Orthogonal Polynomials - 18.9 Recurrence Relations and Derivatives

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18.9#Ex1 A n = ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) 2 ⁢ ( n + 1 ) ⁢ ( n + α + β + 1 ) subscript 𝐴 𝑛 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 2 𝑛 1 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle A_{n}=\dfrac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+% 2)}{2(n+1)(n+\alpha+\beta+1)}}}
A_{n} = \dfrac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{2(n+1)(n+\alpha+\beta+1)}		 

A[n] = ((2*n + alpha + beta + 1)*(2*n + alpha + beta + 2))/(2*(n + 1)*(n + alpha + beta + 1))
 
Subscript[A, n] == Divide[(2*n + \[Alpha]+ \[Beta]+ 1)*(2*n + \[Alpha]+ \[Beta]+ 2),2*(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)]
 Skipped - no semantic math Skipped - no semantic math - -
18.9#Ex2 B n = ( α 2 - β 2 ) ⁢ ( 2 ⁢ n + α + β + 1 ) 2 ⁢ ( n + 1 ) ⁢ ( n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β ) subscript 𝐵 𝑛 superscript 𝛼 2 superscript 𝛽 2 2 𝑛 𝛼 𝛽 1 2 𝑛 1 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 {\displaystyle{\displaystyle B_{n}=\dfrac{(\alpha^{2}-\beta^{2})(2n+\alpha+% \beta+1)}{2(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}}}
B_{n} = \dfrac{(\alpha^{2}-\beta^{2})(2n+\alpha+\beta+1)}{2(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}		 

B[n] = (((alpha)^(2)- (beta)^(2))*(2*n + alpha + beta + 1))/(2*(n + 1)*(n + alpha + beta + 1)*(2*n + alpha + beta))
 
Subscript[B, n] == Divide[(\[Alpha]^(2)- \[Beta]^(2))*(2*n + \[Alpha]+ \[Beta]+ 1),2*(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)*(2*n + \[Alpha]+ \[Beta])]
 Skipped - no semantic math Skipped - no semantic math - -
18.9#Ex3 C n = ( n + α ) ⁢ ( n + β ) ⁢ ( 2 ⁢ n + α + β + 2 ) ( n + 1 ) ⁢ ( n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β ) subscript 𝐶 𝑛 𝑛 𝛼 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 1 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 {\displaystyle{\displaystyle C_{n}=\dfrac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+% 2)}{(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}}}
C_{n} = \dfrac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}		 

C[n] = ((n + alpha)*(n + beta)*(2*n + alpha + beta + 2))/((n + 1)*(n + alpha + beta + 1)*(2*n + alpha + beta))
 
Subscript[C, n] == Divide[(n + \[Alpha])*(n + \[Beta])*(2*n + \[Alpha]+ \[Beta]+ 2),(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)*(2*n + \[Alpha]+ \[Beta])]
 Skipped - no semantic math Skipped - no semantic math - -
18.9.E3 P n ( α , β - 1 ) ⁡ ( x ) - P n ( α - 1 , β ) ⁡ ( x ) = P n - 1 ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 1 𝑛 𝑥 Jacobi-polynomial-P 𝛼 1 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 𝑥 {\displaystyle{\displaystyle P^{(\alpha,\beta-1)}_{n}\left(x\right)-P^{(\alpha% -1,\beta)}_{n}\left(x\right)=P^{(\alpha,\beta)}_{n-1}\left(x\right)}}
\JacobipolyP{\alpha}{\beta-1}{n}@{x}-\JacobipolyP{\alpha-1}{\beta}{n}@{x} = \JacobipolyP{\alpha}{\beta}{n-1}@{x}		 

JacobiP(n, alpha, beta - 1, x)- JacobiP(n, alpha - 1, beta, x) = JacobiP(n - 1, alpha, beta, x)

