Orthogonal Polynomials - 18.17 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.17.E1 2 ⁢ n ⁢ ∫ 0 x ( 1 - y ) α ⁢ ( 1 + y ) β ⁢ P n ( α , β ) ⁡ ( y ) ⁢ d y = P n - 1 ( α + 1 , β + 1 ) ⁡ ( 0 ) - ( 1 - x ) α + 1 ⁢ ( 1 + x ) β + 1 ⁢ P n - 1 ( α + 1 , β + 1 ) ⁡ ( x ) 2 𝑛 superscript subscript 0 𝑥 superscript 1 𝑦 𝛼 superscript 1 𝑦 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 𝑦 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 0 superscript 1 𝑥 𝛼 1 superscript 1 𝑥 𝛽 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle 2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}P^{(% \alpha,\beta)}_{n}\left(y\right)\mathrm{d}y=P^{(\alpha+1,\beta+1)}_{n-1}\left(% 0\right)-(1-x)^{\alpha+1}(1+x)^{\beta+1}P^{(\alpha+1,\beta+1)}_{n-1}\left(x% \right)}}
2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{y}\diff{y} = \JacobipolyP{\alpha+1}{\beta+1}{n-1}@{0}-(1-x)^{\alpha+1}(1+x)^{\beta+1}\JacobipolyP{\alpha+1}{\beta+1}{n-1}@{x}		 

2*n*int((1 - y)^(alpha)*(1 + y)^(beta)* JacobiP(n, alpha, beta, y), y = 0..x) = JacobiP(n - 1, alpha + 1, beta + 1, 0)-(1 - x)^(alpha + 1)*(1 + x)^(beta + 1)* JacobiP(n - 1, alpha + 1, beta + 1, x)

2*n*Integrate[(1 - y)^\[Alpha]*(1 + y)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], y], {y, 0, x}, GenerateConditions->None] == JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, 0]-(1 - x)^(\[Alpha]+ 1)*(1 + x)^(\[Beta]+ 1)* JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x]
Failure Successful Manual Skip! Successful [Tested: 81]
18.17.E2 ∫ 0 x L m ⁡ ( y ) ⁢ L n ⁡ ( x - y ) ⁢ d y = ∫ 0 x L m + n ⁡ ( y ) ⁢ d y superscript subscript 0 𝑥 shorthand-Laguerre-polynomial-L 𝑚 𝑦 shorthand-Laguerre-polynomial-L 𝑛 𝑥 𝑦 𝑦 superscript subscript 0 𝑥 shorthand-Laguerre-polynomial-L 𝑚 𝑛 𝑦 𝑦 {\displaystyle{\displaystyle\int_{0}^{x}L_{m}\left(y\right)L_{n}\left(x-y% \right)\mathrm{d}y=\int_{0}^{x}L_{m+n}\left(y\right)\mathrm{d}y}}
\int_{0}^{x}\LaguerrepolyL[]{m}@{y}\LaguerrepolyL[]{n}@{x-y}\diff{y} = \int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y}		 

int(LaguerreL(m, y)*LaguerreL(n, x - y), y = 0..x) = int(LaguerreL(m + n, y), y = 0..x)

Integrate[LaguerreL[m, y]*LaguerreL[n, x - y], {y, 0, x}, GenerateConditions->None] == Integrate[LaguerreL[m + n, y], {y, 0, x}, GenerateConditions->None]
Failure Failure Successful [Tested: 27] Successful [Tested: 27]
18.17.E2 ∫ 0 x L m + n ⁡ ( y ) ⁢ d y = L m + n ⁡ ( x ) - L m + n + 1 ⁡ ( x ) superscript subscript 0 𝑥 shorthand-Laguerre-polynomial-L 𝑚 𝑛 𝑦 𝑦 shorthand-Laguerre-polynomial-L 𝑚 𝑛 𝑥 shorthand-Laguerre-polynomial-L 𝑚 𝑛 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}L_{m+n}\left(y\right)\mathrm{d}y=L_{m+% n}\left(x\right)-L_{m+n+1}\left(x\right)}}
\int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y} = \LaguerrepolyL[]{m+n}@{x}-\LaguerrepolyL[]{m+n+1}@{x}		 

int(LaguerreL(m + n, y), y = 0..x) = LaguerreL(m + n, x)- LaguerreL(m + n + 1, x)

Integrate[LaguerreL[m + n, y], {y, 0, x}, GenerateConditions->None] == LaguerreL[m + n, x]- LaguerreL[m + n + 1, x]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 27]
18.17.E3 ∫ 0 x H n ⁡ ( y ) ⁢ d y = 1 2 ⁢ ( n + 1 ) ⁢ ( H n + 1 ⁡ ( x ) - H n + 1 ⁡ ( 0 ) ) superscript subscript 0 𝑥 Hermite-polynomial-H 𝑛 𝑦 𝑦 1 2 𝑛 1 Hermite-polynomial-H 𝑛 1 𝑥 Hermite-polynomial-H 𝑛 1 0 {\displaystyle{\displaystyle\int_{0}^{x}H_{n}\left(y\right)\mathrm{d}y=\frac{1% }{2(n+1)}(H_{n+1}\left(x\right)-H_{n+1}\left(0\right))}}
\int_{0}^{x}\HermitepolyH{n}@{y}\diff{y} = \frac{1}{2(n+1)}(\HermitepolyH{n+1}@{x}-\HermitepolyH{n+1}@{0})		 

int(HermiteH(n, y), y = 0..x) = (1)/(2*(n + 1))*(HermiteH(n + 1, x)- HermiteH(n + 1, 0))

Integrate[HermiteH[n, y], {y, 0, x}, GenerateConditions->None] == Divide[1,2*(n + 1)]*(HermiteH[n + 1, x]- HermiteH[n + 1, 0])
Failure Successful
Failed [9 / 9]
Result: Float(undefined)+Float(undefined)*I
Test Values: {x = 3/2, n = 1}


Result: -1.500000000+0.*I
Test Values: {x = 3/2, n = 2}

... skip entries to safe data
 Successful [Tested: 9]
18.17.E4 ∫ 0 x e - y 2 ⁢ H n ⁡ ( y ) ⁢ d y = H n - 1 ⁡ ( 0 ) - e - x 2 ⁢ H n - 1 ⁡ ( x ) superscript subscript 0 𝑥 superscript 𝑒 superscript 𝑦 2 Hermite-polynomial-H 𝑛 𝑦 𝑦 Hermite-polynomial-H 𝑛 1 0 superscript 𝑒 superscript 𝑥 2 Hermite-polynomial-H 𝑛 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{-y^{2}}H_{n}\left(y\right)\mathrm{d% }y=H_{n-1}\left(0\right)-e^{-x^{2}}H_{n-1}\left(x\right)}}
\int_{0}^{x}e^{-y^{2}}\HermitepolyH{n}@{y}\diff{y} = \HermitepolyH{n-1}@{0}-e^{-x^{2}}\HermitepolyH{n-1}@{x}		 

int(exp(- (y)^(2))*HermiteH(n, y), y = 0..x) = HermiteH(n - 1, 0)- exp(- (x)^(2))*HermiteH(n - 1, x)

Integrate[Exp[- (y)^(2)]*HermiteH[n, y], {y, 0, x}, GenerateConditions->None] == HermiteH[n - 1, 0]- Exp[- (x)^(2)]*HermiteH[n - 1, x]
Failure Successful Successful [Tested: 9]
Failed [3 / 9]
Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 1.5]}


Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 0.5]}

... skip entries to safe data
18.17.E5 C n ( λ ) ⁡ ( cos ⁡ θ 1 ) C n ( λ ) ⁡ ( 1 ) ⁢ C n ( λ ) ⁡ ( cos ⁡ θ 2 ) C n ( λ ) ⁡ ( 1 ) = Γ ⁡ ( λ + 1 2 ) π 1 2 ⁢ Γ ⁡ ( λ ) ⁢ ∫ 0 π C n ( λ ) ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) C n ( λ ) ⁡ ( 1 ) ⁢ ( sin ⁡ ϕ ) 2 ⁢ λ - 1 ⁢ d ϕ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 subscript 𝜃 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 subscript 𝜃 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 Euler-Gamma 𝜆 1 2 superscript 𝜋 1 2 Euler-Gamma 𝜆 superscript subscript 0 𝜋 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 superscript italic-ϕ 2 𝜆 1 italic-ϕ {\displaystyle{\displaystyle\frac{C^{(\lambda)}_{n}\left(\cos\theta_{1}\right)% }{C^{(\lambda)}_{n}\left(1\right)}\frac{C^{(\lambda)}_{n}\left(\cos\theta_{2}% \right)}{C^{(\lambda)}_{n}\left(1\right)}=\frac{\Gamma\left(\lambda+\frac{1}{2% }\right)}{\pi^{\frac{1}{2}}\Gamma\left(\lambda\right)}\*\int_{0}^{\pi}\frac{C^% {(\lambda)}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\phi\right)}{C^{(\lambda)}_{n}\left(1\right)}(\sin\phi)^{2\lambda-1}% \mathrm{d}\phi}}
\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}}}{\ultrasphpoly{\lambda}{n}@{1}}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{2}}}}{\ultrasphpoly{\lambda}{n}@{1}} = \frac{\EulerGamma@{\lambda+\frac{1}{2}}}{\pi^{\frac{1}{2}}\EulerGamma@{\lambda}}\*\int_{0}^{\pi}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}}{\ultrasphpoly{\lambda}{n}@{1}}(\sin@@{\phi})^{2\lambda-1}\diff{\phi}		 λ > 0 , ℜ ⁡ ( λ + 1 2 ) > 0 , ℜ ⁡ ( λ ) > 0 formulae-sequence 𝜆 0 formulae-sequence 𝜆 1 2 0 𝜆 0 {\displaystyle{\displaystyle\lambda>0,\Re(\lambda+\frac{1}{2})>0,\Re(\lambda)>% 0}} 
(GegenbauerC(n, lambda, cos(theta[1])))/(GegenbauerC(n, lambda, 1))*(GegenbauerC(n, lambda, cos(theta[2])))/(GegenbauerC(n, lambda, 1)) = (GAMMA(lambda +(1)/(2)))/((Pi)^((1)/(2))* GAMMA(lambda))* int((GegenbauerC(n, lambda, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)))/(GegenbauerC(n, lambda, 1))*(sin(phi))^(2*lambda - 1), phi = 0..Pi)

Divide[GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 1]]],GegenbauerC[n, \[Lambda], 1]]*Divide[GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 2]]],GegenbauerC[n, \[Lambda], 1]] == Divide[Gamma[\[Lambda]+Divide[1,2]],(Pi)^(Divide[1,2])* Gamma[\[Lambda]]]* Integrate[Divide[GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]],GegenbauerC[n, \[Lambda], 1]]*(Sin[\[Phi]])^(2*\[Lambda]- 1), {\[Phi], 0, Pi}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
18.17.E6 P n ⁡ ( cos ⁡ θ 1 ) ⁢ P n ⁡ ( cos ⁡ θ 2 ) = 1 π ⁢ ∫ 0 π P n ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) ⁢ d ϕ Legendre-spherical-polynomial 𝑛 subscript 𝜃 1 Legendre-spherical-polynomial 𝑛 subscript 𝜃 2 1 𝜋 superscript subscript 0 𝜋 Legendre-spherical-polynomial 𝑛 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ italic-ϕ {\displaystyle{\displaystyle P_{n}\left(\cos\theta_{1}\right)P_{n}\left(\cos% \theta_{2}\right)=\frac{1}{\pi}\int_{0}^{\pi}P_{n}\left(\cos\theta_{1}\cos% \theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)\mathrm{d}\phi}}
\LegendrepolyP{n}@{\cos@@{\theta_{1}}}\LegendrepolyP{n}@{\cos@@{\theta_{2}}} = \frac{1}{\pi}\int_{0}^{\pi}\LegendrepolyP{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}\diff{\phi}		 

