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Statement

Notation as on the relations (7)--(11) page.

(14) (p. 296). As $n\to\infty$,

$$x'_{nj}\downarrow j-\tfrac12\quad(j=1,2,\ldots);\qquad \xi'_{nj}\downarrow j-\tfrac12\quad(j=2,3,\ldots).$$

That is, for each fixed rank $j$ the sequences decrease in $n$ and converge to $j-\tfrac12$, the $j$th positive zero of the derivative of $\sin\pi x$. The paper presents (14) as showing that (12) is best possible in a certain sense (p. 296). Remark (i) (p. 297) rephrases the first part: the $j$th positive zero of the derivative of the partial product

$$P_n(x)=\frac{(-1)^n\pi\,p_n(x)}{(n!)^2}=\pi x\prod_{k=1}^{n}\Bigl(1-\frac{x^2}{k^2}\Bigr)$$

of $\sin\pi x$ decreases to $j-\tfrac12$ as $n\to\infty$, $j=1,2,\ldots$.

(15) (p. 297), a corollary of (14): for $j=1,2,\ldots$,

$$\lim_{n\to\infty}\bigl(x'_{n,j+1}-x'_{nj}\bigr) =\lim_{n\to\infty}\bigl(\xi'_{n,j+1}-\xi'_{nj}\bigr)=1,$$

so the left members of (13) tend to the right members.

Open questions (Remark (ii), p. 297; restated in Section 5, p. 299). The paper leaves unanswered whether either convergence in (15) is monotonic in $n$. It likewise notes that (14) makes the second differences $\Delta^2x'_{nj}=x'_{n,j+2}-2x'_{n,j+1}+x'_{nj}$, and the corresponding ones for $\xi'$, converge to $0$ as $n\to\infty$, and asks whether that convergence is monotonic. Section 5 says that the monotonicity in question is, in that context, decreasing.

Read depth. Claims checked: (14), (15) and Remarks (i) and (ii) were read on the page images of pp. 296--297, and the restatement on p. 299. The proof was followed but not checked step by step.

Proof pointer

Pp. 296--297. By (9) and (7) the limits $x'_j$ and $\xi'_j$ exist, and by (7) and (10) it suffices to identify them. The polynomial $P_n$ has the same critical points $x'_{nj}$ as $p_n$ and converges to $\sin\pi x$ uniformly on every finite interval, so $x'_j$ is the unique extremum point of $\sin\pi x$ between $j-1$ and $j$, namely $j-\tfrac12$. The same reasoning applies to $q_n$.

Dependencies

(7), (9) and (10) on the relations (7)--(11) page; the product formula for $\sin\pi x$.

Bears on

Problem 1114: context only. Remark (ii) restates Bálint's verification of Erdős's conjecture as $\Delta^2x'_{nj}>0$ for $j=1,\ldots,n$ and each fixed $n$, with the corresponding inequality for $\xi'$ (p. 297), and asks whether the convergence of these second differences as $n\to\infty$ is monotonic. That question is the paper's own; the problem fixes the polynomial and compares gaps in $j$.