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Source. Theorem 2.4, Section 2, p. 5 of the author's version named on the source card; proof and the special case on p. 6. Read on the PDF page images.

Statement

Setting as on the Theorem 2.2 and Theorem 2.3 pages.

Theorem 2.4 (p. 5). For fixed odd $(q,r)$ and each $m,n\ge1$,

$$\oint_{|x|=2}f_{n,q,r}(x)x^{m-1}dx=2\pi iT_{-,q,r}^{(n)}(m).$$

This is display (3). With $m=1$ and $(q,r)=(3,1)$ it gives $\oint_{|x|=2}f_{n,3,1}(x)\,dx=2\pi i$ for every $n$ (display (2), p. 4), since $1$ is fixed by the $3x-1$ map; the paper reports finding (2) first by numerical integration (p. 4) and derives it from (3) on p. 6.

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

Substitute $x=1/y$, use Theorem 2.3 to replace $f_{n,q,r}(1/y)$ by the power series of $T_{-,q,r}^{(n)}$ in $y$, convergent on $|y|=1/2$, and read off the coefficient (p. 6). Section 3 notes (p. 6) that the partial-fraction formula of Theorem 3.1 gives a second derivation.

Dependencies

Theorem 2.3.

Bears on

Problem 1135: with $(q,r)=(3,1)$ it is an identity for the generating function of the problem's map. It says nothing about whether orbits reach $1$.