  For a given orthonormal basis $(f_n)$ on a probability measure space, we want
to describe all Markov operators which have the $f_n$ as eigenvectors. We
introduce for that what we call the hypergroup property. We study this property
in three different cases. On finite sets, this property appears as the dual of
the GKS property linked with correlation inequalities in statistical mechanics.
The representation theory of groups provides generic examples where these two
properties are satisfied, although this group structure is not necessary in
general. The hypergroup property also holds for Sturm--Liouville bases
associated with log-concave symmetric measures on a compact interval, as stated
in Achour--Trim\`eche's theorem. We give some criteria to relax this symmetry
condition in view of extensions to a more general context. In the case of
Jacobi polynomials with non-symmetric parameters, the hypergroup property is
nothing else than Gasper's theorem. The proof we present is based on a natural
interpretation of these polynomials as harmonic functions and is related to
analysis on spheres. The proof relies on the representation of the polynomials
as the moments of a complex variable.