LegendreP(n, cos(theta[1]))*LegendreP(n, cos(theta[2])) = (1)/(Pi)*int(LegendreP(n, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)), phi = 0..Pi)

LegendreP[n, Cos[Subscript[\[Theta], 1]]]*LegendreP[n, Cos[Subscript[\[Theta], 2]]] == Divide[1,Pi]*Integrate[LegendreP[n, Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]], {\[Phi], 0, Pi}, GenerateConditions->None]
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
18.17.E7 ( P n ⁡ ( x ) ) 2 + 4 ⁢ π - 2 ⁢ ( 𝖰 n ⁡ ( x ) ) 2 = 4 ⁢ π - 2 ⁢ ∫ 1 ∞ Q n ⁡ ( x 2 + ( 1 - x 2 ) ⁢ t ) ⁢ ( t 2 - 1 ) - 1 2 ⁢ d t superscript Legendre-spherical-polynomial 𝑛 𝑥 2 4 superscript 𝜋 2 superscript shorthand-Ferrers-Legendre-Q-first-kind 𝑛 𝑥 2 4 superscript 𝜋 2 superscript subscript 1 shorthand-Legendre-Q-second-kind 𝑛 superscript 𝑥 2 1 superscript 𝑥 2 𝑡 superscript superscript 𝑡 2 1 1 2 𝑡 {\displaystyle{\displaystyle\left(P_{n}\left(x\right)\right)^{2}+4\pi^{-2}% \left(\mathsf{Q}_{n}\left(x\right)\right)^{2}=4\pi^{-2}\*\int_{1}^{\infty}Q_{n% }\left(x^{2}+(1-x^{2})t\right)(t^{2}-1)^{-\frac{1}{2}}\mathrm{d}t}}
\left(\LegendrepolyP{n}@{x}\right)^{2}+4\pi^{-2}\left(\FerrersQ[]{n}@{x}\right)^{2} = 4\pi^{-2}\*\int_{1}^{\infty}\assLegendreQ[]{n}@{x^{2}+(1-x^{2})t}(t^{2}-1)^{-\frac{1}{2}}\diff{t}		 - 1 < x , x < 1 formulae-sequence 1 𝑥 𝑥 1 {\displaystyle{\displaystyle-1<x,x<1}} 
(LegendreP(n, x))^(2)+ 4*(Pi)^(- 2)*(LegendreQ(n, x))^(2) = 4*(Pi)^(- 2)* int(LegendreQ(n, (x)^(2)+(1 - (x)^(2))*t)*((t)^(2)- 1)^(-(1)/(2)), t = 1..infinity)

(LegendreP[n, x])^(2)+ 4*(Pi)^(- 2)*(LegendreQ[n, x])^(2) == 4*(Pi)^(- 2)* Integrate[LegendreQ[n, 0, 3, (x)^(2)+(1 - (x)^(2))*t]*((t)^(2)- 1)^(-Divide[1,2]), {t, 1, Infinity}, GenerateConditions->None]
Failure Failure
Failed [3 / 3]
Result: 0.+Float(infinity)*I
Test Values: {x = 1/2, n = 1}


Result: 0.+Float(infinity)*I
Test Values: {x = 1/2, n = 2}

... skip entries to safe data
 Successful [Tested: 3]
18.17.E8 ( H n ⁡ ( x ) ) 2 + 2 n ⁢ ( n ! ) 2 ⁢ e x 2 ⁢ ( V ⁡ ( - n - 1 2 , 2 1 2 ⁢ x ) ) 2 = 2 n + 3 2 ⁢ n ! ⁢ e x 2 π ⁢ ∫ 0 ∞ e - ( 2 ⁢ n + 1 ) ⁢ t + x 2 ⁢ tanh ⁡ t ( sinh ⁡ 2 ⁢ t ) 1 2 ⁢ d t superscript Hermite-polynomial-H 𝑛 𝑥 2 superscript 2 𝑛 superscript 𝑛 2 superscript 𝑒 superscript 𝑥 2 superscript parabolic-V 𝑛 1 2 superscript 2 1 2 𝑥 2 superscript 2 𝑛 3 2 𝑛 superscript 𝑒 superscript 𝑥 2 𝜋 superscript subscript 0 superscript 𝑒 2 𝑛 1 𝑡 superscript 𝑥 2 𝑡 superscript 2 𝑡 1 2 𝑡 {\displaystyle{\displaystyle\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}% e^{x^{2}}\left(V\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac% {2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}% \tanh t}}{(\sinh 2t)^{\frac{1}{2}}}\mathrm{d}t}}
\left(\HermitepolyH{n}@{x}\right)^{2}+2^{n}(n!)^{2}e^{x^{2}}\left(\paraV@{-n-\tfrac{1}{2}}{2^{\frac{1}{2}}x}\right)^{2} = \frac{2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}\tanh@@{t}}}{(\sinh@@{2t})^{\frac{1}{2}}}\diff{t}		 

(HermiteH(n, x))^(2)+ (2)^(n)*(factorial(n))^(2)* exp((x)^(2))*(CylinderV(- n -(1)/(2), (2)^((1)/(2))* x))^(2) = ((2)^(n +(3)/(2))* factorial(n)*exp((x)^(2)))/(Pi)*int((exp(-(2*n + 1)*t + (x)^(2)* tanh(t)))/((sinh(2*t))^((1)/(2))), t = 0..infinity)

(HermiteH[n, x])^(2)+ (2)^(n)*((n)!)^(2)* Exp[(x)^(2)]*(Divide[GAMMA[1/2 + - n -Divide[1,2]], Pi]*(Sin[Pi*(- n -Divide[1,2])] * ParabolicCylinderD[-(- n -Divide[1,2]) - 1/2, (2)^(Divide[1,2])* x] + ParabolicCylinderD[-(- n -Divide[1,2]) - 1/2, -((2)^(Divide[1,2])* x)]))^(2) == Divide[(2)^(n +Divide[3,2])* (n)!*Exp[(x)^(2)],Pi]*Integrate[Divide[Exp[-(2*n + 1)*t + (x)^(2)* Tanh[t]],(Sinh[2*t])^(Divide[1,2])], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Successful [Tested: 9] Skipped - Because timed out
18.17.E9 ( 1 - x ) α + μ ⁢ P n ( α + μ , β - μ ) ⁡ ( x ) Γ ⁡ ( α + μ + n + 1 ) = ∫ x 1 ( 1 - y ) α ⁢ P n ( α , β ) ⁡ ( y ) Γ ⁡ ( α + n + 1 ) ⁢ ( y - x ) μ - 1 Γ ⁡ ( μ ) ⁢ d y superscript 1 𝑥 𝛼 𝜇 Jacobi-polynomial-P 𝛼 𝜇 𝛽 𝜇 𝑛 𝑥 Euler-Gamma 𝛼 𝜇 𝑛 1 superscript subscript 𝑥 1 superscript 1 𝑦 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 Euler-Gamma 𝛼 𝑛 1 superscript 𝑦 𝑥 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{(1-x)^{\alpha+\mu}P^{(\alpha+\mu,\beta-\mu)}% _{n}\left(x\right)}{\Gamma\left(\alpha+\mu+n+1\right)}=\int_{x}^{1}\frac{(1-y)% ^{\alpha}P^{(\alpha,\beta)}_{n}\left(y\right)}{\Gamma\left(\alpha+n+1\right)}% \frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right)}\mathrm{d}y}}
\frac{(1-x)^{\alpha+\mu}\JacobipolyP{\alpha+\mu}{\beta-\mu}{n}@{x}}{\EulerGamma@{\alpha+\mu+n+1}} = \int_{x}^{1}\frac{(1-y)^{\alpha}\JacobipolyP{\alpha}{\beta}{n}@{y}}{\EulerGamma@{\alpha+n+1}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}		 μ > 0 , - 1 < x , x < 1 , ℜ ⁡ ( α + μ + n + 1 ) > 0 , ℜ ⁡ ( α + n + 1 ) > 0 , ℜ ⁡ ( μ ) > 0 formulae-sequence 𝜇 0 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 𝛼 𝜇 𝑛 1 0 formulae-sequence 𝛼 𝑛 1 0 𝜇 0 {\displaystyle{\displaystyle\mu>0,-1<x,x<1,\Re(\alpha+\mu+n+1)>0,\Re(\alpha+n+% 1)>0,\Re(\mu)>0}} 
((1 - x)^(alpha + mu)* JacobiP(n, alpha + mu, beta - mu, x))/(GAMMA(alpha + mu + n + 1)) = int(((1 - y)^(alpha)* JacobiP(n, alpha, beta, y))/(GAMMA(alpha + n + 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..1)

Divide[(1 - x)^(\[Alpha]+ \[Mu])* JacobiP[n, \[Alpha]+ \[Mu], \[Beta]- \[Mu], x],Gamma[\[Alpha]+ \[Mu]+ n + 1]] == Integrate[Divide[(1 - y)^\[Alpha]* JacobiP[n, \[Alpha], \[Beta], y],Gamma[\[Alpha]+ n + 1]]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, 1}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
18.17.E10 x β + μ ⁢ ( x + 1 ) n Γ ⁡ ( β + μ + n + 1 ) ⁢ P n ( α , β + μ ) ⁡ ( x - 1 x + 1 ) = ∫ 0 x y β ⁢ ( y + 1 ) n Γ ⁡ ( β + n + 1 ) ⁢ P n ( α , β ) ⁡ ( y - 1 y + 1 ) ⁢ ( x - y ) μ - 1 Γ ⁡ ( μ ) ⁢ d y superscript 𝑥 𝛽 𝜇 superscript 𝑥 1 𝑛 Euler-Gamma 𝛽 𝜇 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝜇 𝑛 𝑥 1 𝑥 1 superscript subscript 0 𝑥 superscript 𝑦 𝛽 superscript 𝑦 1 𝑛 Euler-Gamma 𝛽 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 1 𝑦 1 superscript 𝑥 𝑦 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{x^{\beta+\mu}(x+1)^{n}}{\Gamma\left(\beta+% \mu+n+1\right)}P^{(\alpha,\beta+\mu)}_{n}\left(\frac{x-1}{x+1}\right)=\int_{0}% ^{x}\frac{y^{\beta}(y+1)^{n}}{\Gamma\left(\beta+n+1\right)}P^{(\alpha,\beta)}_% {n}\left(\frac{y-1}{y+1}\right)\*\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}% \mathrm{d}y}}
\frac{x^{\beta+\mu}(x+1)^{n}}{\EulerGamma@{\beta+\mu+n+1}}\JacobipolyP{\alpha}{\beta+\mu}{n}@{\frac{x-1}{x+1}} = \int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\EulerGamma@{\beta+n+1}}\JacobipolyP{\alpha}{\beta}{n}@{\frac{y-1}{y+1}}\*\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}		 μ > 0 , x > 0 , ℜ ⁡ ( β + μ + n + 1 ) > 0 , ℜ ⁡ ( β + n + 1 ) > 0 , ℜ ⁡ ( μ ) > 0 formulae-sequence 𝜇 0 formulae-sequence 𝑥 0 formulae-sequence 𝛽 𝜇 𝑛 1 0 formulae-sequence 𝛽 𝑛 1 0 𝜇 0 {\displaystyle{\displaystyle\mu>0,x>0,\Re(\beta+\mu+n+1)>0,\Re(\beta+n+1)>0,% \Re(\mu)>0}} 
((x)^(beta + mu)*(x + 1)^(n))/(GAMMA(beta + mu + n + 1))*JacobiP(n, alpha, beta + mu, (x - 1)/(x + 1)) = int(((y)^(beta)*(y + 1)^(n))/(GAMMA(beta + n + 1))*JacobiP(n, alpha, beta, (y - 1)/(y + 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 0..x)

