Recent work on return properties of quantum walks (the quantum analogue of random walks) has identified their generating functions of first returns as Schur functions. This is connected with a representation of Schur functions in terms of the operators governing the evolution of quantum walks, i.e. the unitary operators on Hilbert spaces. In this paper we propose a generalization of Schur functions by extending the above operator representation to arbitrary closed operators on Banach spaces. Such generalized ''Schur functions'' meet the formal structure of first return generating functions, thus we call them FR-functions. We derive some general properties of FR-functions, among them a simple relation with an operator version of Stieltjes functions which generalizes the renewal equation already known for random and quantum walks. We also prove that FR-functions satisfy splitting properties which extend useful factorizations of Schur functions. When specialized to self-adjoint operators on Hilbert spaces, we show that FR-functions become Nevanlinna functions. This allows us to obtain properties of Nevanlinna functions which, as far as we know, seem to be new. The FR-function structure leads to a new operator representation of Nevanlinna functions in terms of self-adjoint operators, whose spectral measures provide also new integral representations of such functions. This allows us to characterize each Nevanlinna function by a measure on the real line, which we refer to as ''the measure of the Nevanlinna function''. In contrast to standard operator and integral representations of Nevanlinna functions, these new ones are exact analogues of those already known for Schur functions. The above results are also the source of a very simple ''Schur algorithm'' for Nevanlinna functions based on interpolations at points on the real line, which we refer to as the ''Schur algorithm on the real line''. 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. (C) 2017 Elsevier Inc. All rights reserved
