d-Chebyshev Polynomial

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 1997ab).

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).