Basic exponential functions on a \(q\)-quadratic grid (Q2760163)

This is a summary of the current state of knowledge of basic exponential functions on a \(q\)-quadratic grid. If \(u\) is a function of \(x = \cos\theta\), we define the \(q\)-divided difference operator NEWLINE\[NEWLINE {\partial u \over \partial x} = {u(x^+)-u(x^-) \over x^+-x^-} NEWLINE\]NEWLINE where \(x^+=(\cos\theta)(q^{1/2}+q^{-1/2})/2 + i (\sin\theta) (q^{1/2}-q^{-1/2})/2\), \(x^-=(\cos\theta)(q^{1/2}+q^{-1/2})/2 - i (\sin\theta) (q^{1/2}-q^{-1/2})/2\). The analog of the exponential function \(e^{\alpha x}\) is the \(q\)-exponential \({\mathcal{E}}_q(x;\alpha)\) which is a solution of the equation NEWLINE\[NEWLINE {\partial u \over \partial x} = {2q^{1/4}\over 1-q} \alpha u.NEWLINE\]NEWLINE The author states many results for these functions including their addition theorems and expansion formulas in terms of continuous \(q\)-Hermite and \(q\)-ultraspherical polynomials. He also gives results on completeness and surveys what is known about analogs of Fourier series expansions.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00053].