Chebychev Polynomials (of the First Kind)

Here are the first seven Chebyshev Polynomials, along with a plot.


  Table[ChebyshevT[k,x],{k,0,6}]


  Plot[Evaluate[Table[ChebyshevT[k,x],{k,0,6}]],{x,0,6}];

The Chebyshev differential equation is
[ Sqrt[(1-x^2)]y' ]' +L 1/Sqrt[(1-x^2)] y=0 on -1<x<1

The eigenvalues are n^2 and the eigenfunctions are the Chebyshev polynomials.

Note that the eigenvalues are real and satisfy L->Infinity.

These polynomials DO solve the differential equation. Check one of them.


  Simplify[D[Sqrt[1-x^2] D[ChebyshevT[2,x],x],x]+2^2 1/Sqrt[1-x^2]ChebyshevT[2,x]]

Check the first seven:


  Table[D[Sqrt[1-x^2] D[ChebyshevT[k,x],x],x]+k^2 1/Sqrt[1-x^2]ChebyshevT[k,x],{k,0,6}]


  Simplify[%]

By one of the major Sturm-Liouville theorems, the Chebyshev polynomials are orthogonal with respect to the weight 1/Sqrt[1-x^2]. Check one.


  Integrate[1/Sqrt[1-x^2] ChebyshevT[1,x]ChebyshevT[2,x],{x,-1,1}]

Check many at a time:


  Table[Integrate[1/Sqrt[1-x^2] ChebyshevT[k,x]ChebyshevT[m,x],{x,-1,1}],{k,0,3},{m,0,3}];
  MatrixForm[%]