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Relationship between the First and Second Kind Chebyshev Polynomials 📂Numerical Analysis

Relationship between the First and Second Kind Chebyshev Polynomials

Theorem

The first kind Chebyshev polynomials Tn(x)=cos⁡(ncos⁡−1x)T_{n} (x) = \cos \left( n \cos^{-1} x \right) and second kind Chebyshev polynomials Un(x)=1n+1Tn+1’(X)\displaystyle U_{n} (x) = {{1} \over {n+1} } T_{n+1} ’ (X) have the following relationship:

  • [1]: Un(x)−Un−2(x)=2Tn(X)U_{n} (x) - U_{n-2} (x) = 2 T_{n} (X)
  • [2]: Tn(x)−Tn−2(x)=2(x2−1)Un−2(x)T_{n} (x) - T_{n-2} (x) = 2( x^2 - 1 ) U_{n-2} (x)
  • Typically, for 0≤θ≤π0 \le \theta \le \pi, it is set to θ:=cos⁡−1x\theta := \cos^{-1} x .

Proof

The following fact is essential for proving the above equations.

Another expression of the second kind Chebyshev polynomials: Un(x)=sin⁡((n+1)θ)sin⁡θU_{n} (x) = {{\sin \left( ( n +1 ) \theta \right)} \over { \sin \theta }}

[1]

By the sum and difference formulas of trigonometric functions, Un(x)−Un−2(x)=1sin⁡θ[sin⁡(n+1)θ−sin⁡(n−1)θ]=1sin⁡θ2cos⁡((n+1)+(n−1)2θ)sin⁡((n+1)−(n−1)2θ)=2sin⁡θcos⁡nθsin⁡θ=2cos⁡nθ=2Tn(x) \begin{align*} & U_{n} (x) - U_{n-2} (x) \\ =& {{1} \over { \sin \theta }} \left[ \sin (n+1) \theta - \sin (n-1) \theta \right] \\ =& {{1} \over { \sin \theta }} 2 \cos \left( {{ (n+1) + (n-1) } \over {2}} \theta \right) \sin \left( {{ (n+1) - (n-1) } \over {2}} \theta \right) \\ =& {{2} \over { \sin \theta }} \cos n \theta \sin \theta \\ =& 2 \cos n \theta \\ =& 2 T_{n} (x) \end{align*}

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[2]

By the addition theorem of trigonometric functions, Tn±1(x)=cos⁡(n±1)θ=cos⁡(nθ)cos⁡θ∓sin⁡(nθ)sin⁡θ T_{n \pm 1} (x) = \cos (n \pm 1) \theta = \cos (n \theta ) \cos \theta \mp \sin ( n \theta ) \sin \theta Therefore, Tn−1(x)−Tn+1(x)=2sin⁡(nθ)sin⁡θ=2sin⁡2θsin⁡(nθ)sin⁡θ=2sin⁡2θUn−1(x)=2(1−x2)Un−1(x) \begin{align*} & T_{n-1} (x) - T_{n+1} (x) \\ =& 2 \sin (n \theta ) \sin \theta \\ =& 2 \sin^2 \theta {{ \sin (n \theta ) } \over { \sin \theta}} \\ =& 2 \sin^2 \theta U_{n-1} (x) \\ =& 2 (1-x^2) U_{n-1} (x) \end{align*}

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