JacobiP[n, \[Alpha], \[Beta]- 1, x]- JacobiP[n, \[Alpha]- 1, \[Beta], x] == JacobiP[n - 1, \[Alpha], \[Beta], x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E4 ( 1 - x ) ⁢ P n ( α + 1 , β ) ⁡ ( x ) + ( 1 + x ) ⁢ P n ( α , β + 1 ) ⁡ ( x ) = 2 ⁢ P n ( α , β ) ⁡ ( x ) 1 𝑥 Jacobi-polynomial-P 𝛼 1 𝛽 𝑛 𝑥 1 𝑥 Jacobi-polynomial-P 𝛼 𝛽 1 𝑛 𝑥 2 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P% ^{(\alpha,\beta+1)}_{n}\left(x\right)=2P^{(\alpha,\beta)}_{n}\left(x\right)}}
(1-x)\JacobipolyP{\alpha+1}{\beta}{n}@{x}+(1+x)\JacobipolyP{\alpha}{\beta+1}{n}@{x} = 2\JacobipolyP{\alpha}{\beta}{n}@{x}		 

(1 - x)*JacobiP(n, alpha + 1, beta, x)+(1 + x)*JacobiP(n, alpha, beta + 1, x) = 2*JacobiP(n, alpha, beta, x)

(1 - x)*JacobiP[n, \[Alpha]+ 1, \[Beta], x]+(1 + x)*JacobiP[n, \[Alpha], \[Beta]+ 1, x] == 2*JacobiP[n, \[Alpha], \[Beta], x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E5 ( 2 ⁢ n + α + β + 1 ) ⁢ P n ( α , β ) ⁡ ( x ) = ( n + α + β + 1 ) ⁢ P n ( α , β + 1 ) ⁡ ( x ) + ( n + α ) ⁢ P n - 1 ( α , β + 1 ) ⁡ ( x ) 2 𝑛 𝛼 𝛽 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛼 𝛽 1 Jacobi-polynomial-P 𝛼 𝛽 1 𝑛 𝑥 𝑛 𝛼 Jacobi-polynomial-P 𝛼 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle(2n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n}\left(x% \right)=(n+\alpha+\beta+1)P^{(\alpha,\beta+1)}_{n}\left(x\right)+(n+\alpha)P^{% (\alpha,\beta+1)}_{n-1}\left(x\right)}}
(2n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta}{n}@{x} = (n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta+1}{n}@{x}+(n+\alpha)\JacobipolyP{\alpha}{\beta+1}{n-1}@{x}		 

(2*n + alpha + beta + 1)*JacobiP(n, alpha, beta, x) = (n + alpha + beta + 1)*JacobiP(n, alpha, beta + 1, x)+(n + alpha)*JacobiP(n - 1, alpha, beta + 1, x)

(2*n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n, \[Alpha], \[Beta], x] == (n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n, \[Alpha], \[Beta]+ 1, x]+(n + \[Alpha])*JacobiP[n - 1, \[Alpha], \[Beta]+ 1, x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E6 ( n + 1 2 ⁢ α + 1 2 ⁢ β + 1 ) ⁢ ( 1 + x ) ⁢ P n ( α , β + 1 ) ⁡ ( x ) = ( n + 1 ) ⁢ P n + 1 ( α , β ) ⁡ ( x ) + ( n + β + 1 ) ⁢ P n ( α , β ) ⁡ ( x ) 𝑛 1 2 𝛼 1 2 𝛽 1 1 𝑥 Jacobi-polynomial-P 𝛼 𝛽 1 𝑛 𝑥 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 𝑥 𝑛 𝛽 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)P^{% (\alpha,\beta+1)}_{n}\left(x\right)=(n+1)P^{(\alpha,\beta)}_{n+1}\left(x\right% )+(n+\beta+1)P^{(\alpha,\beta)}_{n}\left(x\right)}}
(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)\JacobipolyP{\alpha}{\beta+1}{n}@{x} = (n+1)\JacobipolyP{\alpha}{\beta}{n+1}@{x}+(n+\beta+1)\JacobipolyP{\alpha}{\beta}{n}@{x}		 

