Roots of Difference Quotient Forms of Chebyshev Polynomials

Abstract

AbstractLet ­
Œ Þ
–s be the Chebyshev polynomials of first kind of degree
Π. In this paper, we show that for
ðI , the polynomial with integer coefficients

˜a
­
Œ Þ
˜sa­
Œ Þ
s has all its roots in c
˜c=
 . Key words : Polynomials, Chebyshev Polynomials, Roots 1. Introduction 1 Let
„Þ
–s and
Þ
—s be polynomials in the single independent variables
– and
— with coefficients in the field
g of complex numbers. Cassels et al . [1,2] studied factorizations of
„Þ
–sa
Þ
—s as the polynomial in the pair of variables
– and
— . They also considered a trivial case when
„ and
are the same polynomial since
„Þ
–sa
„Þ
—s is divisible by
–a
— . In this case, obtaining the factors of the polynomial

–a
—
„Þ
–sa
„Þ
—s (1)is in general rather complicated. The polynomial of the form (1) also arises in Bezout matrices that have appeared in the literature for a long time. Given
“Þ
–sa


ˆa×
Œ
“
ˆ
–
ˆ Þ
“
Œ d×si
”Þ
–sa


ˆa×
Œ
”
ˆ
–
ˆ ,let
FÞ
–i
—sa

–a
—
“Þ
–s
”Þ
—sa
“Þ
—s
”Þ
–sa