Remarks on recurrences on streams

The notion of sequences on streams intuitively present in constructions of [2] and [3] is here defined and then a new kind of recurrences for these sequences labelled by elements (“streams”) of linearly ordered set are studied. A peculiar set of solutions of such recurrences consists of Tchebyshev-like special functions, which are polynomials in co-ordinates of a certain curve on hypersurface in determined by quasi-numbers of determinant one. These Tchebyshev-like special functions are shown to satisfy an ordinary m-th order recurrence with parameter dependent coefficients. For m = 2 one gets classical Tchebyshev polynomials of both kinds. The main content of this note is accompanied with an overview comparison with other “non-trigonometric” approaches.