Averaging structure in the \(3x+1\) problem (Q479605)

Consider the function \(T\): \(\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}\) defined by \(T(n): = 3n + 1\) if \(n\) is odd and \(T(n): = n/2\) if \(n\) is even. The \(3n + 1\) problem states that for any given positive integer \(n\), there exists a \(k\) positive integer such that the \(k\)th iterate of \(T(n)\) assumes the number \(1\) i.e., \(T^{(k)}(n) = 1\). This problem is also known as Collatz conjecture. In this paper the author extends the problem investigating the more general function \(T_{q,r}(n): = (qn + r)/2\) if \(n\) is odd and \(T_{q,r}(n): = n/2\) if \(n\) is even, where \(q\) and \(r\) are odd numbers. He prove some properties of the generating functions of the iterates of \(T_{q,r}(n)\) by using analytic tools. These results help to understand better the structure of the set of \(T\)-orbits of a positive integer.