The conumerator and the codenominator

Abstract

In this paper, we answer the question: “what is the qth Fibonacci number, where q is a positive rational?”. The answer is the codenominator function, which is an integral-valued map. It is defined via a system of functional equations, analogous to a system satisfied by the numerator function. Many Fibonacci identities in the literature can be generalized to the codenominator. It is related to Bird's tree and the involution Jimm of the reals induced by the outer automorphism of PGL(2,Z). The “corational” function Jimm satisfies a set of modular functional equations and maps quadratic irrationals to quadratic irrationals in a non-trivial manner. Although having jumps at rationals, it has a derivative which vanishes almost everywhere. We give a formula which gives the amount of jump of Jimm in terms of the codiscriminant function, which is defined in terms of the codenominator.