Cheby(X) = Chebyshev polynomial of an integer


Cheby(X) = Chebyshev polynomial of an integer:

The Chebyshev polynomials of the first kind are defined by the recurrence relation:

Cheby(0) = 1 = {1}
Cheby(1) = x = {1; 0}
Cheby(n+1) = 2*x*Cheby(n) − Cheby(n−1) = {2; 0}*Cheby(n) − Cheby(n−1).

The first few Chebyshev polynomials of the first kind are:

Cheby(0) = 1 = {1}
Cheby(1) = x = {1; 0}
Cheby(2) = 2*x^2 − 1 = {2; 0; −1}
Cheby(3) = 4*x^3 − 3*x = {4; 0; −3; 0}
Cheby(4) = 8*x^4 − 8*x^2 + 1 = {8; 0; −8; 0; 1}
Cheby(5) = 16*x^5 − 20*x^3 + 5*x = {16; 0; −20; 0; 5; 0}
Cheby(6) = 32*x^6 − 48*x^4 + 18*x^2 − 1 = {32; 0; −48; 0; 18; 0; −1}
Cheby(7) = 64*x^7 − 112*x^5 + 56*x^3 − 7*x = {64; 0; −112; 0; 56; 0; −7; 0}
Cheby(8) = 128*x^8 − 256*x^6 + 160*x^4 − 32*x^2 + 1 = {128; 0; −256; 0; 160; 0; −32; 0; 1}
Cheby(9) = 256*x^9 − 576*x^7 + 432*x^5 − 120*x^3 + 9*x = {256; 0; −576; 0; 432; 0; −120; 0; 9; 0}

Cheby(−n) = Cheby(n).

See: Chebyshev Polynomial of the First Kind -- From Mathworld
See: Chebyshev polynomials -- From Wikipedia

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Changes last made on Friday, 13-Mar-09 11:28:56 PDT

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