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Differential equations satisfied by the components with respect to the cyclic group of order \(n\) of some special functions. (English) Zbl 0982.33002

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.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
30B10 Power series (including lacunary series) in one complex variable
33D60 Basic hypergeometric integrals and functions defined by them
34A30 Linear ordinary differential equations and systems
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