(n +(1)/(2)*alpha +(1)/(2)*beta + 1)*(1 + x)*JacobiP(n, alpha, beta + 1, x) = (n + 1)*JacobiP(n + 1, alpha, beta, x)+(n + beta + 1)*JacobiP(n, alpha, beta, x)

(n +Divide[1,2]*\[Alpha]+Divide[1,2]*\[Beta]+ 1)*(1 + x)*JacobiP[n, \[Alpha], \[Beta]+ 1, x] == (n + 1)*JacobiP[n + 1, \[Alpha], \[Beta], x]+(n + \[Beta]+ 1)*JacobiP[n, \[Alpha], \[Beta], x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E7 ( n + λ ) ⁢ C n ( λ ) ⁡ ( x ) = λ ⁢ ( C n ( λ + 1 ) ⁡ ( x ) - C n - 2 ( λ + 1 ) ⁡ ( x ) ) 𝑛 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 2 𝑥 {\displaystyle{\displaystyle(n+\lambda)C^{(\lambda)}_{n}\left(x\right)=\lambda% \left(C^{(\lambda+1)}_{n}\left(x\right)-C^{(\lambda+1)}_{n-2}\left(x\right)% \right)}}
(n+\lambda)\ultrasphpoly{\lambda}{n}@{x} = \lambda\left(\ultrasphpoly{\lambda+1}{n}@{x}-\ultrasphpoly{\lambda+1}{n-2}@{x}\right)		 

(n + lambda)*GegenbauerC(n, lambda, x) = lambda*(GegenbauerC(n, lambda + 1, x)- GegenbauerC(n - 2, lambda + 1, x))

(n + \[Lambda])*GegenbauerC[n, \[Lambda], x] == \[Lambda]*(GegenbauerC[n, \[Lambda]+ 1, x]- GegenbauerC[n - 2, \[Lambda]+ 1, x])
Successful Successful -
Failed [6 / 90]
Result: 0.9374999999999998
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[λ, -1.5]}


Result: -0.5
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[λ, -0.5]}

... skip entries to safe data
18.9.E8 4 ⁢ λ ⁢ ( n + λ + 1 ) ⁢ ( 1 - x 2 ) ⁢ C n ( λ + 1 ) ⁡ ( x ) = - ( n + 1 ) ⁢ ( n + 2 ) ⁢ C n + 2 ( λ ) ⁡ ( x ) + ( n + 2 ⁢ λ ) ⁢ ( n + 2 ⁢ λ + 1 ) ⁢ C n ( λ ) ⁡ ( x ) 4 𝜆 𝑛 𝜆 1 1 superscript 𝑥 2 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 𝑥 𝑛 1 𝑛 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 2 𝑥 𝑛 2 𝜆 𝑛 2 𝜆 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle 4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}% \left(x\right)=-(n+1)(n+2)C^{(\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2% \lambda+1)C^{(\lambda)}_{n}\left(x\right)}}
4\lambda(n+\lambda+1)(1-x^{2})\ultrasphpoly{\lambda+1}{n}@{x} = -(n+1)(n+2)\ultrasphpoly{\lambda}{n+2}@{x}+(n+2\lambda)(n+2\lambda+1)\ultrasphpoly{\lambda}{n}@{x}		 

4*lambda*(n + lambda + 1)*(1 - (x)^(2))*GegenbauerC(n, lambda + 1, x) = -(n + 1)*(n + 2)*GegenbauerC(n + 2, lambda, x)+(n + 2*lambda)*(n + 2*lambda + 1)*GegenbauerC(n, lambda, x)

