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Chebychev Polynomials were used by Bill Daly and Steven Stadnicki to solve a problem. I've built a TRIANGLE page for the results, with new contributions by Bob Harris, Roger Phillips, and many others.
The MacTutor History of Mathematics archive gives Pafnuty Lvovich Chebyshev's biography. At mathworld.wolfram.com, there are twenty different entries for Chebyshev, including Chebyshev Polynomial of the First Kind and Chebyshev Polynomial of the Second Kind. What do they mean? I gained my first insight when I plugged cos(Pi/9) into the Inverse Symbolic Calculator. The Sixth Chebyshev Polynomial of the Second Kind is -1 + 24 x2 - 80 x4 + 64 x6. In Mathematica, ChebyshevU[6,x]. This polynomial is also expressed as U6(x).