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On Discrete q-Extensions of Chebyshev Polynomials

Commun. Math. Anal.
Volume 14, Number 2 (2013), 1 - 12

On Discrete q-Extensions of Chebyshev Polynomials

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Abstract

We study in detail main properties of two families of the basic hypergeometric 2φ1-polynomials, which are natural q-extensions of the classical Chebyshev polynomials Tn(x) andUn(x). In particular, we show that they are expressible as special cases of the big q-Jacobi polynomials Pn(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 pn(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.