Divide[(x)^(\[Beta]+ \[Mu])*(x + 1)^(n),Gamma[\[Beta]+ \[Mu]+ n + 1]]*JacobiP[n, \[Alpha], \[Beta]+ \[Mu], Divide[x - 1,x + 1]] == Integrate[Divide[(y)^\[Beta]*(y + 1)^(n),Gamma[\[Beta]+ n + 1]]*JacobiP[n, \[Alpha], \[Beta], Divide[y - 1,y + 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 0, x}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
18.17.E11 Γ ⁡ ( n + α + β - μ + 1 ) x n + α + β - μ + 1 ⁢ P n ( α , β - μ ) ⁡ ( 1 - 2 ⁢ x - 1 ) = ∫ x ∞ Γ ⁡ ( n + α + β + 1 ) y n + α + β + 1 ⁢ P n ( α , β ) ⁡ ( 1 - 2 ⁢ y - 1 ) ⁢ ( y - x ) μ - 1 Γ ⁡ ( μ ) ⁢ d y Euler-Gamma 𝑛 𝛼 𝛽 𝜇 1 superscript 𝑥 𝑛 𝛼 𝛽 𝜇 1 Jacobi-polynomial-P 𝛼 𝛽 𝜇 𝑛 1 2 superscript 𝑥 1 superscript subscript 𝑥 Euler-Gamma 𝑛 𝛼 𝛽 1 superscript 𝑦 𝑛 𝛼 𝛽 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 superscript 𝑦 1 superscript 𝑦 𝑥 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{\Gamma\left(n+\alpha+\beta-\mu+1\right)}{x^{% n+\alpha+\beta-\mu+1}}P^{(\alpha,\beta-\mu)}_{n}\left(1-2x^{-1}\right)=\int_{x% }^{\infty}\frac{\Gamma\left(n+\alpha+\beta+1\right)}{y^{n+\alpha+\beta+1}}P^{(% \alpha,\beta)}_{n}\left(1-2y^{-1}\right)\*\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu% \right)}\mathrm{d}y}}
\frac{\EulerGamma@{n+\alpha+\beta-\mu+1}}{x^{n+\alpha+\beta-\mu+1}}\JacobipolyP{\alpha}{\beta-\mu}{n}@{1-2x^{-1}} = \int_{x}^{\infty}\frac{\EulerGamma@{n+\alpha+\beta+1}}{y^{n+\alpha+\beta+1}}\JacobipolyP{\alpha}{\beta}{n}@{1-2y^{-1}}\*\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}		 α + β + 1 > μ , μ > 0 , x > 1 , ℜ ⁡ ( n + α + β - μ + 1 ) > 0 , ℜ ⁡ ( n + α + β + 1 ) > 0 , ℜ ⁡ ( μ ) > 0 formulae-sequence 𝛼 𝛽 1 𝜇 formulae-sequence 𝜇 0 formulae-sequence 𝑥 1 formulae-sequence 𝑛 𝛼 𝛽 𝜇 1 0 formulae-sequence 𝑛 𝛼 𝛽 1 0 𝜇 0 {\displaystyle{\displaystyle\alpha+\beta+1>\mu,\mu>0,x>1,\Re(n+\alpha+\beta-% \mu+1)>0,\Re(n+\alpha+\beta+1)>0,\Re(\mu)>0}} 
(GAMMA(n + alpha + beta - mu + 1))/((x)^(n + alpha + beta - mu + 1))*JacobiP(n, alpha, beta - mu, 1 - 2*(x)^(- 1)) = int((GAMMA(n + alpha + beta + 1))/((y)^(n + alpha + beta + 1))*JacobiP(n, alpha, beta, 1 - 2*(y)^(- 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)

Divide[Gamma[n + \[Alpha]+ \[Beta]- \[Mu]+ 1],(x)^(n + \[Alpha]+ \[Beta]- \[Mu]+ 1)]*JacobiP[n, \[Alpha], \[Beta]- \[Mu], 1 - 2*(x)^(- 1)] == Integrate[Divide[Gamma[n + \[Alpha]+ \[Beta]+ 1],(y)^(n + \[Alpha]+ \[Beta]+ 1)]*JacobiP[n, \[Alpha], \[Beta], 1 - 2*(y)^(- 1)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
18.17.E12 Γ ⁡ ( λ - μ ) ⁢ C n ( λ - μ ) ⁡ ( x - 1 2 ) x λ - μ + 1 2 ⁢ n = ∫ x ∞ Γ ⁡ ( λ ) ⁢ C n ( λ ) ⁡ ( y - 1 2 ) y λ + 1 2 ⁢ n ⁢ ( y - x ) μ - 1 Γ ⁡ ( μ ) ⁢ d y Euler-Gamma 𝜆 𝜇 ultraspherical-Gegenbauer-polynomial 𝜆 𝜇 𝑛 superscript 𝑥 1 2 superscript 𝑥 𝜆 𝜇 1 2 𝑛 superscript subscript 𝑥 Euler-Gamma 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 superscript 𝑦 1 2 superscript 𝑦 𝜆 1 2 𝑛 superscript 𝑦 𝑥 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-% \mu)}_{n}\left(x^{-\frac{1}{2}}\right)}{x^{\lambda-\mu+\frac{1}{2}n}}=\int_{x}% ^{\infty}\frac{\Gamma\left(\lambda\right)C^{(\lambda)}_{n}\left(y^{-\frac{1}{2% }}\right)}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right% )}\mathrm{d}y}}
\frac{\EulerGamma@{\lambda-\mu}\ultrasphpoly{\lambda-\mu}{n}@{x^{-\frac{1}{2}}}}{x^{\lambda-\mu+\frac{1}{2}n}} = \int_{x}^{\infty}\frac{\EulerGamma@{\lambda}\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}		 λ > μ , μ > 0 , x > 0 , ℜ ⁡ ( λ - μ ) > 0 , ℜ ⁡ ( λ ) > 0 , ℜ ⁡ ( μ ) > 0 formulae-sequence 𝜆 𝜇 formulae-sequence 𝜇 0 formulae-sequence 𝑥 0 formulae-sequence 𝜆 𝜇 0 formulae-sequence 𝜆 0 𝜇 0 {\displaystyle{\displaystyle\lambda>\mu,\mu>0,x>0,\Re(\lambda-\mu)>0,\Re(% \lambda)>0,\Re(\mu)>0}} 
(GAMMA(lambda - mu)*GegenbauerC(n, lambda - mu, (x)^(-(1)/(2))))/((x)^(lambda - mu +(1)/(2)*n)) = int((GAMMA(lambda)*GegenbauerC(n, lambda, (y)^(-(1)/(2))))/((y)^(lambda +(1)/(2)*n))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)

Divide[Gamma[\[Lambda]- \[Mu]]*GegenbauerC[n, \[Lambda]- \[Mu], (x)^(-Divide[1,2])],(x)^(\[Lambda]- \[Mu]+Divide[1,2]*n)] == Integrate[Divide[Gamma[\[Lambda]]*GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],(y)^(\[Lambda]+Divide[1,2]*n)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
18.17.E13 x 1 2 ⁢ n ⁢ ( x - 1 ) λ + μ - 1 2 Γ ⁡ ( λ + μ + 1 2 ) ⁢ C n ( λ + μ ) ⁡ ( x - 1 2 ) C n ( λ + μ ) ⁡ ( 1 ) = ∫ 1 x y 1 2 ⁢ n ⁢ ( y - 1 ) λ - 1 2 Γ ⁡ ( λ + 1 2 ) ⁢ C n ( λ ) ⁡ ( y - 1 2 ) C n ( λ ) ⁡ ( 1 ) ⁢ ( x - y ) μ - 1 Γ ⁡ ( μ ) ⁢ d y superscript 𝑥 1 2 𝑛 superscript 𝑥 1 𝜆 𝜇 1 2 Euler-Gamma 𝜆 𝜇 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝜇 𝑛 superscript 𝑥 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝜇 𝑛 1 superscript subscript 1 𝑥 superscript 𝑦 1 2 𝑛 superscript 𝑦 1 𝜆 1 2 Euler-Gamma 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 superscript 𝑦 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 superscript 𝑥 𝑦 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{% 2}}}{\Gamma\left(\lambda+\mu+\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}% \left(x^{-\frac{1}{2}}\right)}{C^{(\lambda+\mu)}_{n}\left(1\right)}=\int_{1}^{% x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{\Gamma\left(\lambda+% \tfrac{1}{2}\right)}\frac{C^{(\lambda)}_{n}\left(y^{-\frac{1}{2}}\right)}{C^{(% \lambda)}_{n}\left(1\right)}\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}% \mathrm{d}y}}
\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\EulerGamma@{\lambda+\mu+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda+\mu}{n}@{x^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda+\mu}{n}@{1}} = \int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{\EulerGamma@{\lambda+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda}{n}@{1}}\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}		 μ > 0 , x > 1 , ℜ ⁡ ( λ + μ + 1 2 ) > 0 , ℜ ⁡ ( λ + 1 2 ) > 0 , ℜ ⁡ ( μ ) > 0 formulae-sequence 𝜇 0 formulae-sequence 𝑥 1 formulae-sequence 𝜆 𝜇 1 2 0 formulae-sequence 𝜆 1 2 0 𝜇 0 {\displaystyle{\displaystyle\mu>0,x>1,\Re(\lambda+\mu+\tfrac{1}{2})>0,\Re(% \lambda+\tfrac{1}{2})>0,\Re(\mu)>0}} 
((x)^((1)/(2)*n)*(x - 1)^(lambda + mu -(1)/(2)))/(GAMMA(lambda + mu +(1)/(2)))*(GegenbauerC(n, lambda + mu, (x)^(-(1)/(2))))/(GegenbauerC(n, lambda + mu, 1)) = int(((y)^((1)/(2)*n)*(y - 1)^(lambda -(1)/(2)))/(GAMMA(lambda +(1)/(2)))*(GegenbauerC(n, lambda, (y)^(-(1)/(2))))/(GegenbauerC(n, lambda, 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 1..x)

