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Source. Theorem 2.2, Section 2, p. 3 of the author's version named on the source card; proof pp. 3--4. Theorem 2.1 on p. 3. Read on the PDF page images.

Statement

Setting (p. 2). For odd integers $q$ and $r$, $T_{q,r}(n)=n/2$ for even $n$ and $T_{q,r}(n)=(qn+r)/2$ for odd $n$; the generating function of its $n$-th iterates is

$$f_{n,q,r}(x)=\sum_{k=1}^{\infty}T_{q,r}^{(n)}(k)x^k ,$$

and $O_{q,r}^{(n)}(k)$ is the number of odd terms among $k,T_{q,r}^{(1)}(k),\dots,T_{q,r}^{(n-1)}(k)$.

Theorem 2.1 (p. 3). For fixed odd $(q,r)$ and all $n,k,j\ge0$,

$$T_{q,r}^{(n)}(2^nk+j)=q^{O_{q,r}^{(n)}(j)}k+T_{q,r}^{(n)}(j).$$

The paper presents this as a generalization of a known fact for the $3x+1$ map, citing Terras and Lagarias.

Theorem 2.2 (p. 3). Each $f_{n,q,r}$ is a rational function converging on the disc $|x|<1$, of the form $P_{n,q,r}(x)/(1-x^{2^n})^2$, where $P_{n,q,r}$ is a polynomial of degree $2^{n+1}-1$ divisible by $x$, and

$$f_{n,q,r}(x)=\frac{1}{(1-x^{2^n})^2}\sum_{j=1}^{2^n}q^{O_{q,r}^{(n)}(j)}x^j +\frac{1}{1-x^{2^n}}\sum_{j=1}^{2^n} \Bigl(T_{q,r}^{(n)}(j)-q^{O_{q,r}^{(n)}(j)}\Bigr)x^j$$

(display (1)). The paper lists $f_{n,3,1}$ for $n=0,1,2,3$ on p. 4 and notes there that the poles of $f_{n,q,r}$ are exactly the $2^n$-th roots of unity.

Read depth. Claims checked: both theorems were read clause by clause on the page images. The proofs were read for structure only, and nothing here is independently reviewed.

Proof pointer

Theorem 2.1 is an induction on $n$ (p. 3). For Theorem 2.2, split the summation index $k$ into residue classes $j\in\{1,\dots,2^n\}$ modulo $2^n$, apply Theorem 2.1 to each class, and sum the resulting arithmetico-geometric series (pp. 3--4).

Dependencies

Theorem 2.1 (p. 3).

Bears on

Problem 1135: with $(q,r)=(3,1)$ the map $T_{3,1}$ is the problem's map $f$, and the theorem describes the generating function of its $n$-th iterates. It says nothing about whether orbits reach $1$.