Orthogonal Polynomials - 18.14 Inequalities

From testwiki
Revision as of 11:45, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.14.E1 | P n ( α , β ) ⁡ ( x ) | ≤ P n ( α , β ) ⁡ ( 1 ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq P^{(% \alpha,\beta)}_{n}\left(1\right)}}
|\JacobipolyP{\alpha}{\beta}{n}@{x}| \leq \JacobipolyP{\alpha}{\beta}{n}@{1}		 - 1 ≤ x , x ≤ 1 , α ≥ β , β > - 1 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 𝛼 𝛽 𝛽 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1,\alpha\geq\beta,\beta>-1}} 
abs(JacobiP(n, alpha, beta, x)) <= JacobiP(n, alpha, beta, 1)

Abs[JacobiP[n, \[Alpha], \[Beta], x]] <= JacobiP[n, \[Alpha], \[Beta], 1]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.14.E1 P n ( α , β ) ⁡ ( 1 ) = ( α + 1 ) n n ! Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 Pochhammer 𝛼 1 𝑛 𝑛 {\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{{\left% (\alpha+1\right)_{n}}}{n!}}}
\JacobipolyP{\alpha}{\beta}{n}@{1} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}		 - 1 ≤ x , x ≤ 1 , α ≥ β , β > - 1 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 𝛼 𝛽 𝛽 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1,\alpha\geq\beta,\beta>-1}} 
JacobiP(n, alpha, beta, 1) = (pochhammer(alpha + 1, n))/(factorial(n))

JacobiP[n, \[Alpha], \[Beta], 1] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 9]
18.14.E2 | P n ( α , β ) ⁡ ( x ) | ≤ | P n ( α , β ) ⁡ ( - 1 ) | = ( β + 1 ) n n ! Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 Pochhammer 𝛽 1 𝑛 𝑛 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq|P^{(% \alpha,\beta)}_{n}\left(-1\right)|=\frac{{\left(\beta+1\right)_{n}}}{n!}}}
|\JacobipolyP{\alpha}{\beta}{n}@{x}| \leq |\JacobipolyP{\alpha}{\beta}{n}@{-1}|=\frac{\Pochhammersym{\beta+1}{n}}{n!}		 - 1 ≤ x , x ≤ 1 , β ≥ α , α > - 1 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 𝛽 𝛼 𝛼 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1,\beta\geq\alpha,\alpha>-1}} 
abs(JacobiP(n, alpha, beta, x)) <= abs(JacobiP(n, alpha, beta, - 1)) = (pochhammer(beta + 1, n))/(factorial(n))

Abs[JacobiP[n, \[Alpha], \[Beta], x]] <= Abs[JacobiP[n, \[Alpha], \[Beta], - 1]] == Divide[Pochhammer[\[Beta]+ 1, n],(n)!]
Failure Failure Error
Failed [1 / 9]
Result: False
Test Values: {Rule[n, 1], Rule[x, 0.5], Rule[α, 2], Rule[β, Rational[1, 2]]}

18.14.E4 | C n ( λ ) ⁡ ( x ) | ≤ C n ( λ ) ⁡ ( 1 ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 {\displaystyle{\displaystyle|C^{(\lambda)}_{n}\left(x\right)|\leq C^{(\lambda)% }_{n}\left(1\right)}}
|\ultrasphpoly{\lambda}{n}@{x}| \leq \ultrasphpoly{\lambda}{n}@{1}		 - 1 ≤ x , x ≤ 1 , λ > 0 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 𝜆 0 {\displaystyle{\displaystyle-1\leq x,x\leq 1,\lambda>0}} 
abs(GegenbauerC(n, lambda, x)) <= GegenbauerC(n, lambda, 1)

Abs[GegenbauerC[n, \[Lambda], x]] <= GegenbauerC[n, \[Lambda], 1]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.14.E4 C n ( λ ) ⁡ ( 1 ) = ( 2 ⁢ λ ) n n ! ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 Pochhammer 2 𝜆 𝑛 𝑛 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(1\right)=\frac{{\left(2% \lambda\right)_{n}}}{n!}}}
\ultrasphpoly{\lambda}{n}@{1} = \frac{\Pochhammersym{2\lambda}{n}}{n!}		 - 1 ≤ x , x ≤ 1 , λ > 0 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 𝜆 0 {\displaystyle{\displaystyle-1\leq x,x\leq 1,\lambda>0}} 
GegenbauerC(n, lambda, 1) = (pochhammer(2*lambda, n))/(factorial(n))

