Self-similarity in the combinatorial theory of orthogonal polynomials (Q688691)

The author considers species of weighted combinatorial structures \(F\) associated to certain families of orthogonal polynomials (i.e. species \(F\) and the corresponding family have the same exponential generating function) and deals with the problem to determine the total weight \(\text{fix} (F[\beta])\) of all \(F\)-structures which are kept fixed by the action of a given permutation \(\beta\). The method used for this calculation is the so-called principle of autosimilarity which is demonstrated for some species associated to the Hermite and Laguerre polynomials, respectively. Furthermore, results relating to \(\text{fix} (F[\beta])\) are presented for species \(F\) associated to polynomials of Krawtchouk, Meixner, Meixner-Pollaczek and Jacobi. In these cases the weights correspond to orthogonal polynomials for suitable parameters.