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Recent work on recurrence in quantum walks has provided a representation ofSchur functions in terms of unitary operators. We propose a generalization ofSchur functions by extending this operator representation to arbitraryoperators on Banach spaces. Such generalized Schur functions meet the formalstructure of first return generating functions, thus we call them FR-functions.We derive general properties of FR-functions, among them a generalization ofthe renewal equation already known for random and quantum walks, as well assplitting properties which extend useful factorizations of Schur functions.  When specialized to self-adjoint operators, FR-functions become Nevanlinnafunctions. This leads to new results on Nevanlinna functions: decompositionrules which are the analogue of useful factorizations of Schur functions, asimple Nevanlinna version of the Schur algorithm and new operator and integralrepresentations of Nevanlinna functions which, in contrast to standard ones,are exact analogues of those already known for Schur functions, givingsimilarly a one-to-one correspondence between Nevanlinna functions and measureson the real line.  The paper is completed with several applications of FR-functions toorthogonal polynomials and random and quantum walks which illustrate their wideinterest: an analogue for orthogonal polynomials on the real line of theKhrushchev formula for orthogonal polynomials on the unit circle, and the useof FR-functions to study recurrence in random walks, quantum walks and openquantum walks. These applications provide numerous explicit examples ofFR-functions and their splittings and show that these new tools, despite beingextensions of very classical ones, play an important role in the study ofphysical problems of a highly topical nature.

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