A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions, clarifying the meaning of these functions -as first return generating functions- and their splittings -which become recurrence splitting rules. They also show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature.