ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS

Authors : Jun Hu, Yabo Wu

Pages : 137-163

Doi:10.24330/ieja.296263

View : 14 | Download : 8

Publication Date : 2017-01-17

Article Type : Research Paper

Abstract :In this paper we investigate certain normalized versions Sk,F insert ignore into journalissuearticles values(x);, Sek,F insert ignore into journalissuearticles values(x); of Chebyshev polynomials of the second kind and the fourth kind over a field F of positive characteristic. Under the assumption that insert ignore into journalissuearticles values(char F, 2m + 1); = 1, we show that Sem,F insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(x); by Sm−1,F insert ignore into journalissuearticles values(x);. As an application, for any commutative ring R which is a Z[1/n, 2 cosinsert ignore into journalissuearticles values(2π/n);, u±1/2 ]-algebra, we construct an explicit cellular basis for the Hecke algebra associated to the dihedral groups I2insert ignore into journalissuearticles values(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,Rinsert ignore into journalissuearticles values(x); or Sk,Rinsert ignore into journalissuearticles values(x);.
Keywords : Chebyshev polynomials, dihedral group, Hecke algebras