Orthogonal Polynomials - 18.16 Zeros

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.16.E1 0 < ΞΈ n , 1 0 subscript πœƒ 𝑛 1 {\displaystyle{\displaystyle 0<\theta_{n,1}}}
0 < \theta_{n,1}		 

0 < theta[n , 1]
 
0 < Subscript[\[Theta], n , 1]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E2 ( m - 1 2 ) ⁒ Ο€ n + 1 2 ≀ ΞΈ n , m π‘š 1 2 πœ‹ 𝑛 1 2 subscript πœƒ 𝑛 π‘š {\displaystyle{\displaystyle\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{1}{2}}\leq% \theta_{n,m}}}
\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{1}{2}} \leq \theta_{n,m}		 

((m -(1)/(2))*Pi)/(n +(1)/(2)) <= theta[n , m]
 
Divide[(m -Divide[1,2])*Pi,n +Divide[1,2]] <= Subscript[\[Theta], n , m]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E3 ( m - 1 2 ) ⁒ Ο€ n ≀ ΞΈ n , m π‘š 1 2 πœ‹ 𝑛 subscript πœƒ 𝑛 π‘š {\displaystyle{\displaystyle\frac{(m-\tfrac{1}{2})\pi}{n}\leq\theta_{n,m}}}
\frac{(m-\tfrac{1}{2})\pi}{n} \leq \theta_{n,m}		 Ξ± = Ξ² 𝛼 𝛽 {\displaystyle{\displaystyle\alpha=\beta}} 
((m -(1)/(2))*Pi)/(n) <= theta[n , m]
 
Divide[(m -Divide[1,2])*Pi,n] <= Subscript[\[Theta], n , m]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E4 ( m + 1 2 ⁒ ( Ξ± + Ξ² - 1 ) ) ⁒ Ο€ ρ < ΞΈ n , m π‘š 1 2 𝛼 𝛽 1 πœ‹ 𝜌 subscript πœƒ 𝑛 π‘š {\displaystyle{\displaystyle\frac{\left(m+\tfrac{1}{2}(\alpha+\beta-1)\right)% \pi}{\rho}<\theta_{n,m}}}
\frac{\left(m+\tfrac{1}{2}(\alpha+\beta-1)\right)\pi}{\rho} < \theta_{n,m}		 

((m +(1)/(2)*(alpha + beta - 1))*Pi)/(n +(1)/(2)*(alpha + beta + 1)) < theta[n , m]
 
Divide[(m +Divide[1,2]*(\[Alpha]+ \[Beta]- 1))*Pi,n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1)] < Subscript[\[Theta], n , m]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E5 ΞΈ n , m > ( m + 1 2 ⁒ Ξ± - 1 4 ) ⁒ Ο€ n + Ξ± + 1 2 subscript πœƒ 𝑛 π‘š π‘š 1 2 𝛼 1 4 πœ‹ 𝑛 𝛼 1 2 {\displaystyle{\displaystyle\theta_{n,m}>\frac{\left(m+\tfrac{1}{2}\alpha-% \tfrac{1}{4}\right){\pi}}{n+\alpha+\tfrac{1}{2}}}}
\theta_{n,m} > \frac{\left(m+\tfrac{1}{2}\alpha-\tfrac{1}{4}\right){\pi}}{n+\alpha+\tfrac{1}{2}}		 Ξ± = Ξ² 𝛼 𝛽 {\displaystyle{\displaystyle\alpha=\beta}} 
theta[n , m] > ((m +(1)/(2)*alpha -(1)/(4))*Pi)/(n + alpha +(1)/(2))
 
Subscript[\[Theta], n , m] > Divide[(m +Divide[1,2]*\[Alpha]-Divide[1,4])*Pi,n + \[Alpha]+Divide[1,2]]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E6 ΞΈ n , m ≀ j Ξ± , m ( ρ 2 + 1 12 ⁒ ( 1 - Ξ± 2 - 3 ⁒ Ξ² 2 ) ) 1 2 subscript πœƒ 𝑛 π‘š subscript 𝑗 𝛼 π‘š superscript superscript 𝜌 2 1 12 1 superscript 𝛼 2 3 superscript 𝛽 2 1 2 {\displaystyle{\displaystyle\theta_{n,m}\leq\frac{j_{\alpha,m}}{\left(\rho^{2}% +\tfrac{1}{12}\left(1-\alpha^{2}-3\beta^{2}\right)\right)^{\frac{1}{2}}}}}
\theta_{n,m} \leq \frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{12}\left(1-\alpha^{2}-3\beta^{2}\right)\right)^{\frac{1}{2}}}		 

theta[n , m] <= (j[alpha , m])/(((n +(1)/(2)*(alpha + beta + 1))^(2)+(1)/(12)*(1 - (alpha)^(2)- 3*(beta)^(2)))^((1)/(2)))
 
