Decomposition of some recurrent polynomials with respect to the cyclic group of order \(n\) (Q2712814)

Let \(f\) be a complex function, and define NEWLINE\[NEWLINE f_{[n,k]}(z) = \frac{1}{n} \sum_{j=0}^{n-1} \omega_n^{-kj} f(\omega_n^j z), NEWLINE\]NEWLINE where \(\omega_n=e^{2\pi i/n}\) is a primitive \(n\)th root of unity, then \(f = \sum_{k=0}^{n-1} f_{[n,k]}\) is the decomposition of \(f\) with respect to the cyclic group of order \(n\). For \(n=2\) this is just the decomposition into a sum of an even and an odd function (which is at the basis of the fast Fourier transform, for instance). In this paper it is shown that when a sequence of polynomials \(P_m\) \((m \geq 0)\) satisfies a linear recurrence relation of finite order, then the components \((P_m)_{[n,k]}\) also satisfy a similar recurrence relation, but of higher order. Several examples are given to illustrate this result, in particular examples involving orthogonal polynomials and \(d\)-orthogonal polynomials (multiple orthogonal polynomials or Hermite-Padé polynomials).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00046].