GegenbauerC[n, \[Lambda], 1] == Divide[Pochhammer[2*\[Lambda], n],(n)!]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 9]
18.14.E5 | C 2 ⁢ m ( λ ) ⁡ ( x ) | ≤ | C 2 ⁢ m ( λ ) ⁡ ( 0 ) | = | ( λ ) m m ! | ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑚 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑚 0 Pochhammer 𝜆 𝑚 𝑚 {\displaystyle{\displaystyle|C^{(\lambda)}_{2m}\left(x\right)|\leq|C^{(\lambda% )}_{2m}\left(0\right)|=\left|\frac{{\left(\lambda\right)_{m}}}{m!}\right|}}
|\ultrasphpoly{\lambda}{2m}@{x}| \leq |\ultrasphpoly{\lambda}{2m}@{0}|=\left|\frac{\Pochhammersym{\lambda}{m}}{m!}\right|		 - 1 ≤ x , x ≤ 1 , - 1 2 < λ , λ < 0 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 1 2 𝜆 𝜆 0 {\displaystyle{\displaystyle-1\leq x,x\leq 1,-\tfrac{1}{2}<\lambda,\lambda<0}} 
abs(GegenbauerC(2*m, lambda, x)) <= abs(GegenbauerC(2*m, lambda, 0)) = abs((pochhammer(lambda, m))/(factorial(m)))

Abs[GegenbauerC[2*m, \[Lambda], x]] <= Abs[GegenbauerC[2*m, \[Lambda], 0]] == Abs[Divide[Pochhammer[\[Lambda], m],(m)!]]
Failure Failure Error Skip - No test values generated
18.14.E6 | C 2 ⁢ m + 1 ( λ ) ⁡ ( x ) | < - 2 ⁢ ( λ ) m + 1 ( ( 2 ⁢ m + 1 ) ⁢ ( 2 ⁢ λ + 2 ⁢ m + 1 ) ) 1 2 ⁢ m ! ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑚 1 𝑥 2 Pochhammer 𝜆 𝑚 1 superscript 2 𝑚 1 2 𝜆 2 𝑚 1 1 2 𝑚 {\displaystyle{\displaystyle|C^{(\lambda)}_{2m+1}\left(x\right)|<\frac{-2{% \left(\lambda\right)_{m+1}}}{\left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m% !}}}
|\ultrasphpoly{\lambda}{2m+1}@{x}| < \frac{-2\Pochhammersym{\lambda}{m+1}}{\left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m!}		 - 1 ≤ x , x ≤ 1 , - 1 2 < λ , λ < 0 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 1 2 𝜆 𝜆 0 {\displaystyle{\displaystyle-1\leq x,x\leq 1,-\tfrac{1}{2}<\lambda,\lambda<0}} 
abs(GegenbauerC(2*m + 1, lambda, x)) < (- 2*pochhammer(lambda, m + 1))/(((2*m + 1)*(2*lambda + 2*m + 1))^((1)/(2))* factorial(m))

Abs[GegenbauerC[2*m + 1, \[Lambda], x]] < Divide[- 2*Pochhammer[\[Lambda], m + 1],((2*m + 1)*(2*\[Lambda]+ 2*m + 1))^(Divide[1,2])* (m)!]
Failure Failure Error Skip - No test values generated
18.14.E7 ( n + λ ) 1 - λ ⁢ ( 1 - x 2 ) 1 2 ⁢ λ ⁢ | C n ( λ ) ⁡ ( x ) | < 2 1 - λ Γ ⁡ ( λ ) superscript 𝑛 𝜆 1 𝜆 superscript 1 superscript 𝑥 2 1 2 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 2 1 𝜆 Euler-Gamma 𝜆 {\displaystyle{\displaystyle(n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}% \lambda}|C^{(\lambda)}_{n}\left(x\right)|<\frac{2^{1-\lambda}}{\Gamma\left(% \lambda\right)}}}
(n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}\lambda}|\ultrasphpoly{\lambda}{n}@{x}| < \frac{2^{1-\lambda}}{\EulerGamma@{\lambda}}		 - 1 ≤ x , x ≤ 1 , 0 < λ , λ < 1 , ℜ ⁡ ( λ ) > 0 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 0 𝜆 formulae-sequence 𝜆 1 𝜆 0 {\displaystyle{\displaystyle-1\leq x,x\leq 1,0<\lambda,\lambda<1,\Re(\lambda)>% 0}} 
(n + lambda)^(1 - lambda)*(1 - (x)^(2))^((1)/(2)*lambda)*abs(GegenbauerC(n, lambda, x)) < ((2)^(1 - lambda))/(GAMMA(lambda))

