On some properties of chebyshev polynomials and their applications

In this paper we investigate certain normalized versions S_k,F(x),S_k,F(x) of Chebyshev polynomials of the second kind and the fourth kind over a field F of positive characteristic. Under the assumption that (char F, 2m + 1) = 1, we show that S_m,F(x) has no multiple roots in any one of its splitting fields. The same is true if we replace 2m + 1 by 2m and S_m,F(x) by S_m−1,F (x). As an application, for any commutative ring R which is a Z[1/n, 2 cos(2π/n), u^±1/2]-algebra, we construct an explicit cellular basis for the Hecke algebra associated to the dihedral groups I₂(n) of order 2n and defined over R by using linear combinations of some Kazhdan-Lusztig bases with coefficients given by certain evaluations of S_k,R(x) or S_k,R(x).