View source for 18.3

{{DISPLAYTITLE:Orthogonal Polynomials - 18.3 Definitions}}

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| [https://dlmf.nist.gov/18.3.E1 18.3.E1] || <math qid="Q5507">\sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0</syntaxhighlight> || <math>0 \leq j, j \leq N, 0 \leq k, k \leq N, j \neq k</math> || <syntaxhighlight lang=mathematica>sum(ChebyshevT(j, x[N + 1 , n])*ChebyshevT(k, x[N + 1 , n]), n = 1..N + 1) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[ChebyshevT[j, Subscript[x, N + 1 , n]]*ChebyshevT[k, Subscript[x, N + 1 , n]], {n, 1, N + 1}, GenerateConditions->None] == 0</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
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| [https://dlmf.nist.gov/18.3.E2 18.3.E2] || <math qid="Q5508">x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x[N + 1 , n] = cos((n -(1)/(2))*Pi/(N + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[x, N + 1 , n] == Cos[(n -Divide[1,2])*Pi/(N + 1)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1432026267+.3500908026*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.718798807+.233214116e-1*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.14320262643759762, 0.350090802645732]
Test Values: {Rule[n, 1], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, N], n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.7187988066024098, 0.023321411689447014]
Test Values: {Rule[n, 2], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, N], n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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