Some expansions in basic Fourier series and related topics (Q696868)

This impressive paper deals with the expansion of \(q\)-analogues of some elementary functions in basic Fourier series introduced in [\textit{J.~Bustoz} and \textit{S. K.~Suslov}, ``Basic analog of Fourier series on a \(q\)-quadratic grid'', Methods Appl. Anal. 5, No. 1, 1-38 (1998; Zbl 0961.33013)], which are based on certain \(q\)-analogues of the trigonometric and exponential functions. Starting with the expansions of the continuous \(q\)-ultraspherical and \(q\)-Lommel polynomials, the basic trigonometric and exponential functions and some basic cosecant and cotangent functions, the results eventually lead to a natural \(q\)-extension of the Bernoulli and Euler polynomials and numbers and the Riemann zeta function. The author expresses his hope that this will eventually lead to a \(q\)-extension of the theory of the Riemann zeta and related functions.