Chamberland 2015 averaging structure

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theorem_2_2: For odd q and r, the generating function of the n-th iterates of the qx+r map is a rational function P(x)/(1-x^{2^n})^2 with P of degree 2^{n+1}-1 divisible by x, given explicitly by the first 2^n iterates.

theorem_2_3: The rational generating function of the n-th iterates of the qx+r map at x equals that of the qx-r map at 1/x, so one function encodes the qx+r map on the positive and on the negative integers.

theorem_2_4: For odd q, r and m, n >= 1, the integral of f_{n,q,r}(x) x^{m-1} around the circle |x| = 2 equals 2 pi i times the n-th iterate of the qx-r map at m; for the 3x+1 map and m = 1 the value is 2 pi i for every n.

theorem_3_1: For odd q, r and n >= 1, the generating function of the n-th iterates of the qx+r map is a sum over the 2^n-th roots of unity s of double-pole and simple-pole terms whose coefficients A_{n,q,r}(s), B_{n,q,r}(s) are explicit finite sums.

theorem_3_2: For odd q > r > 0, the generating function of the (n+1)-th iterates of the qx+r map at x^q is expressed through that of the n-th iterates at x^{2q} and at mu^k x^2 for the q-th roots of unity mu^k, generalizing Berg and Meinardus.

theorem_4_1: At a fixed 2^N-th root of unity s, the double-pole coefficient B_{n,q,r}(s), and for s other than 1 the residue coefficient A_{n,q,r}(s), are multiplied by (q+1)/4 at each step n >= N, so they are constant when q = 3; A_{n,q,r}(1) is given in closed form and equals -rn/4 when q = 3.

theorem_4_2: For m >= 1 the n-th iterate of the qx+r map at m equals the sum, over the 2^n-th roots of unity s, of s^{-m} times m B_{n,q,r}(s) - A_{n,q,r}(s).

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Marc Chamberland, Averaging Structure in the 3x+1 Problem, J. Number Theory 148 (2015) 384-397 (author's version; 14 pp.).

Studies, for the maps $T_{q,r}$ ($n/2$ for even $n$, $(qn+r)/2$ for odd $n$, with $q,r$ odd), the generating functions $f_{n,q,r}(x)=\sum_{k\ge1}T_{q,r}^{(n)}(k)x^k$ of the $n$-th iterates. They are rational of the form $P_{n,q,r}(x)/(1-x^{2^n})^2$, so their poles are at most double, at the $2^n$-th roots of unity (Theorem 2.2, p. 3); they satisfy $f_{n,q,r}(x)=f_{n,q,-r}(1/x)$ (Theorem 2.3, p. 4) and a contour identity (Theorem 2.4, p. 5), and have a partial-fraction expansion with coefficients $A_{n,q,r}(s)$, $B_{n,q,r}(s)$ at each root $s$ (Theorem 3.1, pp. 6-7). Theorem 3.2 (p. 8) generalizes Berg and Meinardus's recursion from $f_n$ to $f_{n+1}$ to odd $q>r>0$. Theorem 4.1 (p. 9) proves $B_{n+1,q,r}(s)=\frac{q+1}{4}B_{n,q,r}(s)$ for $s^{2^N}=1$ and $n\ge N$, and the same for $A$ when $s\ne1$, so these coefficients are frozen when $q=3$, while $A_{n,3,r}(1)=-rn/4$ changes with $n$; Theorem 4.2 (p. 12) recovers $T_{q,r}^{(n)}(m)$ from them. The paper uses this only heuristically: for $q=3$ it suggests that every orbit is bounded, and for $q\ge5$ that orbits diverge (Section 4, pp. 12-13); nothing is proved about the 3x+1 conjecture. Relevance: Its results are identities for the generating functions of the iterates of qx+r maps, whose polar coefficients stay fixed in n when q = 3, except the residue at x = 1, supporting only a heuristic for bounded orbits in problem 1135.

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Bears on. #1135: with $(q,r)=(3,1)$ the map $T_{3,1}$ is the problem's map, and Theorems 4.1 and 4.2 (pp. 9, 12) show that its generating functions' polar coefficients stay fixed in $n$, except the residue at $x=1$, and recover each iterate from them; the paper draws from this only a heuristic (pp. 12-13) that every orbit is bounded, and proves nothing about whether orbits reach 1 (theorem_4_1, theorem_4_2).

Results. Theorem 2.2 (p. 3, with Theorem 2.1): $f_{n,q,r}$ is rational, $P_{n,q,r}(x)/(1-x^{2^n})^2$ with an explicit numerator; Theorem 2.3 (p. 4): $f_{n,q,r}(x)=f_{n,q,-r}(1/x)$; Theorem 2.4 (p. 5): $\oint_{|x|=2}f_{n,q,r}(x)x^{m-1}dx=2\pi iT_{-,q,r}^{(n)}(m)$; Theorem 3.1 (pp. 6-7): the partial-fraction expansion with coefficients $A_{n,q,r}(s)$, $B_{n,q,r}(s)$; Theorem 3.2 (p. 8): $f_{n+1,q,r}(x^q)$ through $f_{n,q,r}$, for odd $q>r>0$; Theorem 4.1 (p. 9): the coefficients scale by $(q+1)/4$ in $n$, with $A_{n,q,r}(1)$ in closed form; Theorem 4.2 (p. 12): $T_{q,r}^{(n)}(m)$ as a sum over the $2^n$-th roots of unity.

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