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Source. Theorem 3.2, Section 3, p. 8 of the author's version named on the source card; proof p. 8. Read on the PDF page image.

Statement

Setting as on the Theorem 2.2 page.

Theorem 3.2 (p. 8). Let $q>r>0$ both be odd and $\mu=e^{2\pi i/q}$. Then

$$f_{n+1,q,r}(x^q)=f_{n,q,r}(x^{2q}) +\frac{1}{qx^r}\sum_{k=0}^{q-1}\mu^{(q-r)k/2}f_{n,q,r}(\mu^kx^2).$$

The statement prints no range for $n$. The paper presents it as a generalization of a main result of Berg and Meinardus (p. 8), the formula connecting $f_n$ to $f_{n+1}$ for the $3x+1$ map (p. 2).

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof was read for structure only, and nothing here is independently reviewed.

Proof pointer

Split $f_{n+1,q,r}(x^q)$ into even and odd arguments, apply one step of $T_{q,r}$ to each, and extract the terms of $f_{n,q,r}(x^2)$ whose index lies in the right residue class modulo $q$ by averaging over the $q$-th roots of unity (p. 8).

Dependencies

None beyond the definitions.

Bears on

Problem 1135: with $(q,r)=(3,1)$ it is a recursion in $n$ for the generating functions of the problem's map. It says nothing about whether orbits reach $1$.