\(q\)-Taylor and interpolation series for Jackson \(q\)-difference operators
It is outside the scope of a review to give the formulae needed to understand the main results of the paper explicitly. For readers knowledgable in \(q\)-theory these main results will be stated below: \textbf{A.} Let \(0<R\leq \infty\) and \(f\) be analytic on \(\Omega_R\) with power series expansion \[ f(x)=\sum_{n=0}^{\infty}\,c_n x^n,\;x\in\Omega_R. \] Then \(f\) has the \(q\)-Taylor expansion \[ f(x)=\sum_{k=0}^{\infty}\,{D_q^k f(a)\over\Gamma_q(k+1)}\,\varphi_k(x,a), \] converging absolutely and uniformly on compact subsets of \(\Omega_R\). \textbf{B.} Let \(f(x)\) be a function with \(q\)-exponential growth of order \(k,\;k<\ln q^{-1}\), and finite type \(\alpha,\;\alpha\in\mathbb R\). Then for \(a\in\mathbb C\setminus\{0\}\), \(f(x)\) has the expansion \[ f(x)=\sum_{n=0}^{\infty}\,(-1)^n q^{-n(n-1)/2}\,{D_n^qf(aq^{-n})\over\Gamma_q(n+1)}\,\varphi_n(a,x), \] converging absolutely and uniformly on compact subsets of \(\mathbb C\).
scientific article; zbMATH DE number 5279807