(n + \[Lambda])^(1 - \[Lambda])*(1 - (x)^(2))^(Divide[1,2]*\[Lambda])*Abs[GegenbauerC[n, \[Lambda], x]] < Divide[(2)^(1 - \[Lambda]),Gamma[\[Lambda]]]
Skipped - Unable to analyze test case: Null Skipped - Unable to analyze test case: Null - -
18.14.E8 e - 1 2 ⁢ x ⁢ | L n ( α ) ⁡ ( x ) | ≤ L n ( α ) ⁡ ( 0 ) superscript 𝑒 1 2 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑥 Laguerre-polynomial-L 𝛼 𝑛 0 {\displaystyle{\displaystyle e^{-\frac{1}{2}x}\left|L^{(\alpha)}_{n}\left(x% \right)\right|\leq L^{(\alpha)}_{n}\left(0\right)}}
e^{-\frac{1}{2}x}\left|\LaguerrepolyL[\alpha]{n}@{x}\right| \leq \LaguerrepolyL[\alpha]{n}@{0}		 0 ≤ x , x < ∞ , α ≥ 0 formulae-sequence 0 𝑥 formulae-sequence 𝑥 𝛼 0 {\displaystyle{\displaystyle 0\leq x,x<\infty,\alpha\geq 0}} 
exp(-(1)/(2)*x)*abs(LaguerreL(n, alpha, x)) <= LaguerreL(n, alpha, 0)

Exp[-Divide[1,2]*x]*Abs[LaguerreL[n, \[Alpha], x]] <= LaguerreL[n, \[Alpha], 0]
Missing Macro Error Failure - Successful [Tested: 27]
18.14.E8 L n ( α ) ⁡ ( 0 ) = ( α + 1 ) n n ! Laguerre-polynomial-L 𝛼 𝑛 0 Pochhammer 𝛼 1 𝑛 𝑛 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(0\right)=\frac{{\left(% \alpha+1\right)_{n}}}{n!}}}
\LaguerrepolyL[\alpha]{n}@{0} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}		 0 ≤ x , x < ∞ , α ≥ 0 formulae-sequence 0 𝑥 formulae-sequence 𝑥 𝛼 0 {\displaystyle{\displaystyle 0\leq x,x<\infty,\alpha\geq 0}} 
LaguerreL(n, alpha, 0) = (pochhammer(alpha + 1, n))/(factorial(n))

LaguerreL[n, \[Alpha], 0] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]
Missing Macro Error Successful - Successful [Tested: 9]
18.14.E9 1 ( 2 n ⁢ n ! ) 1 2 ⁢ e - 1 2 ⁢ x 2 ⁢ | H n ⁡ ( x ) | ≤ 1 1 superscript superscript 2 𝑛 𝑛 1 2 superscript 𝑒 1 2 superscript 𝑥 2 Hermite-polynomial-H 𝑛 𝑥 1 {\displaystyle{\displaystyle\frac{1}{(2^{n}n!)^{\frac{1}{2}}}e^{-\frac{1}{2}x^% {2}}|H_{n}\left(x\right)|\leq 1}}
\frac{1}{(2^{n}n!)^{\frac{1}{2}}}e^{-\frac{1}{2}x^{2}}|\HermitepolyH{n}@{x}| \leq 1		 - ∞ < x , x < ∞ formulae-sequence 𝑥 𝑥 {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
(1)/(((2)^(n)* factorial(n))^((1)/(2)))*exp(-(1)/(2)*(x)^(2))*abs(HermiteH(n, x)) <= 1

