The 
-Chebyshev
polynomials form the polynomial sequence 
defined by
 |
(1)
|
and
 |
(2)
|
where 
is a positive integer and 
is constant. They form a 
-symmetric d-orthogonal
polynomial sequence (Douak and Maroni 1997a, 1997b).
Their ordinary generating function is
 |
(3)
|
and expansion in the canonical basis gives
 |
(4)
|
In particular, only powers congruent to 
modulo 
occur. For 
and 
,
 |
(5)
|
where 
is a Chebyshev polynomial of the
second kind (Mesquita and da Rocha 2026). More generally, if 
and 
, then zero is a root of multiplicity
;
it is not a root when 
. For 
, the other roots occur 
at a time on each of the 
rays with arguments 
, where 
, 1, ..., 
(Mesquita and da Rocha 2026).
-Orthogonal Tchebychev Polynomials. I." Appl. Numer.
Math. 24, 23-53, 1997a. 
-Orthogonal Tchebychev Polynomials. II." Methods Appl.
Anal. 4, 404-429, 1997b. 
-Orthogonal Polynomials in Terms of Orthogonal Polynomials
and the Canonical Basis." Math. Comput. Sci. 20, #15, 2026.