Divide[(x)^(Divide[1,2]*n)*(x - 1)^(\[Lambda]+ \[Mu]-Divide[1,2]),Gamma[\[Lambda]+ \[Mu]+Divide[1,2]]]*Divide[GegenbauerC[n, \[Lambda]+ \[Mu], (x)^(-Divide[1,2])],GegenbauerC[n, \[Lambda]+ \[Mu], 1]] == Integrate[Divide[(y)^(Divide[1,2]*n)*(y - 1)^(\[Lambda]-Divide[1,2]),Gamma[\[Lambda]+Divide[1,2]]]*Divide[GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],GegenbauerC[n, \[Lambda], 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 1, x}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
18.17.E14 x α + μ ⁢ L n ( α + μ ) ⁡ ( x ) Γ ⁡ ( α + μ + n + 1 ) = ∫ 0 x y α ⁢ L n ( α ) ⁡ ( y ) Γ ⁡ ( α + n + 1 ) ⁢ ( x - y ) μ - 1 Γ ⁡ ( μ ) ⁢ d y superscript 𝑥 𝛼 𝜇 Laguerre-polynomial-L 𝛼 𝜇 𝑛 𝑥 Euler-Gamma 𝛼 𝜇 𝑛 1 superscript subscript 0 𝑥 superscript 𝑦 𝛼 Laguerre-polynomial-L 𝛼 𝑛 𝑦 Euler-Gamma 𝛼 𝑛 1 superscript 𝑥 𝑦 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{x^{\alpha+\mu}L^{(\alpha+\mu)}_{n}\left(x% \right)}{\Gamma\left(\alpha+\mu+n+1\right)}=\int_{0}^{x}\frac{y^{\alpha}L^{(% \alpha)}_{n}\left(y\right)}{\Gamma\left(\alpha+n+1\right)}\frac{(x-y)^{\mu-1}}% {\Gamma\left(\mu\right)}\mathrm{d}y}}
\frac{x^{\alpha+\mu}\LaguerrepolyL[\alpha+\mu]{n}@{x}}{\EulerGamma@{\alpha+\mu+n+1}} = \int_{0}^{x}\frac{y^{\alpha}\LaguerrepolyL[\alpha]{n}@{y}}{\EulerGamma@{\alpha+n+1}}\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}		 μ > 0 , x > 0 , ℜ ⁡ ( α + μ + n + 1 ) > 0 , ℜ ⁡ ( α + n + 1 ) > 0 , ℜ ⁡ ( μ ) > 0 formulae-sequence 𝜇 0 formulae-sequence 𝑥 0 formulae-sequence 𝛼 𝜇 𝑛 1 0 formulae-sequence 𝛼 𝑛 1 0 𝜇 0 {\displaystyle{\displaystyle\mu>0,x>0,\Re(\alpha+\mu+n+1)>0,\Re(\alpha+n+1)>0,% \Re(\mu)>0}} 
((x)^(alpha + mu)* LaguerreL(n, alpha + mu, x))/(GAMMA(alpha + mu + n + 1)) = int(((y)^(alpha)* LaguerreL(n, alpha, y))/(GAMMA(alpha + n + 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 0..x)

Divide[(x)^(\[Alpha]+ \[Mu])* LaguerreL[n, \[Alpha]+ \[Mu], x],Gamma[\[Alpha]+ \[Mu]+ n + 1]] == Integrate[Divide[(y)^\[Alpha]* LaguerreL[n, \[Alpha], y],Gamma[\[Alpha]+ n + 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 0, x}, GenerateConditions->None]
Missing Macro Error Failure - Manual Skip!
18.17.E15 e - x ⁢ L n ( α ) ⁡ ( x ) = ∫ x ∞ e - y ⁢ L n ( α + μ ) ⁡ ( y ) ⁢ ( y - x ) μ - 1 Γ ⁡ ( μ ) ⁢ d y superscript 𝑒 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript subscript 𝑥 superscript 𝑒 𝑦 Laguerre-polynomial-L 𝛼 𝜇 𝑛 𝑦 superscript 𝑦 𝑥 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle e^{-x}L^{(\alpha)}_{n}\left(x\right)=\int_{x}^{% \infty}e^{-y}L^{(\alpha+\mu)}_{n}\left(y\right)\frac{(y-x)^{\mu-1}}{\Gamma% \left(\mu\right)}\mathrm{d}y}}
e^{-x}\LaguerrepolyL[\alpha]{n}@{x} = \int_{x}^{\infty}e^{-y}\LaguerrepolyL[\alpha+\mu]{n}@{y}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}		 μ > 0 , ℜ ⁡ ( μ ) > 0 formulae-sequence 𝜇 0 𝜇 0 {\displaystyle{\displaystyle\mu>0,\Re(\mu)>0}} 
exp(- x)*LaguerreL(n, alpha, x) = int(exp(- y)*LaguerreL(n, alpha + mu, y)*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)

Exp[- x]*LaguerreL[n, \[Alpha], x] == Integrate[Exp[- y]*LaguerreL[n, \[Alpha]+ \[Mu], y]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}, GenerateConditions->None]
Missing Macro Error Aborted - Skipped - Because timed out
18.17.E16 ∫ - 1 1 ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) ⁢ e i ⁢ x ⁢ y ⁢ d x = ( i ⁢ y ) n ⁢ e i ⁢ y n ! ⁢ 2 n + α + β + 1 ⁢ B ⁡ ( n + α + 1 , n + β + 1 ) ⁢ F 1 1 ⁡ ( n + α + 1 ; 2 ⁢ n + α + β + 2 ; - 2 ⁢ i ⁢ y ) superscript subscript 1 1 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑒 𝑖 𝑥 𝑦 𝑥 superscript 𝑖 𝑦 𝑛 superscript 𝑒 𝑖 𝑦 𝑛 superscript 2 𝑛 𝛼 𝛽 1 Euler-Beta 𝑛 𝛼 1 𝑛 𝛽 1 Kummer-confluent-hypergeometric-M-as-1F1 𝑛 𝛼 1 2 𝑛 𝛼 𝛽 2 2 𝑖 𝑦 {\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha% ,\beta)}_{n}\left(x\right)e^{ixy}\mathrm{d}x=\frac{(iy)^{n}e^{iy}}{n!}2^{n+% \alpha+\beta+1}\mathrm{B}\left(n+\alpha+1,n+\beta+1\right){{}_{1}F_{1}}\left(n% +\alpha+1;2n+\alpha+\beta+2;-2iy\right)}}
\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}e^{ixy}\diff{x} = \frac{(iy)^{n}e^{iy}}{n!}2^{n+\alpha+\beta+1}\EulerBeta@{n+\alpha+1}{n+\beta+1}\genhyperF{1}{1}@{n+\alpha+1}{2n+\alpha+\beta+2}{-2iy}		 ℜ ⁡ ( n + α + 1 ) > 0 , ℜ ⁡ ( n + β + 1 ) > 0 , ℜ ⁡ ( ( n + α + 1 ) + b ) > 0 , ℜ ⁡ ( a + ( n + β + 1 ) ) > 0 formulae-sequence 𝑛 𝛼 1 0 formulae-sequence 𝑛 𝛽 1 0 formulae-sequence 𝑛 𝛼 1 𝑏 0 𝑎 𝑛 𝛽 1 0 {\displaystyle{\displaystyle\Re(n+\alpha+1)>0,\Re(n+\beta+1)>0,\Re((n+\alpha+1% )+b)>0,\Re(a+(n+\beta+1))>0}} 
int((1 - x)^(alpha)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x)*exp(I*x*y), x = - 1..1) = ((I*y)^(n)* exp(I*y))/(factorial(n))*(2)^(n + alpha + beta + 1)* Beta(n + alpha + 1, n + beta + 1)*hypergeom([n + alpha + 1], [2*n + alpha + beta + 2], - 2*I*y)

Integrate[(1 - x)^\[Alpha]*(1 + x)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], x]*Exp[I*x*y], {x, - 1, 1}, GenerateConditions->None] == Divide[(I*y)^(n)* Exp[I*y],(n)!]*(2)^(n + \[Alpha]+ \[Beta]+ 1)* Beta[n + \[Alpha]+ 1, n + \[Beta]+ 1]*HypergeometricPFQ[{n + \[Alpha]+ 1}, {2*n + \[Alpha]+ \[Beta]+ 2}, - 2*I*y]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
18.17.E17 ∫ 0 1 ( 1 - x 2 ) λ - 1 2 ⁢ C 2 ⁢ n ( λ ) ⁡ ( x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ π ⁢ Γ ⁡ ( 2 ⁢ n + 2 ⁢ λ ) ⁢ J λ + 2 ⁢ n ⁡ ( y ) ( 2 ⁢ n ) ! ⁢ Γ ⁡ ( λ ) ⁢ ( 2 ⁢ y ) λ superscript subscript 0 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 𝜋 Euler-Gamma 2 𝑛 2 𝜆 Bessel-J 𝜆 2 𝑛 𝑦 2 𝑛 Euler-Gamma 𝜆 superscript 2 𝑦 𝜆 {\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{2n}\left(x\right)\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi% \Gamma\left(2n+2\lambda\right)J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(% \lambda\right)(2y)^{\lambda}}}}
\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n}@{x}\cos@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda}\BesselJ{\lambda+2n}@{y}}{(2n)!\EulerGamma@{\lambda}(2y)^{\lambda}}		 ℜ ⁡ ( ( λ + 2 ⁢ n ) + k + 1 ) > 0 , ℜ ⁡ ( 2 ⁢ n + 2 ⁢ λ ) > 0 , ℜ ⁡ ( λ ) > 0 formulae-sequence 𝜆 2 𝑛 𝑘 1 0 formulae-sequence 2 𝑛 2 𝜆 0 𝜆 0 {\displaystyle{\displaystyle\Re((\lambda+2n)+k+1)>0,\Re(2n+2\lambda)>0,\Re(% \lambda)>0}} 
int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2*n, lambda, x)*cos(x*y), x = 0..1) = ((- 1)^(n)* Pi*GAMMA(2*n + 2*lambda)*BesselJ(lambda + 2*n, y))/(factorial(2*n)*GAMMA(lambda)*(2*y)^(lambda))

Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2*n, \[Lambda], x]*Cos[x*y], {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pi*Gamma[2*n + 2*\[Lambda]]*BesselJ[\[Lambda]+ 2*n, y],(2*n)!*Gamma[\[Lambda]]*(2*y)^\[Lambda]]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
18.17.E18 ∫ 0 1 ( 1 - x 2 ) λ - 1 2 ⁢ C 2 ⁢ n + 1 ( λ ) ⁡ ( x ) ⁢ sin ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ π ⁢ Γ ⁡ ( 2 ⁢ n + 2 ⁢ λ + 1 ) ⁢ J 2 ⁢ n + λ + 1 ⁡ ( y ) ( 2 ⁢ n + 1 ) ! ⁢ Γ ⁡ ( λ ) ⁢ ( 2 ⁢ y ) λ superscript subscript 0 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 1 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 𝜋 Euler-Gamma 2 𝑛 2 𝜆 1 Bessel-J 2 𝑛 𝜆 1 𝑦 2 𝑛 1 Euler-Gamma 𝜆 superscript 2 𝑦 𝜆 {\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{2n+1}\left(x\right)\sin\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi% \Gamma\left(2n+2\lambda+1\right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma% \left(\lambda\right)(2y)^{\lambda}}}}
\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n+1}@{x}\sin@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda+1}\BesselJ{2n+\lambda+1}@{y}}{(2n+1)!\EulerGamma@{\lambda}(2y)^{\lambda}}		 ℜ ⁡ ( ( 2 ⁢ n + λ + 1 ) + k + 1 ) > 0 , ℜ ⁡ ( 2 ⁢ n + 2 ⁢ λ + 1 ) > 0 , ℜ ⁡ ( λ ) > 0 formulae-sequence 2 𝑛 𝜆 1 𝑘 1 0 formulae-sequence 2 𝑛 2 𝜆 1 0 𝜆 0 {\displaystyle{\displaystyle\Re((2n+\lambda+1)+k+1)>0,\Re(2n+2\lambda+1)>0,\Re% (\lambda)>0}} 
int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2*n + 1, lambda, x)*sin(x*y), x = 0..1) = ((- 1)^(n)* Pi*GAMMA(2*n + 2*lambda + 1)*BesselJ(2*n + lambda + 1, y))/(factorial(2*n + 1)*GAMMA(lambda)*(2*y)^(lambda))

Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2*n + 1, \[Lambda], x]*Sin[x*y], {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pi*Gamma[2*n + 2*\[Lambda]+ 1]*BesselJ[2*n + \[Lambda]+ 1, y],(2*n + 1)!*Gamma[\[Lambda]]*(2*y)^\[Lambda]]
Failure Failure Skipped - Because timed out Skipped - Because timed out
18.17.E19 ∫ - 1 1 P n ⁡ ( x ) ⁢ e i ⁢ x ⁢ y ⁢ d x = i n ⁢ 2 ⁢ π y ⁢ J n + 1 2 ⁡ ( y ) superscript subscript 1 1 Legendre-spherical-polynomial 𝑛 𝑥 superscript 𝑒 𝑖 𝑥 𝑦 𝑥 superscript 𝑖 𝑛 2 𝜋 𝑦 Bessel-J 𝑛 1 2 𝑦 {\displaystyle{\displaystyle\int_{-1}^{1}P_{n}\left(x\right)e^{ixy}\mathrm{d}x% =i^{n}\sqrt{\frac{2\pi}{y}}J_{n+\frac{1}{2}}\left(y\right)}}
\int_{-1}^{1}\LegendrepolyP{n}@{x}e^{ixy}\diff{x} = i^{n}\sqrt{\frac{2\pi}{y}}\BesselJ{n+\frac{1}{2}}@{y}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}} 
int(LegendreP(n, x)*exp(I*x*y), x = - 1..1) = (I)^(n)*sqrt((2*Pi)/(y))*BesselJ(n +(1)/(2), y)

Integrate[LegendreP[n, x]*Exp[I*x*y], {x, - 1, 1}, GenerateConditions->None] == (I)^(n)*Sqrt[Divide[2*Pi,y]]*BesselJ[n +Divide[1,2], y]
Failure Failure
Failed [9 / 18]
Result: -.1455515881e-15-1.584691883*I
Test Values: {y = -3/2, n = 1}


Result: -.5093971348+.7797894631e-16*I
Test Values: {y = -3/2, n = 2}

... skip entries to safe data
Failed [9 / 18]
Result: Complex[0.0, -1.584691882848889]
Test Values: {Rule[n, 1], Rule[y, -1.5]}


Result: Complex[-0.5093971347536326, -3.3306690738754696*^-16]
Test Values: {Rule[n, 2], Rule[y, -1.5]}

... skip entries to safe data
18.17.E20 ∫ 0 1 P n ⁡ ( 1 - 2 ⁢ x 2 ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ 1 2 ⁢ π ⁢ J n + 1 2 ⁡ ( 1 2 ⁢ y ) ⁢ J - n - 1 2 ⁡ ( 1 2 ⁢ y ) superscript subscript 0 1 Legendre-spherical-polynomial 𝑛 1 2 superscript 𝑥 2 𝑥 𝑦 𝑥 superscript 1 𝑛 1 2 𝜋 Bessel-J 𝑛 1 2 1 2 𝑦 Bessel-J 𝑛 1 2 1 2 𝑦 {\displaystyle{\displaystyle\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\cos\left(xy% \right)\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\pi J_{n+\frac{1}{2}}\left(\tfrac{1}{2}% y\right)J_{-n-\frac{1}{2}}\left(\tfrac{1}{2}y\right)}}
\int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\cos@{xy}\diff{x} = (-1)^{n}\tfrac{1}{2}\pi\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\BesselJ{-n-\frac{1}{2}}@{\tfrac{1}{2}y}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}} 
int(LegendreP(n, 1 - 2*(x)^(2))*cos(x*y), x = 0..1) = (- 1)^(n)*(1)/(2)*Pi*BesselJ(n +(1)/(2), (1)/(2)*y)*BesselJ(- n -(1)/(2), (1)/(2)*y)

Integrate[LegendreP[n, 1 - 2*(x)^(2)]*Cos[x*y], {x, 0, 1}, GenerateConditions->None] == (- 1)^(n)*Divide[1,2]*Pi*BesselJ[n +Divide[1,2], Divide[1,2]*y]*BesselJ[- n -Divide[1,2], Divide[1,2]*y]
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
18.17.E21 ∫ 0 1 P n ⁡ ( 1 - 2 ⁢ x 2 ) ⁢ sin ⁡ ( x ⁢ y ) ⁢ d x = 1 2 ⁢ π ⁢ ( J n + 1 2 ⁡ ( 1 2 ⁢ y ) ) 2 superscript subscript 0 1 Legendre-spherical-polynomial 𝑛 1 2 superscript 𝑥 2 𝑥 𝑦 𝑥 1 2 𝜋 superscript Bessel-J 𝑛 1 2 1 2 𝑦 2 {\displaystyle{\displaystyle\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\sin\left(xy% \right)\mathrm{d}x=\tfrac{1}{2}\pi\left(J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y% \right)\right)^{2}}}
\int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\sin@{xy}\diff{x} = \tfrac{1}{2}\pi\left(\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\right)^{2}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}} 
int(LegendreP(n, 1 - 2*(x)^(2))*sin(x*y), x = 0..1) = (1)/(2)*Pi*(BesselJ(n +(1)/(2), (1)/(2)*y))^(2)

Integrate[LegendreP[n, 1 - 2*(x)^(2)]*Sin[x*y], {x, 0, 1}, GenerateConditions->None] == Divide[1,2]*Pi*(BesselJ[n +Divide[1,2], Divide[1,2]*y])^(2)
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
18.17.E30 ∫ 0 ∞ x 2 ⁢ n ⁢ e - 1 2 ⁢ x 2 ⁢ L n ( n - 1 2 ) ⁡ ( 1 2 ⁢ x 2 ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = 1 2 ⁢ π ⁢ y 2 ⁢ n ⁢ e - 1 2 ⁢ y 2 ⁢ L n ( n - 1 2 ) ⁡ ( 1 2 ⁢ y 2 ) superscript subscript 0 superscript 𝑥 2 𝑛 superscript 𝑒 1 2 superscript 𝑥 2 Laguerre-polynomial-L 𝑛 1 2 𝑛 1 2 superscript 𝑥 2 𝑥 𝑦 𝑥 1 2 𝜋 superscript 𝑦 2 𝑛 superscript 𝑒 1 2 superscript 𝑦 2 Laguerre-polynomial-L 𝑛 1 2 𝑛 1 2 superscript 𝑦 2 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{2n}e^{-\frac{1}{2}x^{2}}L^{(n-% \frac{1}{2})}_{n}\left(\tfrac{1}{2}x^{2}\right)\cos\left(xy\right)\mathrm{d}x=% \sqrt{\tfrac{1}{2}\pi}y^{2n}e^{-\frac{1}{2}y^{2}}L^{(n-\frac{1}{2})}_{n}\left(% \tfrac{1}{2}y^{2}\right)}}
\int_{0}^{\infty}x^{2n}e^{-\frac{1}{2}x^{2}}\LaguerrepolyL[n-\frac{1}{2}]{n}@{\tfrac{1}{2}x^{2}}\cos@{xy}\diff{x} = \sqrt{\tfrac{1}{2}\pi}y^{2n}e^{-\frac{1}{2}y^{2}}\LaguerrepolyL[n-\frac{1}{2}]{n}@{\tfrac{1}{2}y^{2}}		 

int((x)^(2*n)* exp(-(1)/(2)*(x)^(2))*LaguerreL(n, n -(1)/(2), (1)/(2)*(x)^(2))*cos(x*y), x = 0..infinity) = sqrt((1)/(2)*Pi)*(y)^(2*n)* exp(-(1)/(2)*(y)^(2))*LaguerreL(n, n -(1)/(2), (1)/(2)*(y)^(2))

Integrate[(x)^(2*n)* Exp[-Divide[1,2]*(x)^(2)]*LaguerreL[n, n -Divide[1,2], Divide[1,2]*(x)^(2)]*Cos[x*y], {x, 0, Infinity}, GenerateConditions->None] == Sqrt[Divide[1,2]*Pi]*(y)^(2*n)* Exp[-Divide[1,2]*(y)^(2)]*LaguerreL[n, n -Divide[1,2], Divide[1,2]*(y)^(2)]
Missing Macro Error Aborted - Skipped - Because timed out
18.17.E31 ∫ 0 ∞ e - a ⁢ x ⁢ x ν - 2 ⁢ n ⁢ L 2 ⁢ n - 1 ( ν - 2 ⁢ n ) ⁡ ( a ⁢ x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = i ⁢ ( - 1 ) n ⁢ Γ ⁡ ( ν ) 2 ⁢ ( 2 ⁢ n - 1 ) ! ⁢ y 2 ⁢ n - 1 ⁢ ( ( a + i ⁢ y ) - ν - ( a - i ⁢ y ) - ν ) superscript subscript 0 superscript 𝑒 𝑎 𝑥 superscript 𝑥 𝜈 2 𝑛 Laguerre-polynomial-L 𝜈 2 𝑛 2 𝑛 1 𝑎 𝑥 𝑥 𝑦 𝑥 𝑖 superscript 1 𝑛 Euler-Gamma 𝜈 2 2 𝑛 1 superscript 𝑦 2 𝑛 1 superscript 𝑎 𝑖 𝑦 𝜈 superscript 𝑎 𝑖 𝑦 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}x^{\nu-2n}L^{(\nu-2n)}_{2n% -1}\left(ax\right)\cos\left(xy\right)\mathrm{d}x=i\frac{(-1)^{n}\Gamma\left(% \nu\right)}{2(2n-1)!}y^{2n-1}\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right)}}
\int_{0}^{\infty}e^{-ax}x^{\nu-2n}\LaguerrepolyL[\nu-2n]{2n-1}@{ax}\cos@{xy}\diff{x} = i\frac{(-1)^{n}\EulerGamma@{\nu}}{2(2n-1)!}y^{2n-1}\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right)		 ν > 2 ⁢ n - 1 , a > 0 , ℜ ⁡ ( ν ) > 0 formulae-sequence 𝜈 2 𝑛 1 formulae-sequence 𝑎 0 𝜈 0 {\displaystyle{\displaystyle\nu>2n-1,a>0,\Re(\nu)>0}} 
int(exp(- a*x)*(x)^(nu - 2*n)* LaguerreL(2*n - 1, nu - 2*n, a*x)*cos(x*y), x = 0..infinity) = I*((- 1)^(n)* GAMMA(nu))/(2*factorial(2*n - 1))*(y)^(2*n - 1)*((a + I*y)^(- nu)-(a - I*y)^(- nu))

Integrate[Exp[- a*x]*(x)^(\[Nu]- 2*n)* LaguerreL[2*n - 1, \[Nu]- 2*n, a*x]*Cos[x*y], {x, 0, Infinity}, GenerateConditions->None] == I*Divide[(- 1)^(n)* Gamma[\[Nu]],2*(2*n - 1)!]*(y)^(2*n - 1)*((a + I*y)^(- \[Nu])-(a - I*y)^(- \[Nu]))
Missing Macro Error Aborted - Skipped - Because timed out
18.17.E32 ∫ 0 ∞ e - a ⁢ x ⁢ x ν - 1 - 2 ⁢ n ⁢ L 2 ⁢ n ( ν - 1 - 2 ⁢ n ) ⁡ ( a ⁢ x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ Γ ⁡ ( ν ) 2 ⁢ ( 2 ⁢ n ) ! ⁢ y 2 ⁢ n ⁢ ( ( a + i ⁢ y ) - ν + ( a - i ⁢ y ) - ν ) superscript subscript 0 superscript 𝑒 𝑎 𝑥 superscript 𝑥 𝜈 1 2 𝑛 Laguerre-polynomial-L 𝜈 1 2 𝑛 2 𝑛 𝑎 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 Euler-Gamma 𝜈 2 2 𝑛 superscript 𝑦 2 𝑛 superscript 𝑎 𝑖 𝑦 𝜈 superscript 𝑎 𝑖 𝑦 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}% _{2n}\left(ax\right)\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(% \nu\right)}{2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right)}}
\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}\LaguerrepolyL[\nu-1-2n]{2n}@{ax}\cos@{xy}\diff{x} = \frac{(-1)^{n}\EulerGamma@{\nu}}{2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right)		 ν > 2 ⁢ n , a > 0 , ℜ ⁡ ( ν ) > 0 formulae-sequence 𝜈 2 𝑛 formulae-sequence 𝑎 0 𝜈 0 {\displaystyle{\displaystyle\nu>2n,a>0,\Re(\nu)>0}} 
int(exp(- a*x)*(x)^(nu - 1 - 2*n)* LaguerreL(2*n, nu - 1 - 2*n, a*x)*cos(x*y), x = 0..infinity) = ((- 1)^(n)* GAMMA(nu))/(2*factorial(2*n))*(y)^(2*n)*((a + I*y)^(- nu)+(a - I*y)^(- nu))

