Applications of \(q\)-Taylor theorems
This nice paper deals with several \(q\)-analogues of Taylor's theorems. One of these deals with the basis \((\varphi_n)_{n\geq 0}\) with \[ \varphi_n(\cos \theta)=(q^{1/4}e^{i \theta}, q^{1/4}e^{-i \theta}; q^{1/2})_n, \] where \( (a; q)_n = \prod^{n-1}_{j=0}(1-aq^j) \) and \( (a, b; q)_n=(a; q)_n (b; q)_n\). Since \[ \lim_{q\rightarrow 1} \varphi_n (x)=2^n (1-x)^n, \] this function can be looked at as a \(q\)-analogue of \((x-1)^n\). Using the Askey-Wilson operator \(D_q\), \[ D_qf(x)={\check{f}(q^{1/2}e^{i\varphi})-\check{f}(q^{-1/2}e^{i\varphi})\over (q^{1/2}-q^{-1/2})i \sin \theta}, \qquad x= \cos \theta, \] \[ \check{f}(z)=f \left( {1 \over 2} \left( z+ {1 \over z} \right) \right), \qquad z=e^{i \theta}, \] it is straightforward to see that \[ D_q \varphi_n (x)= -2\;q^{1/2} \;{1-q^n \over 1-q} \;\varphi_{n-1} (x). \] From this, the authors obtain the following \(q\)-analogue of Taylor's theorem. Theorem 2.1: If \(f(x)\) is a polynomial in \(x\) of degree \(n\), then \[ f(x)= \sum^n_{k=0} f_k \varphi_k (x), \quad \text{where}\quad f_k = {(q-1)^k \over 2^kq^{k/4}(q;q)_k} (D_q^k f)(\zeta_0) \quad\text{and}\quad \zeta_0= {q^{1/4}+\;q^{-1/4} \over 2}. \] The authors combine these expansions with earlier expansion theorems to deduce inverse relations and evaluations of certain linearization coefficients. By products include new summation theorems, new results on a \(q\)-exponential function, and quadratic transformations for \(q\)-series.
scientific article; zbMATH DE number 1915240