4*\[Lambda]*(n + \[Lambda]+ 1)*(1 - (x)^(2))*GegenbauerC[n, \[Lambda]+ 1, x] == -(n + 1)*(n + 2)*GegenbauerC[n + 2, \[Lambda], x]+(n + 2*\[Lambda])*(n + 2*\[Lambda]+ 1)*GegenbauerC[n, \[Lambda], x]
Successful Successful - Successful [Tested: 90]
18.9.E9 T n ⁡ ( x ) = 1 2 ⁢ ( U n ⁡ ( x ) - U n - 2 ⁡ ( x ) ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 1 2 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 2 𝑥 {\displaystyle{\displaystyle T_{n}\left(x\right)=\tfrac{1}{2}\left(U_{n}\left(% x\right)-U_{n-2}\left(x\right)\right)}}
\ChebyshevpolyT{n}@{x} = \tfrac{1}{2}\left(\ChebyshevpolyU{n}@{x}-\ChebyshevpolyU{n-2}@{x}\right)		 

ChebyshevT(n, x) = (1)/(2)*(ChebyshevU(n, x)- ChebyshevU(n - 2, x))

ChebyshevT[n, x] == Divide[1,2]*(ChebyshevU[n, x]- ChebyshevU[n - 2, x])
Successful Failure - Successful [Tested: 9]
18.9.E10 ( 1 - x 2 ) ⁢ U n ⁡ ( x ) = - 1 2 ⁢ ( T n + 2 ⁡ ( x ) - T n ⁡ ( x ) ) 1 superscript 𝑥 2 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 1 2 Chebyshev-polynomial-first-kind-T 𝑛 2 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle(1-x^{2})U_{n}\left(x\right)=-\tfrac{1}{2}\left(T_% {n+2}\left(x\right)-T_{n}\left(x\right)\right)}}
(1-x^{2})\ChebyshevpolyU{n}@{x} = -\tfrac{1}{2}\left(\ChebyshevpolyT{n+2}@{x}-\ChebyshevpolyT{n}@{x}\right)		 

(1 - (x)^(2))*ChebyshevU(n, x) = -(1)/(2)*(ChebyshevT(n + 2, x)- ChebyshevT(n, x))

(1 - (x)^(2))*ChebyshevU[n, x] == -Divide[1,2]*(ChebyshevT[n + 2, x]- ChebyshevT[n, x])
Successful Failure - Successful [Tested: 9]
18.9.E13 L n ( α ) ⁡ ( x ) = L n ( α + 1 ) ⁡ ( x ) - L n - 1 ( α + 1 ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 Laguerre-polynomial-L 𝛼 1 𝑛 𝑥 Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑥 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=L^{(\alpha+1)}_{n}% \left(x\right)-L^{(\alpha+1)}_{n-1}\left(x\right)}}
\LaguerrepolyL[\alpha]{n}@{x} = \LaguerrepolyL[\alpha+1]{n}@{x}-\LaguerrepolyL[\alpha+1]{n-1}@{x}		 

LaguerreL(n, alpha, x) = LaguerreL(n, alpha + 1, x)- LaguerreL(n - 1, alpha + 1, x)

LaguerreL[n, \[Alpha], x] == LaguerreL[n, \[Alpha]+ 1, x]- LaguerreL[n - 1, \[Alpha]+ 1, x]
Missing Macro Error Successful - Successful [Tested: 27]
18.9.E14 x ⁢ L n ( α + 1 ) ⁡ ( x ) = - ( n + 1 ) ⁢ L n + 1 ( α ) ⁡ ( x ) + ( n + α + 1 ) ⁢ L n ( α ) ⁡ ( x ) 𝑥 Laguerre-polynomial-L 𝛼 1 𝑛 𝑥 𝑛 1 Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 𝑛 𝛼 1 Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle xL^{(\alpha+1)}_{n}\left(x\right)=-(n+1)L^{(% \alpha)}_{n+1}\left(x\right)+(n+\alpha+1)L^{(\alpha)}_{n}\left(x\right)}}
x\LaguerrepolyL[\alpha+1]{n}@{x} = -(n+1)\LaguerrepolyL[\alpha]{n+1}@{x}+(n+\alpha+1)\LaguerrepolyL[\alpha]{n}@{x}		 