Integrate[Exp[- a*x]*(x)^(\[Nu]- 1 - 2*n)* LaguerreL[2*n, \[Nu]- 1 - 2*n, a*x]*Cos[x*y], {x, 0, Infinity}, GenerateConditions->None] == Divide[(- 1)^(n)* Gamma[\[Nu]],2*(2*n)!]*(y)^(2*n)*((a + I*y)^(- \[Nu])+(a - I*y)^(- \[Nu]))
Missing Macro Error Aborted - Skipped - Because timed out
18.17.E33 ∫ - 1 1 e - ( x + 1 ) ⁢ z ⁢ P n ( α , β ) ⁡ ( x ) ⁢ ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ d x = ( - 1 ) n ⁢ 2 α + β + n + 1 ⁢ Γ ⁡ ( α + n + 1 ) ⁢ Γ ⁡ ( β + n + 1 ) Γ ⁡ ( α + β + 2 ⁢ n + 2 ) ⁢ n ! ⁢ z n ⁢ F 1 1 ⁡ ( β + n + 1 α + β + 2 ⁢ n + 2 ; - 2 ⁢ z ) superscript subscript 1 1 superscript 𝑒 𝑥 1 𝑧 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 𝑥 superscript 1 𝑛 superscript 2 𝛼 𝛽 𝑛 1 Euler-Gamma 𝛼 𝑛 1 Euler-Gamma 𝛽 𝑛 1 Euler-Gamma 𝛼 𝛽 2 𝑛 2 𝑛 superscript 𝑧 𝑛 Kummer-confluent-hypergeometric-M-as-1F1 𝛽 𝑛 1 𝛼 𝛽 2 𝑛 2 2 𝑧 {\displaystyle{\displaystyle\int_{-1}^{1}e^{-(x+1)z}P^{(\alpha,\beta)}_{n}% \left(x\right)(1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+% \beta+n+1}\Gamma\left(\alpha+n+1\right)\Gamma\left(\beta+n+1\right)}{\Gamma% \left(\alpha+\beta+2n+2\right)n!}z^{n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha% +\beta+2n+2};-2z\right)}}
\int_{-1}^{1}e^{-(x+1)z}\JacobipolyP{\alpha}{\beta}{n}@{x}(1-x)^{\alpha}(1+x)^{\beta}\diff{x} = \frac{(-1)^{n}2^{\alpha+\beta+n+1}\EulerGamma@{\alpha+n+1}\EulerGamma@{\beta+n+1}}{\EulerGamma@{\alpha+\beta+2n+2}n!}z^{n}\genhyperF{1}{1}@@{\beta+n+1}{\alpha+\beta+2n+2}{-2z}		 ℜ ⁡ ( α + n + 1 ) > 0 , ℜ ⁡ ( β + n + 1 ) > 0 , ℜ ⁡ ( α + β + 2 ⁢ n + 2 ) > 0 formulae-sequence 𝛼 𝑛 1 0 formulae-sequence 𝛽 𝑛 1 0 𝛼 𝛽 2 𝑛 2 0 {\displaystyle{\displaystyle\Re(\alpha+n+1)>0,\Re(\beta+n+1)>0,\Re(\alpha+% \beta+2n+2)>0}} 
int(exp(-(x + 1)*(x + y*I))*JacobiP(n, alpha, beta, x)*(1 - x)^(alpha)*(1 + x)^(beta), x = - 1..1) = ((- 1)^(n)* (2)^(alpha + beta + n + 1)* GAMMA(alpha + n + 1)*GAMMA(beta + n + 1))/(GAMMA(alpha + beta + 2*n + 2)*factorial(n))*(x + y*I)^(n)* hypergeom([beta + n + 1], [alpha + beta + 2*n + 2], - 2*(x + y*I))

Integrate[Exp[-(x + 1)*(x + y*I)]*JacobiP[n, \[Alpha], \[Beta], x]*(1 - x)^\[Alpha]*(1 + x)^\[Beta], {x, - 1, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* (2)^(\[Alpha]+ \[Beta]+ n + 1)* Gamma[\[Alpha]+ n + 1]*Gamma[\[Beta]+ n + 1],Gamma[\[Alpha]+ \[Beta]+ 2*n + 2]*(n)!]*(x + y*I)^(n)* HypergeometricPFQ[{\[Beta]+ n + 1}, {\[Alpha]+ \[Beta]+ 2*n + 2}, - 2*(x + y*I)]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
18.17.E34 ∫ 0 ∞ e - x ⁢ z ⁢ L n ( α ) ⁡ ( x ) ⁢ e - x ⁢ x α ⁢ d x = Γ ⁡ ( α + n + 1 ) ⁢ z n n ! ⁢ ( z + 1 ) α + n + 1 superscript subscript 0 superscript 𝑒 𝑥 𝑧 Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 𝑒 𝑥 superscript 𝑥 𝛼 𝑥 Euler-Gamma 𝛼 𝑛 1 superscript 𝑧 𝑛 𝑛 superscript 𝑧 1 𝛼 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-xz}L^{(\alpha)}_{n}\left(x% \right)e^{-x}x^{\alpha}\mathrm{d}x=\frac{\Gamma\left(\alpha+n+1\right)z^{n}}{n% !(z+1)^{\alpha+n+1}}}}
\int_{0}^{\infty}e^{-xz}\LaguerrepolyL[\alpha]{n}@{x}e^{-x}x^{\alpha}\diff{x} = \frac{\EulerGamma@{\alpha+n+1}z^{n}}{n!(z+1)^{\alpha+n+1}}		 ℜ ⁡ z > - 1 , ℜ ⁡ ( α + n + 1 ) > 0 formulae-sequence 𝑧 1 𝛼 𝑛 1 0 {\displaystyle{\displaystyle\Re z>-1,\Re(\alpha+n+1)>0}} 
int(exp(- x*(x + y*I))*LaguerreL(n, alpha, x)*exp(- x)*(x)^(alpha), x = 0..infinity) = (GAMMA(alpha + n + 1)*(x + y*I)^(n))/(factorial(n)*((x + y*I)+ 1)^(alpha + n + 1))

Integrate[Exp[- x*(x + y*I)]*LaguerreL[n, \[Alpha], x]*Exp[- x]*(x)^\[Alpha], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]+ n + 1]*(x + y*I)^(n),(n)!*((x + y*I)+ 1)^(\[Alpha]+ n + 1)]
Missing Macro Error Failure -
Failed [162 / 162]
Result: Plus[Complex[-0.07467065623203636, -0.1489394690482153], NIntegrate[Complex[-0.027140152128725715, 0.033616541935162864]
Test Values: {1.5, 0, DirectedInfinity[1]}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5]}


Result: Plus[Complex[-0.13823623490446432, -0.16092399439966643], NIntegrate[Complex[-0.006785038032181429, 0.008404135483790716]
Test Values: {1.5, 0, DirectedInfinity[1]}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5]}

... skip entries to safe data
18.17.E35 ∫ - ∞ ∞ e - x ⁢ z ⁢ H n ⁡ ( x ) ⁢ e - x 2 ⁢ d x = π 1 2 ⁢ ( - z ) n ⁢ e 1 4 ⁢ z 2 superscript subscript superscript 𝑒 𝑥 𝑧 Hermite-polynomial-H 𝑛 𝑥 superscript 𝑒 superscript 𝑥 2 𝑥 superscript 𝜋 1 2 superscript 𝑧 𝑛 superscript 𝑒 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-xz}H_{n}\left(x\right)e% ^{-x^{2}}\mathrm{d}x=\pi^{\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}}}}
\int_{-\infty}^{\infty}e^{-xz}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \pi^{\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}}		 

int(exp(- x*(x + y*I))*HermiteH(n, x)*exp(- (x)^(2)), x = - infinity..infinity) = (Pi)^((1)/(2))*(-(x + y*I))^(n)* exp((1)/(4)*(x + y*I)^(2))

Integrate[Exp[- x*(x + y*I)]*HermiteH[n, x]*Exp[- (x)^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == (Pi)^(Divide[1,2])*(-(x + y*I))^(n)* Exp[Divide[1,4]*(x + y*I)^(2)]
Failure Failure
Failed [54 / 54]
Result: -1.252480791-2.835663866*I
Test Values: {x = 3/2, y = -3/2, n = 1, z = 1+I}


Result: 5.718319609+3.439082150*I
Test Values: {x = 3/2, y = -3/2, n = 2, z = 1+I}

... skip entries to safe data
Failed [54 / 54]
Result: Plus[Complex[-1.25248079113256, -3.5452022239920282], NIntegrate[Complex[-0.020935135800726114, 0.025930837352181123]
Test Values: {1.5, DirectedInfinity[-1], DirectedInfinity[1]}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[z, Complex[1, 1]]}


Result: Plus[Complex[7.196524522686883, 3.4390821492892023], NIntegrate[Complex[-0.048848650201694266, 0.060505287155089287]
Test Values: {1.5, DirectedInfinity[-1], DirectedInfinity[1]}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[z, Complex[1, 1]]}

... skip entries to safe data
18.17.E36 ∫ - 1 1 ( 1 - x ) z - 1 ⁢ ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) ⁢ d x = 2 β + z ⁢ Γ ⁡ ( z ) ⁢ Γ ⁡ ( 1 + β + n ) ⁢ ( 1 + α - z ) n n ! ⁢ Γ ⁡ ( 1 + β + z + n ) superscript subscript 1 1 superscript 1 𝑥 𝑧 1 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑥 superscript 2 𝛽 𝑧 Euler-Gamma 𝑧 Euler-Gamma 1 𝛽 𝑛 Pochhammer 1 𝛼 𝑧 𝑛 𝑛 Euler-Gamma 1 𝛽 𝑧 𝑛 {\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,% \beta)}_{n}\left(x\right)\mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)% \Gamma\left(1+\beta+n\right){\left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+% \beta+z+n\right)}}}
\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}\diff{x} = \frac{2^{\beta+z}\EulerGamma@{z}\EulerGamma@{1+\beta+n}\Pochhammersym{1+\alpha-z}{n}}{n!\EulerGamma@{1+\beta+z+n}}		 ℜ ⁡ z > 0 , ℜ ⁡ ( 1 + β + n ) > 0 , ℜ ⁡ ( 1 + β + z + n ) > 0 formulae-sequence 𝑧 0 formulae-sequence 1 𝛽 𝑛 0 1 𝛽 𝑧 𝑛 0 {\displaystyle{\displaystyle\Re z>0,\Re(1+\beta+n)>0,\Re(1+\beta+z+n)>0}} 
int((1 - x)^((x + y*I)- 1)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x), x = - 1..1) = ((2)^(beta +(x + y*I))* GAMMA(x + y*I)*GAMMA(1 + beta + n)*pochhammer(1 + alpha -(x + y*I), n))/(factorial(n)*GAMMA(1 + beta +(x + y*I)+ n))
 