Divide[1,((2)^(n)* (n)!)^(Divide[1,2])]*Exp[-Divide[1,2]*(x)^(2)]*Abs[HermiteH[n, x]] <= 1
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.14.E10 ( P n ⁡ ( x ) ) 2 ≥ P n - 1 ⁡ ( x ) ⁢ P n + 1 ⁡ ( x ) superscript Legendre-spherical-polynomial 𝑛 𝑥 2 Legendre-spherical-polynomial 𝑛 1 𝑥 Legendre-spherical-polynomial 𝑛 1 𝑥 {\displaystyle{\displaystyle(P_{n}\left(x\right))^{2}\geq P_{n-1}\left(x\right% )P_{n+1}\left(x\right)}}
(\LegendrepolyP{n}@{x})^{2} \geq \LegendrepolyP{n-1}@{x}\LegendrepolyP{n+1}@{x}		 - 1 ≤ x , x ≤ 1 formulae-sequence 1 𝑥 𝑥 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1}} 
(LegendreP(n, x))^(2) >= LegendreP(n - 1, x)*LegendreP(n + 1, x)

(LegendreP[n, x])^(2) >= LegendreP[n - 1, x]*LegendreP[n + 1, x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
18.14.E11 ( R n ⁢ ( x ) ) 2 ≥ R n - 1 ⁢ ( x ) ⁢ R n + 1 ⁢ ( x ) superscript subscript 𝑅 𝑛 𝑥 2 subscript 𝑅 𝑛 1 𝑥 subscript 𝑅 𝑛 1 𝑥 {\displaystyle{\displaystyle(R_{n}(x))^{2}\geq R_{n-1}(x)R_{n+1}(x)}}
(R_{n}(x))^{2} \geq R_{n-1}(x)R_{n+1}(x)		 - 1 ≤ x , x ≤ 1 , β ≥ α , α > - 1 formulae-sequence 1 𝑥 formulae-sequence 𝑥 1 formulae-sequence 𝛽 𝛼 𝛼 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1,\beta\geq\alpha,\alpha>-1}} 
(R[n](x))^(2) >= R[n - 1](x)* R[n + 1](x)
 
(Subscript[R, n][x])^(2) >= Subscript[R, n - 1][x]* Subscript[R, n + 1][x]
 Skipped - no semantic math Skipped - no semantic math - -
18.14.E12 ( L n ( α ) ⁡ ( x ) ) 2 ≥ L n - 1 ( α ) ⁡ ( x ) ⁢ L n + 1 ( α ) ⁡ ( x ) superscript Laguerre-polynomial-L 𝛼 𝑛 𝑥 2 Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 {\displaystyle{\displaystyle(L^{(\alpha)}_{n}\left(x\right))^{2}\geq L^{(% \alpha)}_{n-1}\left(x\right)L^{(\alpha)}_{n+1}\left(x\right)}}
(\LaguerrepolyL[\alpha]{n}@{x})^{2} \geq \LaguerrepolyL[\alpha]{n-1}@{x}\LaguerrepolyL[\alpha]{n+1}@{x}		 0 ≤ x , x < ∞ , α ≥ 0 formulae-sequence 0 𝑥 formulae-sequence 𝑥 𝛼 0 {\displaystyle{\displaystyle 0\leq x,x<\infty,\alpha\geq 0}} 
(LaguerreL(n, alpha, x))^(2) >= LaguerreL(n - 1, alpha, x)*LaguerreL(n + 1, alpha, x)

