

Wronskian[{LegendreP[\[Nu], - \[Mu], 3, x], Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1]}, x] == -Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]*((x)^(2)- 1)]
Failure Failure Successful [Tested: 195] Successful [Tested: 207]
14.2.E9 𝒲 ⁡ { 𝑸 ν μ ⁡ ( x ) , 𝑸 - ν - 1 μ ⁡ ( x ) } = cos ⁡ ( ν ⁢ π ) x 2 - 1 Wronskian associated-Legendre-black-Q 𝜇 𝜈 𝑥 associated-Legendre-black-Q 𝜇 𝜈 1 𝑥 𝜈 𝜋 superscript 𝑥 2 1 {\displaystyle{\displaystyle\mathscr{W}\left\{\boldsymbol{Q}^{\mu}_{\nu}\left(% x\right),\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)\right\}=\frac{\cos\left(% \nu\pi\right)}{x^{2}-1}}}
\Wronskian@{\assLegendreOlverQ[\mu]{\nu}@{x},\assLegendreOlverQ[\mu]{-\nu-1}@{x}} = \frac{\cos@{\nu\pi}}{x^{2}-1}		 

(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1))*diff(exp(-(mu)*Pi*I)*LegendreQ(- nu - 1,mu,x)/GAMMA(- nu - 1+mu+1), x)-diff(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1), x)*(exp(-(mu)*Pi*I)*LegendreQ(- nu - 1,mu,x)/GAMMA(- nu - 1+mu+1)) = (cos(nu*Pi))/((x)^(2)- 1)

Wronskian[{Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1], Exp[-(\[Mu]) Pi I] LegendreQ[- \[Nu]- 1, \[Mu], 3, x]/Gamma[- \[Nu]- 1 + \[Mu] + 1]}, x] == Divide[Cos[\[Nu]*Pi],(x)^(2)- 1]
Failure Aborted
Failed [39 / 300]
Result: 1.832150333+.7522048283*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -3.053583887-1.253674714*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [57 / 300]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.2.E10 𝒲 ⁡ { P ν μ ⁡ ( x ) , Q ν μ ⁡ ( x ) } = - e μ ⁢ π ⁢ i ⁢ Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 1 ) ⁢ ( x 2 - 1 ) Wronskian Legendre-P-first-kind 𝜇 𝜈 𝑥 Legendre-Q-second-kind 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 1 superscript 𝑥 2 1 {\displaystyle{\displaystyle\mathscr{W}\left\{P^{\mu}_{\nu}\left(x\right),Q^{% \mu}_{\nu}\left(x\right)\right\}=-e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)\left(x^{2}-1\right)}}}
\Wronskian@{\assLegendreP[\mu]{\nu}@{x},\assLegendreQ[\mu]{\nu}@{x}} = -e^{\mu\pi i}\frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\left(x^{2}-1\right)}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}} 
(LegendreP(nu, mu, x))*diff(LegendreQ(nu, mu, x), x)-diff(LegendreP(nu, mu, x), x)*(LegendreQ(nu, mu, x)) = - exp(mu*Pi*I)*(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*((x)^(2)- 1))

Wronskian[{LegendreP[\[Nu], \[Mu], 3, x], LegendreQ[\[Nu], \[Mu], 3, x]}, x] == - Exp[\[Mu]*Pi*I]*Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*((x)^(2)- 1)]
Failure Failure Successful [Tested: 120] Successful [Tested: 135]
14.2.E11 P ν + 1 μ ⁡ ( x ) ⁢ Q ν μ ⁡ ( x ) - P ν μ ⁡ ( x ) ⁢ Q ν + 1 μ ⁡ ( x ) = e μ ⁢ π ⁢ i ⁢ Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 2 ) Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Legendre-Q-second-kind 𝜇 𝜈 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 Legendre-Q-second-kind 𝜇 𝜈 1 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 2 {\displaystyle{\displaystyle P^{\mu}_{\nu+1}\left(x\right)Q^{\mu}_{\nu}\left(x% \right)-P^{\mu}_{\nu}\left(x\right)Q^{\mu}_{\nu+1}\left(x\right)=e^{\mu\pi i}% \frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)}}}
\assLegendreP[\mu]{\nu+1}@{x}\assLegendreQ[\mu]{\nu}@{x}-\assLegendreP[\mu]{\nu}@{x}\assLegendreQ[\mu]{\nu+1}@{x} = e^{\mu\pi i}\frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+2}}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 2 ) > 0 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 2 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+2)>0}} 
LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x)- LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x) = exp(mu*Pi*I)*(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 2))

LegendreP[\[Nu]+ 1, \[Mu], 3, x]*LegendreQ[\[Nu], \[Mu], 3, x]- LegendreP[\[Nu], \[Mu], 3, x]*LegendreQ[\[Nu]+ 1, \[Mu], 3, x] == Exp[\[Mu]*Pi*I]*Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 2]]
Failure Failure Successful [Tested: 162] Successful [Tested: 174]
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