Integrate[(1 - x)^((x + y*I)- 1)*(1 + x)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], x], {x, - 1, 1}, GenerateConditions->None] == Divide[(2)^(\[Beta]+(x + y*I))* Gamma[x + y*I]*Gamma[1 + \[Beta]+ n]*Pochhammer[1 + \[Alpha]-(x + y*I), n],(n)!*Gamma[1 + \[Beta]+(x + y*I)+ n]]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
18.17.E37 ∫ 0 1 ( 1 - x 2 ) λ - 1 2 ⁢ C n ( λ ) ⁡ ( x ) ⁢ x z - 1 ⁢ d x = π ⁢  2 1 - 2 ⁢ λ - z ⁢ Γ ⁡ ( n + 2 ⁢ λ ) ⁢ Γ ⁡ ( z ) n ! ⁢ Γ ⁡ ( λ ) ⁢ Γ ⁡ ( 1 2 + 1 2 ⁢ n + λ + 1 2 ⁢ z ) ⁢ Γ ⁡ ( 1 2 + 1 2 ⁢ z - 1 2 ⁢ n ) superscript subscript 0 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑥 𝑧 1 𝑥 𝜋 superscript  2 1 2 𝜆 𝑧 Euler-Gamma 𝑛 2 𝜆 Euler-Gamma 𝑧 𝑛 Euler-Gamma 𝜆 Euler-Gamma 1 2 1 2 𝑛 𝜆 1 2 𝑧 Euler-Gamma 1 2 1 2 𝑧 1 2 𝑛 {\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{n}\left(x\right)x^{z-1}\mathrm{d}x=\frac{\pi\,2^{1-2\lambda-z}% \Gamma\left(n+2\lambda\right)\Gamma\left(z\right)}{n!\Gamma\left(\lambda\right% )\Gamma\left(\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z\right)\Gamma\left(% \frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n\right)}}}
\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{n}@{x}x^{z-1}\diff{x} = \frac{\pi\,2^{1-2\lambda-z}\EulerGamma@{n+2\lambda}\EulerGamma@{z}}{n!\EulerGamma@{\lambda}\EulerGamma@{\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z}\EulerGamma@{\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n}}		 ℜ ⁡ z > 0 , ℜ ⁡ ( n + 2 ⁢ λ ) > 0 , ℜ ⁡ ( λ ) > 0 , ℜ ⁡ ( 1 2 + 1 2 ⁢ n + λ + 1 2 ⁢ z ) > 0 , ℜ ⁡ ( 1 2 + 1 2 ⁢ z - 1 2 ⁢ n ) > 0 formulae-sequence 𝑧 0 formulae-sequence 𝑛 2 𝜆 0 formulae-sequence 𝜆 0 formulae-sequence 1 2 1 2 𝑛 𝜆 1 2 𝑧 0 1 2 1 2 𝑧 1 2 𝑛 0 {\displaystyle{\displaystyle\Re z>0,\Re(n+2\lambda)>0,\Re(\lambda)>0,\Re(\frac% {1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z)>0,\Re(\frac{1}{2}+\frac{1}{2}z-\frac% {1}{2}n)>0}} 
int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x)*(x)^((x + y*I)- 1), x = 0..1) = (Pi*(2)^(1 - 2*lambda -(x + y*I))* GAMMA(n + 2*lambda)*GAMMA(x + y*I))/(factorial(n)*GAMMA(lambda)*GAMMA((1)/(2)+(1)/(2)*n + lambda +(1)/(2)*(x + y*I))*GAMMA((1)/(2)+(1)/(2)*(x + y*I)-(1)/(2)*n))
 
Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x]*(x)^((x + y*I)- 1), {x, 0, 1}, GenerateConditions->None] == Divide[Pi*(2)^(1 - 2*\[Lambda]-(x + y*I))* Gamma[n + 2*\[Lambda]]*Gamma[x + y*I],(n)!*Gamma[\[Lambda]]*Gamma[Divide[1,2]+Divide[1,2]*n + \[Lambda]+Divide[1,2]*(x + y*I)]*Gamma[Divide[1,2]+Divide[1,2]*(x + y*I)-Divide[1,2]*n]]
 Failure Aborted Skipped - Because timed out
Failed [270 / 270]
Result: Plus[Complex[-0.2612561594092788, -0.2567131462958256], NIntegrate[Complex[0.3181035727957409, 0.7653241874975689]
Test Values: {1.5, 0, 1}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[-0.264978322932814, -0.1130252321165333], NIntegrate[Complex[0.21035635691874377, 2.1256411810993385]
Test Values: {1.5, 0, 1}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.17.E38 ∫ 0 1 P 2 ⁢ n ⁡ ( x ) ⁢ x z - 1 ⁢ d x = ( - 1 ) n ⁢ ( 1 2 - 1 2 ⁢ z ) n 2 ⁢ ( 1 2 ⁢ z ) n + 1 superscript subscript 0 1 Legendre-spherical-polynomial 2 𝑛 𝑥 superscript 𝑥 𝑧 1 𝑥 superscript 1 𝑛 Pochhammer 1 2 1 2 𝑧 𝑛 2 Pochhammer 1 2 𝑧 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\mathrm{d}x% =\frac{(-1)^{n}{\left(\frac{1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2% }z\right)_{n+1}}}}}
\int_{0}^{1}\LegendrepolyP{2n}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{\frac{1}{2}-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}z}{n+1}}		 ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} 
int(LegendreP(2*n, x)*(x)^((x + y*I)- 1), x = 0..1) = ((- 1)^(n)* pochhammer((1)/(2)-(1)/(2)*(x + y*I), n))/(2*pochhammer((1)/(2)*(x + y*I), n + 1))
 
Integrate[LegendreP[2*n, x]*(x)^((x + y*I)- 1), {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pochhammer[Divide[1,2]-Divide[1,2]*(x + y*I), n],2*Pochhammer[Divide[1,2]*(x + y*I), n + 1]]
 Failure Failure Skipped - Because timed out
Failed [54 / 54]
Result: Plus[Complex[-0.19540229885057472, 0.011494252873563225], NIntegrate[Complex[2.8897275468024644, -2.0119423961065603]
Test Values: {1.5, 0, 1}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5]}

Result: Plus[Complex[0.03978779840848807, 0.061007957559681705], NIntegrate[Complex[14.158094475230552, -9.85742429396774]
Test Values: {1.5, 0, 1}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]}

... skip entries to safe data
18.17.E39 ∫ 0 1 P 2 ⁢ n + 1 ⁡ ( x ) ⁢ x z - 1 ⁢ d x = ( - 1 ) n ⁢ ( 1 - 1 2 ⁢ z ) n 2 ⁢ ( 1 2 + 1 2 ⁢ z ) n + 1 superscript subscript 0 1 Legendre-spherical-polynomial 2 𝑛 1 𝑥 superscript 𝑥 𝑧 1 𝑥 superscript 1 𝑛 Pochhammer 1 1 2 𝑧 𝑛 2 Pochhammer 1 2 1 2 𝑧 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\mathrm{d% }x=\frac{(-1)^{n}{\left(1-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{% 1}{2}z\right)_{n+1}}}}}
\int_{0}^{1}\LegendrepolyP{2n+1}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{1-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}+\frac{1}{2}z}{n+1}}		 ℜ ⁡ z > - 1 𝑧 1 {\displaystyle{\displaystyle\Re z>-1}} 
int(LegendreP(2*n + 1, x)*(x)^((x + y*I)- 1), x = 0..1) = ((- 1)^(n)* pochhammer(1 -(1)/(2)*(x + y*I), n))/(2*pochhammer((1)/(2)+(1)/(2)*(x + y*I), n + 1))
 
Integrate[LegendreP[2*n + 1, x]*(x)^((x + y*I)- 1), {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pochhammer[1 -Divide[1,2]*(x + y*I), n],2*Pochhammer[Divide[1,2]+Divide[1,2]*(x + y*I), n + 1]]
 Failure Failure
Failed [54 / 54]
Result: .1141366199-.1434447856*I
Test Values: {x = 3/2, y = -3/2, n = 1}

Result: -.1797435469+.6231194668e-1*I
Test Values: {x = 3/2, y = -3/2, n = 2}

... skip entries to safe data
Failed [54 / 54]
Result: Plus[Complex[-0.058823529411764705, 0.0980392156862745], NIntegrate[Complex[6.21919624203139, -4.330049939446727]
Test Values: {1.5, 0, 1}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5]}

Result: Plus[Complex[0.04824851288830139, -0.012998457810090328], NIntegrate[Complex[33.25149808949738, -23.151005642155518]
Test Values: {1.5, 0, 1}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]}

... skip entries to safe data
18.17.E40 ∫ 0 ∞ e - a ⁢ x ⁢ L n ( α ) ⁡ ( b ⁢ x ) ⁢ x z - 1 ⁢ d x = Γ ⁡ ( z + n ) n ! ⁢ ( a - b ) n ⁢ a - n - z ⁢ F 1 2 ⁡ ( - n , 1 + α - z 1 - n - z ; a a - b ) superscript subscript 0 superscript 𝑒 𝑎 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑏 𝑥 superscript 𝑥 𝑧 1 𝑥 Euler-Gamma 𝑧 𝑛 𝑛 superscript 𝑎 𝑏 𝑛 superscript 𝑎 𝑛 𝑧 Gauss-hypergeometric-F-as-2F1 𝑛 1 𝛼 𝑧 1 𝑛 𝑧 𝑎 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}L^{(\alpha)}_{n}\left(bx% \right)x^{z-1}\mathrm{d}x=\frac{\Gamma\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{-n-% z}\*{{}_{2}F_{1}}\left({-n,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right)}}
\int_{0}^{\infty}e^{-ax}\LaguerrepolyL[\alpha]{n}@{bx}x^{z-1}\diff{x} = \frac{\EulerGamma@{z+n}}{n!}\*{(a-b)^{n}}a^{-n-z}\*\genhyperF{2}{1}@@{-n,1+\alpha-z}{1-n-z}{\frac{a}{a-b}}		 ℜ ⁡ a > 0 , ℜ ⁡ z > 0 , ℜ ⁡ ( z + n ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑧 0 𝑧 𝑛 0 {\displaystyle{\displaystyle\Re a>0,\Re z>0,\Re(z+n)>0}} 
int(exp(- a*x)*LaguerreL(n, alpha, b*x)*(x)^((x + y*I)- 1), x = 0..infinity) = (GAMMA((x + y*I)+ n))/(factorial(n))*(a - b)^(n)*(a)^(- n -(x + y*I))* hypergeom([- n , 1 + alpha -(x + y*I)], [1 - n -(x + y*I)], (a)/(a - b))
 
Integrate[Exp[- a*x]*LaguerreL[n, \[Alpha], b*x]*(x)^((x + y*I)- 1), {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[(x + y*I)+ n],(n)!]*(a - b)^(n)*(a)^(- n -(x + y*I))* HypergeometricPFQ[{- n , 1 + \[Alpha]-(x + y*I)}, {1 - n -(x + y*I)}, Divide[a,a - b]]
 Missing Macro Error Aborted - Skipped - Because timed out
18.17.E45 ( n + 1 2 ) ⁢ ( 1 + x ) 1 2 ⁢ ∫ - 1 x ( x - t ) - 1 2 ⁢ P n ⁡ ( t ) ⁢ d t = T n ⁡ ( x ) + T n + 1 ⁡ ( x ) 𝑛 1 2 superscript 1 𝑥 1 2 superscript subscript 1 𝑥 superscript 𝑥 𝑡 1 2 Legendre-spherical-polynomial 𝑛 𝑡 𝑡 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 {\displaystyle{\displaystyle(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x% -t)^{-\frac{1}{2}}P_{n}\left(t\right)\mathrm{d}t=T_{n}\left(x\right)+T_{n+1}% \left(x\right)}}
(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n+1}@{x}		 

(n +(1)/(2))*(1 + x)^((1)/(2))* int((x - t)^(-(1)/(2))* LegendreP(n, t), t = - 1..x) = ChebyshevT(n, x)+ ChebyshevT(n + 1, x)
 