(LaguerreL[n, \[Alpha], x])^(2) >= LaguerreL[n - 1, \[Alpha], x]*LaguerreL[n + 1, \[Alpha], x]
Missing Macro Error Failure - Successful [Tested: 27]
18.14.E13 ( H n ⁡ ( x ) ) 2 ≥ H n - 1 ⁡ ( x ) ⁢ H n + 1 ⁡ ( x ) superscript Hermite-polynomial-H 𝑛 𝑥 2 Hermite-polynomial-H 𝑛 1 𝑥 Hermite-polynomial-H 𝑛 1 𝑥 {\displaystyle{\displaystyle(H_{n}\left(x\right))^{2}\geq H_{n-1}\left(x\right% )H_{n+1}\left(x\right)}}
(\HermitepolyH{n}@{x})^{2} \geq \HermitepolyH{n-1}@{x}\HermitepolyH{n+1}@{x}		 - ∞ < x , x < ∞ formulae-sequence 𝑥 𝑥 {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
(HermiteH(n, x))^(2) >= HermiteH(n - 1, x)*HermiteH(n + 1, x)

(HermiteH[n, x])^(2) >= HermiteH[n - 1, x]*HermiteH[n + 1, x]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
18.14.E14 - 1 = x n , 0 1 subscript 𝑥 𝑛 0 {\displaystyle{\displaystyle-1=x_{n,0}}}
-1 = x_{n,0}		 

- 1 = x[n , 0]
 
- 1 == Subscript[x, n , 0]
 Skipped - no semantic math Skipped - no semantic math - -
18.14.E15 x n , m ≤ ( β - α ) / ( α + β + 1 ) ≤ x n , m + 1 subscript 𝑥 𝑛 𝑚 𝛽 𝛼 𝛼 𝛽 1 subscript 𝑥 𝑛 𝑚 1 {\displaystyle{\displaystyle x_{n,m}\leq(\beta-\alpha)/(\alpha+\beta+1)\leq x_% {n,m+1}}}
x_{n,m} \leq (\beta-\alpha)/(\alpha+\beta+1)\leq x_{n,m+1}		 

x[n , m] <= (beta - alpha)/(alpha + beta + 1) <= x[n , m + 1]
 
Subscript[x, n , m] <= (\[Beta]- \[Alpha])/(\[Alpha]+ \[Beta]+ 1) <= Subscript[x, n , m + 1]
 Skipped - no semantic math Skipped - no semantic math - -
18.14#Ex1 | P n ( α , β ) ⁡ ( x n , 0 ) | > | P n ( α , β ) ⁡ ( x n , 1 ) | Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 0 Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 1 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|>|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)|}}
|\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|		 

abs(JacobiP(n, alpha, beta, x[n , 0])) > abs(JacobiP(n, alpha, beta, x[n , 1]))

Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] > Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]
Failure Failure
Failed [184 / 300]
Result: 2.500000000 < 2.500000000
Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 4.871793818 < 4.871793818
Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [184 / 300]
Result: False
Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.14#Ex2 | P n ( α , β ) ⁡ ( x n , n ) | > | P n ( α , β ) ⁡ ( x n , n - 1 ) | Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 𝑛 1 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|>|P^{(% \alpha,\beta)}_{n}\left(x_{n,n-1}\right)|}}
|\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-1}}|		 α > - 1 2 , β > - 1 2 . formulae-sequence 𝛼 1 2 𝛽 1 2 {\displaystyle{\displaystyle\alpha>-\tfrac{1}{2},\beta>-\tfrac{1}{2}.}} 
abs(JacobiP(n, alpha, beta, x[n , n])) > abs(JacobiP(n, alpha, beta, x[n , n - 1]))

Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]] > Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]]
Error Failure - Skip - No test values generated
18.14#Ex3 | P n ( α , β ) ⁡ ( x n , 0 ) | < | P n ( α , β ) ⁡ ( x n , 1 ) | Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 0 Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 1 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|<|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)|}}
|\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|		 

abs(JacobiP(n, alpha, beta, x[n , 0])) < abs(JacobiP(n, alpha, beta, x[n , 1]))

Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] < Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]
Failure Failure
Failed [184 / 300]
Result: 2.500000000 < 2.500000000
Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 4.871793820 < 4.871793820
Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [184 / 300]
Result: False
Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.14#Ex4 | P n ( α , β ) ⁡ ( x n , n ) | < | P n ( α , β ) ⁡ ( x n , n - 1 ) | Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 𝑛 1 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|<|P^{(% \alpha,\beta)}_{n}\left(x_{n,n-1}\right)|}}
|\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-1}}|		 - 1 < α , α < - 1 2 , - 1 < β , β < - 1 2 . formulae-sequence 1 𝛼 formulae-sequence 𝛼 1 2 formulae-sequence 1 𝛽 𝛽 1 2 {\displaystyle{\displaystyle-1<\alpha,\alpha<-\tfrac{1}{2},-1<\beta,\beta<-% \tfrac{1}{2}.}} 
abs(JacobiP(n, alpha, beta, x[n , n])) < abs(JacobiP(n, alpha, beta, x[n , n - 1]))

Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]] < Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]]
Error Failure - Skip - No test values generated
18.14.E18 | P n ( α , β ) ⁡ ( x n , 0 ) | < | P n ( α , β ) ⁡ ( x n , 1 ) | Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 0 Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 1 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|<|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)|}}
|\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|		 - 1 < β , β ≤ - 1 2 formulae-sequence 1 𝛽 𝛽 1 2 {\displaystyle{\displaystyle-1<\beta,\beta\leq-\tfrac{1}{2}}} 
abs(JacobiP(n, alpha, beta, x[n , 0])) < abs(JacobiP(n, alpha, beta, x[n , 1]))

Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] < Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]
Failure Failure Error Skip - No test values generated
18.14.E19 | P n ( α , β ) ⁡ ( x n , 0 ) | > | P n ( α , β ) ⁡ ( x n , 1 ) | Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 0 Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 1 {\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|>|P^{(% \alpha,\beta)}_{n}\left(x_{n,1}\right)|}}
|\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|		 - 1 < α , α ≤ - 1 2 formulae-sequence 1 𝛼 𝛼 1 2 {\displaystyle{\displaystyle-1<\alpha,\alpha\leq-\tfrac{1}{2}}} 
abs(JacobiP(n, alpha, beta, x[n , 0])) > abs(JacobiP(n, alpha, beta, x[n , 1]))

Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] > Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]
Failure Failure Error Skip - No test values generated
18.14.E20 | P n ( α , β ) ⁡ ( x n , n - m ) P n ( α , β ) ⁡ ( 1 ) | > | P n + 1 ( α , β ) ⁡ ( x n + 1 , n - m + 1 ) P n + 1 ( α , β ) ⁡ ( 1 ) | Jacobi-polynomial-P 𝛼 𝛽 𝑛 subscript 𝑥 𝑛 𝑛 𝑚 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 subscript 𝑥 𝑛 1 𝑛 𝑚 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 1 {\displaystyle{\displaystyle\left|\frac{P^{(\alpha,\beta)}_{n}\left(x_{n,n-m}% \right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}\right|>\left|\frac{P^{(\alpha,% \beta)}_{n+1}\left(x_{n+1,n-m+1}\right)}{P^{(\alpha,\beta)}_{n+1}\left(1\right% )}\right|}}
\left|\frac{\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-m}}}{\JacobipolyP{\alpha}{\beta}{n}@{1}}\right| > \left|\frac{\JacobipolyP{\alpha}{\beta}{n+1}@{x_{n+1,n-m+1}}}{\JacobipolyP{\alpha}{\beta}{n+1}@{1}}\right|		 α = β , β > - 1 2 , m = 1 formulae-sequence 𝛼 𝛽 formulae-sequence 𝛽 1 2 𝑚 1 {\displaystyle{\displaystyle\alpha=\beta,\beta>-\tfrac{1}{2},m=1}} 
abs((JacobiP(n, alpha, beta, x[n , n - m]))/(JacobiP(n, alpha, beta, 1))) > abs((JacobiP(n + 1, alpha, beta, x[n + 1 , n - m + 1]))/(JacobiP(n + 1, alpha, beta, 1)))

Abs[Divide[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - m]],JacobiP[n, \[Alpha], \[Beta], 1]]] > Abs[Divide[JacobiP[n + 1, \[Alpha], \[Beta], Subscript[x, n + 1 , n - m + 1]],JacobiP[n + 1, \[Alpha], \[Beta], 1]]]
Failure Failure
Failed [188 / 300]
Result: 1.113552873 < 1.000000000
Test Values: {alpha = 3/2, beta = 3/2, x[n,n-m] = 1/2*3^(1/2)+1/2*I, x[n+1,n-m+1] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}