Subscript[\[Theta], n , m] <= Divide[Subscript[j, \[Alpha], m],((n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2)+Divide[1,12]*(1 - \[Alpha]^(2)- 3*\[Beta]^(2)))^(Divide[1,2])]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E7 ΞΈ n , m β‰₯ j Ξ± , m ( ρ 2 + 1 4 - 1 2 ⁒ ( Ξ± 2 + Ξ² 2 ) - Ο€ - 2 ⁒ ( 1 - 4 ⁒ Ξ± 2 ) ) 1 2 subscript πœƒ 𝑛 π‘š subscript 𝑗 𝛼 π‘š superscript superscript 𝜌 2 1 4 1 2 superscript 𝛼 2 superscript 𝛽 2 superscript πœ‹ 2 1 4 superscript 𝛼 2 1 2 {\displaystyle{\displaystyle\theta_{n,m}\geq\frac{j_{\alpha,m}}{\left(\rho^{2}% +\tfrac{1}{4}-\tfrac{1}{2}(\alpha^{2}+\beta^{2})-\pi^{-2}(1-4\alpha^{2})\right% )^{\frac{1}{2}}}}}
\theta_{n,m} \geq \frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{4}-\tfrac{1}{2}(\alpha^{2}+\beta^{2})-\pi^{-2}(1-4\alpha^{2})\right)^{\frac{1}{2}}}		 

theta[n , m] >= (j[alpha , m])/(((n +(1)/(2)*(alpha + beta + 1))^(2)+(1)/(4)-(1)/(2)*((alpha)^(2)+ (beta)^(2))- (Pi)^(- 2)*(1 - 4*(alpha)^(2)))^((1)/(2)))
 
Subscript[\[Theta], n , m] >= Divide[Subscript[j, \[Alpha], m],((n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2)+Divide[1,4]-Divide[1,2]*(\[Alpha]^(2)+ \[Beta]^(2))- (Pi)^(- 2)*(1 - 4*\[Alpha]^(2)))^(Divide[1,2])]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E9 0 < x n , 1 0 subscript π‘₯ 𝑛 1 {\displaystyle{\displaystyle 0<x_{n,1}}}
0 < x_{n,1}		 

0 < x[n , 1]
 
0 < Subscript[x, n , 1]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E10 x n , m > j Ξ± , m 2 / Ξ½ subscript π‘₯ 𝑛 π‘š superscript subscript 𝑗 𝛼 π‘š 2 𝜈 {\displaystyle{\displaystyle x_{n,m}>\ifrac{j_{\alpha,m}^{2}}{\nu}}}
x_{n,m} > \ifrac{j_{\alpha,m}^{2}}{\nu}		 

x[n , m] > ((j[alpha , m])^(2))/(4*n + 2*alpha + 2)
 
Subscript[x, n , m] > Divide[(Subscript[j, \[Alpha], m])^(2),4*n + 2*\[Alpha]+ 2]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E11 x n , m < ( 4 ⁒ m + 2 ⁒ Ξ± + 2 ) ⁒ ( 2 ⁒ m + Ξ± + 1 + ( ( 2 ⁒ m + Ξ± + 1 ) 2 + 1 4 - Ξ± 2 ) 1 2 ) / Ξ½ subscript π‘₯ 𝑛 π‘š 4 π‘š 2 𝛼 2 2 π‘š 𝛼 1 superscript superscript 2 π‘š 𝛼 1 2 1 4 superscript 𝛼 2 1 2 𝜈 {\displaystyle{\displaystyle x_{n,m}<(4m+2\alpha+2)\left(2m+\alpha+1+\left((2m% +\alpha+1)^{2}+\tfrac{1}{4}-\alpha^{2}\right)^{\frac{1}{2}}\right)\Big{/}\nu}}
x_{n,m} < (4m+2\alpha+2)\left(2m+\alpha+1+\left((2m+\alpha+1)^{2}+\tfrac{1}{4}-\alpha^{2}\right)^{\frac{1}{2}}\right)\Big{/}\nu		 

x[n , m] < (4*m + 2*alpha + 2)*(2*m + alpha + 1 +((2*m + alpha + 1)^(2)+(1)/(4)- (alpha)^(2))^((1)/(2)))/(4*n + 2*alpha + 2)
 
