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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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8.94 The Chebyshev polynomials Tn(x) and Un(x)

8.940 Definition

1. Chebyshev's polynomials of the first kind

Tn(x)=cos(n arccos x)=12[(x+i1x2)n+(xi1x2)n]=xn|(2n)xn2(1x2)+(4n)x(1x2)2(6n)xn6(1x2)3+|

si1677_e  NA 66, 71

2. Chebyshev's polynomials of the second kind:

Un(x)=sin[(n+1)arccosx]sin[arccosx]=12i1x2[(x+i1x2)n+1(xi1x2)n+1]=(1n+1)xn(3n+1)xn2(1x2)+(5n+1)xn4(1x2)2

si1678_e

Functional relations

8.941 Recursion formulas:

1. 

Tn+1(x)2x Tn(x)+Tn1(x)=0

  NA 358

2. 

Un+1(x)2x Un(x)+Un1(x)=0

3. 

Tn(x)=Un(x)x Un1(x)

  EH II 184(3) ...

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