On Discrete $q$-Extensions of Chebyshev Polynomials

Abstract

We study in detail main properties of two families of the basic hypergeometric ${}_2\phi_1$-polynomials, which are natural $q$-extensions of the classical Chebyshev polynomials $T_n(x)$ and $U_n(x)$. In particular, we show that they are expressible as special cases of the big $q$-Jacobi polynomials $P_n(x;a,b,c;q)$ with some chosen parameters $a$, $b$ and $c$. We derive quadratic transformations that relate these polynomials to the little $q$-Jacobi polynomials $p_n(x;a,b\,|q)$. Explicit forms of discrete orthogonality relations on a finite interval, $q$-difference equations and Rodrigues-type difference formulas for these $q$-Chebyshev polynomials are also given.