ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS

Abstract

In this paper we investigate certain normalized versions Sk,F (x), Sek,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 Sem,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 Sem,F (x) by Sm−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 Sek,R(x) or Sk,R(x).