x*LaguerreL(n, alpha + 1, x) = -(n + 1)*LaguerreL(n + 1, alpha, x)+(n + alpha + 1)*LaguerreL(n, alpha, x)

x*LaguerreL[n, \[Alpha]+ 1, x] == -(n + 1)*LaguerreL[n + 1, \[Alpha], x]+(n + \[Alpha]+ 1)*LaguerreL[n, \[Alpha], x]
Missing Macro Error Successful - Successful [Tested: 27]
18.9.E15 d d x ⁡ P n ( α , β ) ⁡ ( x ) = 1 2 ⁢ ( n + α + β + 1 ) ⁢ P n - 1 ( α + 1 , β + 1 ) ⁡ ( x ) derivative 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 1 2 𝑛 𝛼 𝛽 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{% n}\left(x\right)=\tfrac{1}{2}(n+\alpha+\beta+1)P^{(\alpha+1,\beta+1)}_{n-1}% \left(x\right)}}
\deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = \tfrac{1}{2}(n+\alpha+\beta+1)\JacobipolyP{\alpha+1}{\beta+1}{n-1}@{x}		 

diff(JacobiP(n, alpha, beta, x), x) = (1)/(2)*(n + alpha + beta + 1)*JacobiP(n - 1, alpha + 1, beta + 1, x)

D[JacobiP[n, \[Alpha], \[Beta], x], x] == Divide[1,2]*(n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E16 d d x ⁡ ( ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) ) = - 2 ⁢ ( n + 1 ) ⁢ ( 1 - x ) α - 1 ⁢ ( 1 + x ) β - 1 ⁢ P n + 1 ( α - 1 , β - 1 ) ⁡ ( x ) derivative 𝑥 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 2 𝑛 1 superscript 1 𝑥 𝛼 1 superscript 1 𝑥 𝛽 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x)^{\alpha}% (1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)\right)=-2(n+1)(1-x)^{\alpha-% 1}(1+x)^{\beta-1}P^{(\alpha-1,\beta-1)}_{n+1}\left(x\right)}}
\deriv{}{x}\left((1-x)^{\alpha}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}\right) = -2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}\JacobipolyP{\alpha-1}{\beta-1}{n+1}@{x}		 

diff((1 - x)^(alpha)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x), x) = - 2*(n + 1)*(1 - x)^(alpha - 1)*(1 + x)^(beta - 1)* JacobiP(n + 1, alpha - 1, beta - 1, x)

D[(1 - x)^\[Alpha]*(1 + x)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], x], x] == - 2*(n + 1)*(1 - x)^(\[Alpha]- 1)*(1 + x)^(\[Beta]- 1)* JacobiP[n + 1, \[Alpha]- 1, \[Beta]- 1, x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E17 ( 2 ⁢ n + α + β ) ⁢ ( 1 - x 2 ) ⁢ d d x ⁡ P n ( α , β ) ⁡ ( x ) = n ⁢ ( α - β - ( 2 ⁢ n + α + β ) ⁢ x ) ⁢ P n ( α , β ) ⁡ ( x ) + 2 ⁢ ( n + α ) ⁢ ( n + β ) ⁢ P n - 1 ( α , β ) ⁡ ( x ) 2 𝑛 𝛼 𝛽 1 superscript 𝑥 2 derivative 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 2 𝑛 𝛼 𝑛 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 𝑥 {\displaystyle{\displaystyle(2n+\alpha+\beta)(1-x^{2})\frac{\mathrm{d}}{% \mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=n\left(\alpha-\beta-(2n+% \alpha+\beta)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)+2(n+\alpha)(n+\beta)% P^{(\alpha,\beta)}_{n-1}\left(x\right)}}
(2n+\alpha+\beta)(1-x^{2})\deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = n\left(\alpha-\beta-(2n+\alpha+\beta)x\right)\JacobipolyP{\alpha}{\beta}{n}@{x}+2(n+\alpha)(n+\beta)\JacobipolyP{\alpha}{\beta}{n-1}@{x}		 