Subscript[x, n , m] < (4*m + 2*\[Alpha]+ 2)*(2*m + \[Alpha]+ 1 +((2*m + \[Alpha]+ 1)^(2)+Divide[1,4]- \[Alpha]^(2))^(Divide[1,2]))/(4*n + 2*\[Alpha]+ 2)
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E12 x n , 1 β‰₯ 2 ⁒ n 2 + Ξ± ⁒ n - n + 2 ⁒ Ξ± + 2 - 2 ⁒ ( n - 1 ) ⁒ n 2 + ( n + 2 ) ⁒ ( Ξ± + 1 ) n + 2 subscript π‘₯ 𝑛 1 2 superscript 𝑛 2 𝛼 𝑛 𝑛 2 𝛼 2 2 𝑛 1 superscript 𝑛 2 𝑛 2 𝛼 1 𝑛 2 {\displaystyle{\displaystyle x_{n,1}\geq\frac{2n^{2}+\alpha n-n+2\alpha+2-2(n-% 1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}}}
x_{n,1} \geq \frac{2n^{2}+\alpha n-n+2\alpha+2-2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}		 

x[n , 1] >= (2*(n)^(2)+ alpha*n - n + 2*alpha + 2 - 2*(n - 1)*sqrt((n)^(2)+(n + 2)*(alpha + 1)))/(n + 2)
 
Subscript[x, n , 1] >= Divide[2*(n)^(2)+ \[Alpha]*n - n + 2*\[Alpha]+ 2 - 2*(n - 1)*Sqrt[(n)^(2)+(n + 2)*(\[Alpha]+ 1)],n + 2]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E13 x n , n ≀ 2 ⁒ n 2 + Ξ± ⁒ n - n + 2 ⁒ Ξ± + 2 + 2 ⁒ ( n - 1 ) ⁒ n 2 + ( n + 2 ) ⁒ ( Ξ± + 1 ) n + 2 subscript π‘₯ 𝑛 𝑛 2 superscript 𝑛 2 𝛼 𝑛 𝑛 2 𝛼 2 2 𝑛 1 superscript 𝑛 2 𝑛 2 𝛼 1 𝑛 2 {\displaystyle{\displaystyle x_{n,n}\leq\frac{2n^{2}+\alpha n-n+2\alpha+2+2(n-% 1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}}}
x_{n,n} \leq \frac{2n^{2}+\alpha n-n+2\alpha+2+2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}		 

x[n , n] <= (2*(n)^(2)+ alpha*n - n + 2*alpha + 2 + 2*(n - 1)*sqrt((n)^(2)+(n + 2)*(alpha + 1)))/(n + 2)
 
Subscript[x, n , n] <= Divide[2*(n)^(2)+ \[Alpha]*n - n + 2*\[Alpha]+ 2 + 2*(n - 1)*Sqrt[(n)^(2)+(n + 2)*(\[Alpha]+ 1)],n + 2]
 Skipped - no semantic math Skipped - no semantic math - -
18.16.E16 ( 2 ⁒ n + 1 ) 1 2 > x n , 1 superscript 2 𝑛 1 1 2 subscript π‘₯ 𝑛 1 {\displaystyle{\displaystyle(2n+1)^{\frac{1}{2}}>x_{n,1}}}
(2n+1)^{\frac{1}{2}} > x_{n,1}		 

(2*n + 1)^((1)/(2)) > x[n , 1]

(2*n + 1)^(Divide[1,2]) > Subscript[x, n , 1]
Failure Failure
Failed [1 / 30]
Result: 2. < 1.732050808
Test Values: {x[n,1] = 2, n = 1}

Failed [13 / 30]
Result: Greater[1.7320508075688772, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[n, 1], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Greater[2.23606797749979, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[n, 2], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.16.E16 x n , 1 > x n , 2 subscript π‘₯ 𝑛 1 subscript π‘₯ 𝑛 2 {\displaystyle{\displaystyle x_{n,1}>x_{n,2}}}
x_{n,1} > x_{n,2}		 

x[n , 1] > x[n , 2]

Subscript[x, n , 1] > Subscript[x, n , 2]
Failure Failure
Failed [75 / 300]
Result: .8660254040+.5000000000*I < .8660254040+.5000000000*I
Test Values: {x[n,1] = 1/2*3^(1/2)+1/2*I, x[n,2] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .8660254040+.5000000000*I < .8660254040+.5000000000*I
Test Values: {x[n,1] = 1/2*3^(1/2)+1/2*I, x[n,2] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [255 / 300]
Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[n, 1], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[n, 2], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.16.E16 x n , ⌊ n / 2 βŒ‹ > 0 subscript π‘₯ 𝑛 𝑛 2 0 {\displaystyle{\displaystyle x_{n,\left\lfloor n/2\right\rfloor}>0}}
x_{n,\floor{n/2}} > 0		 

x[n , floor(n/2)] > 0

Subscript[x, n , Floor[n/2]] > 0
Failure Failure
Failed [9 / 30]
Result: 0. < -1.500000000
Test Values: {x[n,floor(1/2*n)] = -3/2, n = 1}


Result: 0. < -1.500000000
Test Values: {x[n,floor(1/2*n)] = -3/2, n = 2}

... skip entries to safe data
Failed [21 / 30]
Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0]
Test Values: {Rule[n, 1], Rule[Subscript[x, n, Floor[Times[Rational[1, 2], n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0]
Test Values: {Rule[n, 2], Rule[Subscript[x, n, Floor[Times[Rational[1, 2], n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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