Result: 1.400000001 < 1.113552873
Test Values: {alpha = 3/2, beta = 3/2, x[n,n-m] = 1/2*3^(1/2)+1/2*I, x[n+1,n-m+1] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [234 / 300]
Result: False
Test Values: {Rule[m, 1], Rule[n, 1], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, n], Plus[1, Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[m, 1], Rule[n, 2], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, n], Plus[1, Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.14.E21 0 = x n , 0 0 subscript 𝑥 𝑛 0 {\displaystyle{\displaystyle 0=x_{n,0}}}
0 = x_{n,0}		 

0 = x[n , 0]
 
0 == Subscript[x, n , 0]
 Skipped - no semantic math Skipped - no semantic math - -
18.14.E22 x n , m ≤ α + 1 2 subscript 𝑥 𝑛 𝑚 𝛼 1 2 {\displaystyle{\displaystyle x_{n,m}\leq\alpha+\tfrac{1}{2}}}
x_{n,m} \leq \alpha+\tfrac{1}{2}		 

x[n , m] <= alpha +(1)/(2)
 
Subscript[x, n , m] <= \[Alpha]+Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
18.14#Ex5 | L n ( α ) ⁡ ( x n , 0 ) | > | L n ( α ) ⁡ ( x n , 1 ) | Laguerre-polynomial-L 𝛼 𝑛 subscript 𝑥 𝑛 0 Laguerre-polynomial-L 𝛼 𝑛 subscript 𝑥 𝑛 1 {\displaystyle{\displaystyle|L^{(\alpha)}_{n}\left(x_{n,0}\right)|>|L^{(\alpha% )}_{n}\left(x_{n,1}\right)|}}
|\LaguerrepolyL[\alpha]{n}@{x_{n,0}}| > |\LaguerrepolyL[\alpha]{n}@{x_{n,1}}|		 

abs(LaguerreL(n, alpha, x[n , 0])) > abs(LaguerreL(n, alpha, x[n , 1]))

Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 0]]] > Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 1]]]
Missing Macro Error Failure -
Failed [165 / 300]
Result: False
Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.14#Ex6 | L n ( α ) ⁡ ( x n , n - 1 ) | > | L n ( α ) ⁡ ( x n , n - 2 ) | Laguerre-polynomial-L 𝛼 𝑛 subscript 𝑥 𝑛 𝑛 1 Laguerre-polynomial-L 𝛼 𝑛 subscript 𝑥 𝑛 𝑛 2 {\displaystyle{\displaystyle|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|>|L^{(% \alpha)}_{n}\left(x_{n,n-2}\right)|}}
|\LaguerrepolyL[\alpha]{n}@{x_{n,n-1}}| > |\LaguerrepolyL[\alpha]{n}@{x_{n,n-2}}|		 

abs(LaguerreL(n, alpha, x[n , n - 1])) > abs(LaguerreL(n, alpha, x[n , n - 2]))

Abs[LaguerreL[n, \[Alpha], Subscript[x, n , n - 1]]] > Abs[LaguerreL[n, \[Alpha], Subscript[x, n , n - 2]]]
Missing Macro Error Failure -
Failed [165 / 300]
Result: False
Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[Subscript[x, n, Plus[-2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[Subscript[x, n, Plus[-2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.14.E24 | L n ( α ) ⁡ ( x n , 0 ) | < | L n ( α ) ⁡ ( x n , 1 ) | Laguerre-polynomial-L 𝛼 𝑛 subscript 𝑥 𝑛 0 Laguerre-polynomial-L 𝛼 𝑛 subscript 𝑥 𝑛 1 {\displaystyle{\displaystyle|L^{(\alpha)}_{n}\left(x_{n,0}\right)|<|L^{(\alpha% )}_{n}\left(x_{n,1}\right)|}}
|\LaguerrepolyL[\alpha]{n}@{x_{n,0}}| < |\LaguerrepolyL[\alpha]{n}@{x_{n,1}}|		 

abs(LaguerreL(n, alpha, x[n , 0])) < abs(LaguerreL(n, alpha, x[n , 1]))

Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 0]]] < Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 1]]]
Missing Macro Error Failure -
Failed [165 / 300]
Result: False
Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=18.14&oldid=58341"