(n +Divide[1,2])*(1 + x)^(Divide[1,2])* Integrate[(x - t)^(-Divide[1,2])* LegendreP[n, t], {t, - 1, x}, GenerateConditions->None] == ChebyshevT[n, x]+ ChebyshevT[n + 1, x]
 Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.17.E46 ( n + 1 2 ) ⁢ ( 1 - x ) 1 2 ⁢ ∫ x 1 ( t - x ) - 1 2 ⁢ P n ⁡ ( t ) ⁢ d t = T n ⁡ ( x ) - T n + 1 ⁡ ( x ) 𝑛 1 2 superscript 1 𝑥 1 2 superscript subscript 𝑥 1 superscript 𝑡 𝑥 1 2 Legendre-spherical-polynomial 𝑛 𝑡 𝑡 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 {\displaystyle{\displaystyle(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-% x)^{-\frac{1}{2}}P_{n}\left(t\right)\mathrm{d}t=T_{n}\left(x\right)-T_{n+1}% \left(x\right)}}
(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-x)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}-\ChebyshevpolyT{n+1}@{x}		 

(n +(1)/(2))*(1 - x)^((1)/(2))* int((t - x)^(-(1)/(2))* LegendreP(n, t), t = x..1) = ChebyshevT(n, x)- ChebyshevT(n + 1, x)
 
(n +Divide[1,2])*(1 - x)^(Divide[1,2])* Integrate[(t - x)^(-Divide[1,2])* LegendreP[n, t], {t, x, 1}, GenerateConditions->None] == ChebyshevT[n, x]- ChebyshevT[n + 1, x]
 Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.17.E47 ∫ 0 x t α ⁢ L m ( α ) ⁡ ( t ) L m ( α ) ⁡ ( 0 ) ⁢ ( x - t ) β ⁢ L n ( β ) ⁡ ( x - t ) L n ( β ) ⁡ ( 0 ) ⁢ d t = Γ ⁡ ( α + 1 ) ⁢ Γ ⁡ ( β + 1 ) Γ ⁡ ( α + β + 2 ) ⁢ x α + β + 1 ⁢ L m + n ( α + β + 1 ) ⁡ ( x ) L m + n ( α + β + 1 ) ⁡ ( 0 ) superscript subscript 0 𝑥 superscript 𝑡 𝛼 Laguerre-polynomial-L 𝛼 𝑚 𝑡 Laguerre-polynomial-L 𝛼 𝑚 0 superscript 𝑥 𝑡 𝛽 Laguerre-polynomial-L 𝛽 𝑛 𝑥 𝑡 Laguerre-polynomial-L 𝛽 𝑛 0 𝑡 Euler-Gamma 𝛼 1 Euler-Gamma 𝛽 1 Euler-Gamma 𝛼 𝛽 2 superscript 𝑥 𝛼 𝛽 1 Laguerre-polynomial-L 𝛼 𝛽 1 𝑚 𝑛 𝑥 Laguerre-polynomial-L 𝛼 𝛽 1 𝑚 𝑛 0 {\displaystyle{\displaystyle\int_{0}^{x}t^{\alpha}\frac{L^{(\alpha)}_{m}\left(% t\right)}{L^{(\alpha)}_{m}\left(0\right)}(x-t)^{\beta}\frac{L^{(\beta)}_{n}% \left(x-t\right)}{L^{(\beta)}_{n}\left(0\right)}\mathrm{d}t=\frac{\Gamma\left(% \alpha+1\right)\Gamma\left(\beta+1\right)}{\Gamma\left(\alpha+\beta+2\right)}x% ^{\alpha+\beta+1}\frac{L^{(\alpha+\beta+1)}_{m+n}\left(x\right)}{L^{(\alpha+% \beta+1)}_{m+n}\left(0\right)}}}
\int_{0}^{x}t^{\alpha}\frac{\LaguerrepolyL[\alpha]{m}@{t}}{\LaguerrepolyL[\alpha]{m}@{0}}(x-t)^{\beta}\frac{\LaguerrepolyL[\beta]{n}@{x-t}}{\LaguerrepolyL[\beta]{n}@{0}}\diff{t} = \frac{\EulerGamma@{\alpha+1}\EulerGamma@{\beta+1}}{\EulerGamma@{\alpha+\beta+2}}x^{\alpha+\beta+1}\frac{\LaguerrepolyL[\alpha+\beta+1]{m+n}@{x}}{\LaguerrepolyL[\alpha+\beta+1]{m+n}@{0}}		 ℜ ⁡ ( α + 1 ) > 0 , ℜ ⁡ ( β + 1 ) > 0 , ℜ ⁡ ( α + β + 2 ) > 0 formulae-sequence 𝛼 1 0 formulae-sequence 𝛽 1 0 𝛼 𝛽 2 0 {\displaystyle{\displaystyle\Re(\alpha+1)>0,\Re(\beta+1)>0,\Re(\alpha+\beta+2)% >0}} 
int((t)^(alpha)*(LaguerreL(m, alpha, t))/(LaguerreL(m, alpha, 0))*(x - t)^(beta)*(LaguerreL(n, beta, x - t))/(LaguerreL(n, beta, 0)), t = 0..x) = (GAMMA(alpha + 1)*GAMMA(beta + 1))/(GAMMA(alpha + beta + 2))*(x)^(alpha + beta + 1)*(LaguerreL(m + n, alpha + beta + 1, x))/(LaguerreL(m + n, alpha + beta + 1, 0))
 
Integrate[(t)^\[Alpha]*Divide[LaguerreL[m, \[Alpha], t],LaguerreL[m, \[Alpha], 0]]*(x - t)^\[Beta]*Divide[LaguerreL[n, \[Beta], x - t],LaguerreL[n, \[Beta], 0]], {t, 0, x}, GenerateConditions->None] == Divide[Gamma[\[Alpha]+ 1]*Gamma[\[Beta]+ 1],Gamma[\[Alpha]+ \[Beta]+ 2]]*(x)^(\[Alpha]+ \[Beta]+ 1)*Divide[LaguerreL[m + n, \[Alpha]+ \[Beta]+ 1, x],LaguerreL[m + n, \[Alpha]+ \[Beta]+ 1, 0]]
 Missing Macro Error Failure - Manual Skip!
18.17.E48 ∫ - ∞ ∞ H m ⁡ ( y ) ⁢ e - y 2 ⁢ H n ⁡ ( x - y ) ⁢ e - ( x - y ) 2 ⁢ d y = π 1 2 ⁢ 2 - 1 2 ⁢ ( m + n + 1 ) ⁢ H m + n ⁡ ( 2 - 1 2 ⁢ x ) ⁢ e - 1 2 ⁢ x 2 superscript subscript Hermite-polynomial-H 𝑚 𝑦 superscript 𝑒 superscript 𝑦 2 Hermite-polynomial-H 𝑛 𝑥 𝑦 superscript 𝑒 superscript 𝑥 𝑦 2 𝑦 superscript 𝜋 1 2 superscript 2 1 2 𝑚 𝑛 1 Hermite-polynomial-H 𝑚 𝑛 superscript 2 1 2 𝑥 superscript 𝑒 1 2 superscript 𝑥 2 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}H_{m}\left(y\right)e^{-y^{2% }}H_{n}\left(x-y\right)e^{-(x-y)^{2}}\mathrm{d}y=\pi^{\frac{1}{2}}2^{-\frac{1}% {2}(m+n+1)}H_{m+n}\left(2^{-\frac{1}{2}}x\right)e^{-\frac{1}{2}x^{2}}}}
\int_{-\infty}^{\infty}\HermitepolyH{m}@{y}e^{-y^{2}}\HermitepolyH{n}@{x-y}e^{-(x-y)^{2}}\diff{y} = \pi^{\frac{1}{2}}2^{-\frac{1}{2}(m+n+1)}\HermitepolyH{m+n}@{2^{-\frac{1}{2}}x}e^{-\frac{1}{2}x^{2}}		 

int(HermiteH(m, y)*exp(- (y)^(2))*HermiteH(n, x - y)*exp(-(x - y)^(2)), y = - infinity..infinity) = (Pi)^((1)/(2))* (2)^(-(1)/(2)*(m + n + 1))* HermiteH(m + n, (2)^(-(1)/(2))* x)*exp(-(1)/(2)*(x)^(2))
 
Integrate[HermiteH[m, y]*Exp[- (y)^(2)]*HermiteH[n, x - y]*Exp[-(x - y)^(2)], {y, - Infinity, Infinity}, GenerateConditions->None] == (Pi)^(Divide[1,2])* (2)^(-Divide[1,2]*(m + n + 1))* HermiteH[m + n, (2)^(-Divide[1,2])* x]*Exp[-Divide[1,2]*(x)^(2)]
 Failure Aborted Successful [Tested: 27] Skipped - Because timed out
18.17.E49 ∫ - ∞ ∞ H ℓ ⁡ ( x ) ⁢ H m ⁡ ( x ) ⁢ H n ⁡ ( x ) ⁢ e - x 2 ⁢ d x = 2 1 2 ⁢ ( ℓ + m + n ) ⁢ ℓ ! ⁢ m ! ⁢ n ! ⁢ π ( 1 2 ⁢ ℓ + 1 2 ⁢ m - 1 2 ⁢ n ) ! ⁢ ( 1 2 ⁢ m + 1 2 ⁢ n - 1 2 ⁢ ℓ ) ! ⁢ ( 1 2 ⁢ n + 1 2 ⁢ ℓ - 1 2 ⁢ m ) ! superscript subscript Hermite-polynomial-H ℓ 𝑥 Hermite-polynomial-H 𝑚 𝑥 Hermite-polynomial-H 𝑛 𝑥 superscript 𝑒 superscript 𝑥 2 𝑥 superscript 2 1 2 ℓ 𝑚 𝑛 ℓ 𝑚 𝑛 𝜋 1 2 ℓ 1 2 𝑚 1 2 𝑛 1 2 𝑚 1 2 𝑛 1 2 ℓ 1 2 𝑛 1 2 ℓ 1 2 𝑚 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}H_{\ell}\left(x\right)H_{m}% \left(x\right)H_{n}\left(x\right)e^{-x^{2}}\mathrm{d}x=\frac{2^{\frac{1}{2}(% \ell+m+n)}\ell\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-% \tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(% \tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!}}}
\int_{-\infty}^{\infty}\HermitepolyH{\ell}@{x}\HermitepolyH{m}@{x}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \frac{2^{\frac{1}{2}(\ell+m+n)}\ell\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!}		 

int(HermiteH(ell, x)*HermiteH(m, x)*HermiteH(n, x)*exp(- (x)^(2)), x = - infinity..infinity) = ((2)^((1)/(2)*(ell + m + n))* factorial(ell)*factorial(m)*factorial(n)*sqrt(Pi))/(factorial((1)/(2)*ell +(1)/(2)*m -(1)/(2)*n)*factorial((1)/(2)*m +(1)/(2)*n -(1)/(2)*ell)*factorial((1)/(2)*n +(1)/(2)*ell -(1)/(2)*m))
 
Integrate[HermiteH[\[ScriptL], x]*HermiteH[m, x]*HermiteH[n, x]*Exp[- (x)^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == Divide[(2)^(Divide[1,2]*(\[ScriptL]+ m + n))* (\[ScriptL])!*(m)!*(n)!*Sqrt[Pi],(Divide[1,2]*\[ScriptL]+Divide[1,2]*m -Divide[1,2]*n)!*(Divide[1,2]*m +Divide[1,2]*n -Divide[1,2]*\[ScriptL])!*(Divide[1,2]*n +Divide[1,2]*\[ScriptL]-Divide[1,2]*m)!]
 Failure Aborted Error Skipped - Because timed out
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