(2*n + alpha + beta)*(1 - (x)^(2))*diff(JacobiP(n, alpha, beta, x), x) = n*(alpha - beta -(2*n + alpha + beta)*x)*JacobiP(n, alpha, beta, x)+ 2*(n + alpha)*(n + beta)*JacobiP(n - 1, alpha, beta, x)

(2*n + \[Alpha]+ \[Beta])*(1 - (x)^(2))*D[JacobiP[n, \[Alpha], \[Beta], x], x] == n*(\[Alpha]- \[Beta]-(2*n + \[Alpha]+ \[Beta])*x)*JacobiP[n, \[Alpha], \[Beta], x]+ 2*(n + \[Alpha])*(n + \[Beta])*JacobiP[n - 1, \[Alpha], \[Beta], x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E18 ( 2 ⁢ n + α + β + 2 ) ⁢ ( 1 - x 2 ) ⁢ d d x ⁡ P n ( α , β ) ⁡ ( x ) = ( n + α + β + 1 ) ⁢ ( α - β + ( 2 ⁢ n + α + β + 2 ) ⁢ x ) ⁢ P n ( α , β ) ⁡ ( x ) - 2 ⁢ ( n + 1 ) ⁢ ( n + α + β + 1 ) ⁢ P n + 1 ( α , β ) ⁡ ( x ) 2 𝑛 𝛼 𝛽 2 1 superscript 𝑥 2 derivative 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛼 𝛽 1 𝛼 𝛽 2 𝑛 𝛼 𝛽 2 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 2 𝑛 1 𝑛 𝛼 𝛽 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 𝑥 {\displaystyle{\displaystyle(2n+\alpha+\beta+2)(1-x^{2})\frac{\mathrm{d}}{% \mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\alpha+\beta+1)\left(% \alpha-\beta+(2n+\alpha+\beta+2)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)-2% (n+1)(n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n+1}\left(x\right)}}
(2n+\alpha+\beta+2)(1-x^{2})\deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = (n+\alpha+\beta+1)\left(\alpha-\beta+(2n+\alpha+\beta+2)x\right)\JacobipolyP{\alpha}{\beta}{n}@{x}-2(n+1)(n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta}{n+1}@{x}		 

(2*n + alpha + beta + 2)*(1 - (x)^(2))*diff(JacobiP(n, alpha, beta, x), x) = (n + alpha + beta + 1)*(alpha - beta +(2*n + alpha + beta + 2)*x)*JacobiP(n, alpha, beta, x)- 2*(n + 1)*(n + alpha + beta + 1)*JacobiP(n + 1, alpha, beta, x)

(2*n + \[Alpha]+ \[Beta]+ 2)*(1 - (x)^(2))*D[JacobiP[n, \[Alpha], \[Beta], x], x] == (n + \[Alpha]+ \[Beta]+ 1)*(\[Alpha]- \[Beta]+(2*n + \[Alpha]+ \[Beta]+ 2)*x)*JacobiP[n, \[Alpha], \[Beta], x]- 2*(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n + 1, \[Alpha], \[Beta], x]
Failure Successful Successful [Tested: 81] Successful [Tested: 81]
18.9.E19 d d x ⁡ C n ( λ ) ⁡ ( x ) = 2 ⁢ λ ⁢ C n - 1 ( λ + 1 ) ⁡ ( x ) derivative 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 2 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}% \left(x\right)=2\lambda C^{(\lambda+1)}_{n-1}\left(x\right)}}
\deriv{}{x}\ultrasphpoly{\lambda}{n}@{x} = 2\lambda\ultrasphpoly{\lambda+1}{n-1}@{x}		 

diff(GegenbauerC(n, lambda, x), x) = 2*lambda*GegenbauerC(n - 1, lambda + 1, x)

