Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm.

Dual addition formulas: the case of continuous q-ultraspherical and q-Hermite polynomials

We settle the dual addition formula for continuous q-ultraspherical polynomials as an expansion in terms of special q-Racah polynomials for which the constant term is given by the linearization formula for the continuous q-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman–Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous q-Hermite polynomials.