Differential equations satisfied by the components with respect to the cyclic group of order \(n\) of some special functions (Q1978811)

Let \(f\) be a complex function of the variable \(z\) admitting a Laurent expansion in an annulus \(C\) with center at the origin. For an arbitrary positive integer \(n\), Ricci's theorem asserts that the function \(f\) can be written as the sum of \(n\) functions \(f_{[n,k]}\), \(k\in \{0,1,\dots, n\}\), defined by \[ f_{[n,k]}(z)= \frac{1}{n} \sum_{l=0}^{n-1} \exp \biggl(- \frac{2i\pi kl}{n}\biggr) f\Biggl( z\exp \biggl( \frac{2i\pi l}{n}\biggr) \Biggr), \qquad z\in \mathbb{C}. \] In this paper, the author presents a technique which, starting from a suitable differential equation satisfied by the function \(f\), provides a differential equation satisfied by the functions \(f_{[n,k]}\). Some known special functions are also treated for illustrations and examples.