D[GegenbauerC[n, \[Lambda], x], x] == 2*\[Lambda]*GegenbauerC[n - 1, \[Lambda]+ 1, x]
Successful Successful - Successful [Tested: 90]
18.9.E20 d d x ⁡ ( ( 1 - x 2 ) λ - 1 2 ⁢ C n ( λ ) ⁡ ( x ) ) = - ( n + 1 ) ⁢ ( n + 2 ⁢ λ - 1 ) 2 ⁢ ( λ - 1 ) ⁢ ( 1 - x 2 ) λ - 3 2 ⁢ C n + 1 ( λ - 1 ) ⁡ ( x ) derivative 𝑥 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 1 𝑛 2 𝜆 1 2 𝜆 1 superscript 1 superscript 𝑥 2 𝜆 3 2 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{% \lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)\right)=-\frac{(n+1)(n+2% \lambda-1)}{2(\lambda-1)}{(1-x^{2})^{\lambda-\frac{3}{2}}}C^{(\lambda-1)}_{n+1% }\left(x\right)}}
\deriv{}{x}\left((1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{n}@{x}\right) = -\frac{(n+1)(n+2\lambda-1)}{2(\lambda-1)}{(1-x^{2})^{\lambda-\frac{3}{2}}}\ultrasphpoly{\lambda-1}{n+1}@{x}		 

diff((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x), x) = -((n + 1)*(n + 2*lambda - 1))/(2*(lambda - 1))*(1 - (x)^(2))^(lambda -(3)/(2))*GegenbauerC(n + 1, lambda - 1, x)

D[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x], x] == -Divide[(n + 1)*(n + 2*\[Lambda]- 1),2*(\[Lambda]- 1)]*(1 - (x)^(2))^(\[Lambda]-Divide[3,2])*GegenbauerC[n + 1, \[Lambda]- 1, x]
Successful Successful - Successful [Tested: 90]
18.9.E21 d d x ⁡ T n ⁡ ( x ) = n ⁢ U n - 1 ⁡ ( x ) derivative 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑛 Chebyshev-polynomial-second-kind-U 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}T_{n}\left(x\right)=% nU_{n-1}\left(x\right)}}
\deriv{}{x}\ChebyshevpolyT{n}@{x} = n\ChebyshevpolyU{n-1}@{x}		 

diff(ChebyshevT(n, x), x) = n*ChebyshevU(n - 1, x)

D[ChebyshevT[n, x], x] == n*ChebyshevU[n - 1, x]
Successful Successful - Successful [Tested: 9]
18.9.E22 d d x ⁡ ( ( 1 - x 2 ) 1 2 ⁢ U n ⁡ ( x ) ) = - ( n + 1 ) ⁢ ( 1 - x 2 ) - 1 2 ⁢ T n + 1 ⁡ ( x ) derivative 𝑥 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{% \frac{1}{2}}U_{n}\left(x\right)\right)=-(n+1){(1-x^{2})^{-\frac{1}{2}}}T_{n+1}% \left(x\right)}}
\deriv{}{x}\left((1-x^{2})^{\frac{1}{2}}\ChebyshevpolyU{n}@{x}\right) = -(n+1){(1-x^{2})^{-\frac{1}{2}}}\ChebyshevpolyT{n+1}@{x}		 

diff((1 - (x)^(2))^((1)/(2))* ChebyshevU(n, x), x) = -(n + 1)*(1 - (x)^(2))^(-(1)/(2))*ChebyshevT(n + 1, x)

