Functional equations connected with the Collatz problem
The still open \(3x+1\)-problem (or Collatz- or Hasse- or Syracuse- or Kakutani-problem) is to prove that for every \(n\in \mathbb{N}\) there exists a \(k\) with \(t_ k(n)= 1\) where the function \(t(n)\) takes odd numbers \(n\) to \((3n+1)/2\) and even numbers \(n\) to \(n/2\) and the iterates of this mapping are defined recursively by \(t_ 0(n) =n\), \(t_ m(n)= t(t_{m_ 1}(n))\) for \(m\geq 1\). In this note the authors consider two generating functions: \(f_ m(z):= \sum_{n=0}^ \infty t_ m(n) z^ n\) and \(g_ n(w):= \sum_{m=0}^ \infty t_ m(n) w^ m\) which converge for complex numbers \(z\), \(w\) with \(| z|<1\), resp. for \(| w|< {3\over 2}\). While it is simple to prove that the \(f_ m(z)\) represent rational functions it turns out that the fact that \(g_ n(w)= {{q_ n(w)} \over {1-w^ 2}}\) with some polynomials \(q_ n(w)\) with integer coefficients is equivalent to the ``Collatz''-conjecture. They also introduce the generating function \(F(z,w):= \sum_{m,n=0}^ \infty t_ m(n) z^ n w^ m\) and show that \(F\) satisfies the linear functional equation \[ F(z^ 3,w)= {{z^ 3} \over {(1-z^ 3)^ 2}}+ wF(z^ 6,w)+ {w\over {3z}} \sum_{\nu=0}^ 2 e^{2\pi\nu i/3} F(e^{2\pi \nu i/3} z^ 2, w). \] Considering the related homogeneous equation and choosing \(w=1\) the authors prove that the ``Collatz''-conjecture holds if and only if in \(z=0\) there exists no holomorphic solution \(h(z)= \sum_{n=0}^ \infty h_ n z^ n\) of the functional equation \(h(z^ 3)= h(z^ 6)+ {1\over {3z}} \sum_{\nu=0}^ 2 e^{2\pi\nu i/3} h(e^{2\pi\nu i/3} z^ 2)\) other than \(h_ 0+ h_ 1 {z\over {1-z}}\). Doubtless this note contains some very interesting analytical aspects of the ``Collatz''-problem, but it still remains open whether these methods will be suitable to prove or disprove this famous conjecture.
scientific article; zbMATH DE number 554043
scientific article