D[(1 - (x)^(2))^(Divide[1,2])* ChebyshevU[n, x], x] == -(n + 1)*(1 - (x)^(2))^(-Divide[1,2])*ChebyshevT[n + 1, x]
Successful Successful - Successful [Tested: 9]
18.9.E23 d d x ⁡ L n ( α ) ⁡ ( x ) = - L n - 1 ( α + 1 ) ⁡ ( x ) derivative 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑥 Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}L^{(\alpha)}_{n}% \left(x\right)=-L^{(\alpha+1)}_{n-1}\left(x\right)}}
\deriv{}{x}\LaguerrepolyL[\alpha]{n}@{x} = -\LaguerrepolyL[\alpha+1]{n-1}@{x}		 

diff(LaguerreL(n, alpha, x), x) = - LaguerreL(n - 1, alpha + 1, x)

D[LaguerreL[n, \[Alpha], x], x] == - LaguerreL[n - 1, \[Alpha]+ 1, x]
Missing Macro Error Successful - Successful [Tested: 27]
18.9.E24 d d x ⁡ ( e - x ⁢ x α ⁢ L n ( α ) ⁡ ( x ) ) = ( n + 1 ) ⁢ e - x ⁢ x α - 1 ⁢ L n + 1 ( α - 1 ) ⁡ ( x ) derivative 𝑥 superscript 𝑒 𝑥 superscript 𝑥 𝛼 Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑛 1 superscript 𝑒 𝑥 superscript 𝑥 𝛼 1 Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}x^{% \alpha}L^{(\alpha)}_{n}\left(x\right)\right)=(n+1)e^{-x}x^{\alpha-1}L^{(\alpha% -1)}_{n+1}\left(x\right)}}
\deriv{}{x}\left(e^{-x}x^{\alpha}\LaguerrepolyL[\alpha]{n}@{x}\right) = (n+1)e^{-x}x^{\alpha-1}\LaguerrepolyL[\alpha-1]{n+1}@{x}		 

diff(exp(- x)*(x)^(alpha)* LaguerreL(n, alpha, x), x) = (n + 1)*exp(- x)*(x)^(alpha - 1)* LaguerreL(n + 1, alpha - 1, x)

D[Exp[- x]*(x)^\[Alpha]* LaguerreL[n, \[Alpha], x], x] == (n + 1)*Exp[- x]*(x)^(\[Alpha]- 1)* LaguerreL[n + 1, \[Alpha]- 1, x]
Missing Macro Error Successful - Successful [Tested: 27]
18.9.E25 d d x ⁡ H n ⁡ ( x ) = 2 ⁢ n ⁢ H n - 1 ⁡ ( x ) derivative 𝑥 Hermite-polynomial-H 𝑛 𝑥 2 𝑛 Hermite-polynomial-H 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}H_{n}\left(x\right)=% 2nH_{n-1}\left(x\right)}}
\deriv{}{x}\HermitepolyH{n}@{x} = 2n\HermitepolyH{n-1}@{x}		 

diff(HermiteH(n, x), x) = 2*n*HermiteH(n - 1, x)

D[HermiteH[n, x], x] == 2*n*HermiteH[n - 1, x]
Successful Successful - Successful [Tested: 9]
18.9.E26 d d x ⁡ ( e - x 2 ⁢ H n ⁡ ( x ) ) = - e - x 2 ⁢ H n + 1 ⁡ ( x ) derivative 𝑥 superscript 𝑒 superscript 𝑥 2 Hermite-polynomial-H 𝑛 𝑥 superscript 𝑒 superscript 𝑥 2 Hermite-polynomial-H 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x^{2}}H_{n% }\left(x\right)\right)=-e^{-x^{2}}H_{n+1}\left(x\right)}}
\deriv{}{x}\left(e^{-x^{2}}\HermitepolyH{n}@{x}\right) = -e^{-x^{2}}\HermitepolyH{n+1}@{x}		 

diff(exp(- (x)^(2))*HermiteH(n, x), x) = - exp(- (x)^(2))*HermiteH(n + 1, x)

D[Exp[- (x)^(2)]*HermiteH[n, x], x] == - Exp[- (x)^(2)]*HermiteH[n + 1, x]
Successful Successful - Successful [Tested: 9]
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